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removing the tutorial 

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README.rst
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@ -3,7 +3,7 @@ Quantum Information Software Kit (QISKit) SDK Python
|Build Status|
Python software development kit (SDK) and Jupyter notebooks for working
Python software development kit (SDK) for working
with OpenQASM and the IBM Q experience (QX).
Philosophy

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tutorial/__init__.py
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{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"<img src=\"images/QISKit-c.gif\" alt=\"Note: In order for images to show up in this jupyter notebook you need to select File => Trusted Notebook\" width=\"250 px\" align=\"left\">"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## _*QISKit (Quantum Information Software Kit) *_ \n",
"\n",
"The latest version of this notebook is available on https://github.com/IBM/qiskit-sdk-py/tree/master/scripts.\n",
"\n",
"For more information about how to use the IBM Q experience (QX), consult the [tutorials](https://quantumexperience.ng.bluemix.net/qstage/#/tutorial?sectionId=c59b3710b928891a1420190148a72cce&pageIndex=0), or check out the [community](https://quantumexperience.ng.bluemix.net/qstage/#/community).\n",
"\n",
"***\n",
"### Contributors (alphabetical)\n",
"Jerry Chow, Antonio Córcoles, Abigail Cross, Andrew Cross, Ismael Faro, Andreas Fuhrer, Jay M. Gambetta, Antonio Mezzacapo, Ramis Movassagh, Anna Phan, Rudy Raymond, Kristan Temme, Chris Wood\n",
"\n",
"In future releases, anyone who contributes to the tutorial can include their name here."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Introduction\n",
"Welcome to QISKit! Interested in quantum computing and programming a real live quantum processor? QISKit is a simple set of tools for anyone to get started with quantum information science. We have put together these Jupyter notebooks to demonstrate how to use our tools and explore the quantum world.\n",
"\n",
"The notebooks are organized into the following topics:\n",
"\n",
"1. [Introducing the tools](#section1)\n",
"2. [Exploring quantum physics](#section2)\n",
"3. [Verification tools for quantum science](#section3)\n",
"4. [Applications of short-depth quantum circuits on quantum computers](#section4)\n",
"5. [Quantum games](#section5) "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 1. Introducing the tools<a id='section1'></a>\n",
"\n",
"In this first topic, we break down the tools in this QISKit SDK, and introduce all the different parts to make this useful. Our list of introductory notebooks:\n",
"* [Getting Started with QISKit SDK](sections/getting_started.ipynb) - how to use the the QISKit SDK tools. \n",
"* [Understanding the Different Backends](sections/working_with_backends.ipynb) - how to get information about the connected backends.\n",
"* [Compiling and Running a Quantum Program](sections/compiling_and_running.ipynb) - how to rewrite circuits to different backends.\n",
"* Loading and Saving a Quantum Program [coming soon]\n",
"* [Visualizing a Quantum State](sections/visualizing_quantum_state.ipynb) - illustrates the different tools we have for visualizing a quantum state"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 2. Exploring quantum information concepts<a id='section2'></a> \n",
"\n",
"The next set of notebooks shows how you can explore some simple concepts of quantum information science. \n",
"\n",
"* [Superposition and Entanglement](sections/superposition_and_entanglement.ipynb) - how to make simple quantum states on one and two qubits, and demonstrates concepts such as quantum superpositions and entanglement. \n",
"* [Single-qubit States: Amplitude and Phase](sections/single_qubit_states_amplitude_and_phase.ipynb) - discusses more complicated single-qubit states. \n",
"* [Single-qubit Quantum Random Access Coding](sections/single-qubit_quantum_random_access_coding.ipynb) - how superpositions of one-qubit quantum states can be used to encode two and three bits into one qubit, and how measurements can be used to decode any one bit with a success probability of more than half. \n",
"* [Two-qubit Quantum Random Access Coding](sections/two-qubit_state_quantum_random_access_coding.ipynb) - how superposition and entanglement can be used to encode seven bits of information into two qubits, such that any one of seven bits can be recovered probabilistically.\n",
"* [Entanglement Revisited](sections/entanglement_revisited.ipynb) - the CHSH inequality, and extensions for three qubits (Mermin). \n",
"* [Quantum Teleportation](sections/quantum_teleportation.ipynb) - introduces quantum teleportation.\n",
"* [Quantum Superdense Coding](sections/superdense_coding.ipynb) - introduces the concept of superdense coding."
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": true
},
"source": [
"## 3. Verification tools for quantum information science<a id='section3'></a>\n",
"\n",
"The third set of notebooks allows you to explore tools for verification of quantum systems with commonly used techniques such as tomography and randomized benchmarking.\n",
"\n",
"### Error Amplications methods\n",
"* [Relaxation and decoherence](sections/Relaxation_and_decoherence.ipynb) - a simple notebook showing how to measure coherence \n",
"\n",
"### Tomography methods\n",
"* [Quantum state tomography](sections/state_tomography.ipynb) - how to run quantum state tomography\n",
"* [Quantum process tomography](sections/process_tomography.ipynb) - how to run quantum process tomography\n",
"\n",
"### Randomization methods\n",
"* Pauli randomized benchmarking [coming soon]\n",
"* Standard randomized benchmarking [coming soon]\n",
"* Purity randomized benchmarking [coming soon]\n",
"* Leakage randomized benchmarking [coming soon]\n",
"* Simultaneous randomized benchmarking [coming soon]\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 4. Applications of quantum information science<a id='section4'></a>\n",
"\n",
"To fully grasp the possibilities, this set of notebooks shows how you can explore applications of short-depth quantum computers.\n",
"\n",
"### Sampling \n",
"* [Quantum optimization](sections/classical_optimization.ipynb) by variational quantum eigensolver method - illustrates how to use a quantum computer to look at optimization problems. \n",
"* [Quantum chemistry](sections/quantum_chemistry.ipynb) by variational quantum eigensolver method - discusses using a quantum computer to look at chemistry problems. \n",
"\n",
"### Non-Sampling \n",
"* [Iterative phase estimation](sections/iterative_phase_estimation_algorithm.ipynb)"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": true
},
"source": [
"## 5. Quantum Games<a id='section5'></a>\n",
"\n",
"The notebooks below show how quantum algorithms can be used to play games and solve puzzles in different ways that cannot be done with their classical counterparts.\n",
"\n",
"* [Quantum counterfeit coin finding](sections/Quantum_counterfeit_coin_problem.ipynb) algorithm: can you solve [the counterfeit coin riddle](https://ed.ted.com/lessons/can-you-solve-the-counterfeit-coin-riddle-jennifer-lu)? You are given a quantum computer and quantum beam balance, and your task is to find a counterfeit coin hidden in a set of coins. Armed with the knowledge of the Bernstein-Vazirani algorithm, you can easily find the counterfeit coin using the beam balance only once.\n"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": []
}
],
"metadata": {
"anaconda-cloud": {},
"kernelspec": {
"display_name": "Python 3",
"language": "python",
"name": "python3"
},
"language_info": {
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"name": "ipython",
"version": 3
},
"file_extension": ".py",
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"version": "3.6.0"
}
},
"nbformat": 4,
"nbformat_minor": 1
}

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# Makefile for Sphinx documentation
#
# You can set these variables from the command line.
SPHINXOPTS =
SPHINXBUILD = sphinx-build
PAPER =
BUILDDIR = _build
# User-friendly check for sphinx-build
ifeq ($(shell which $(SPHINXBUILD) >/dev/null 2>&1; echo $$?), 1)
$(error The '$(SPHINXBUILD)' command was not found. Make sure you have Sphinx installed, then set the SPHINXBUILD environment variable to point to the full path of the '$(SPHINXBUILD)' executable. Alternatively you can add the directory with the executable to your PATH. If you don\'t have Sphinx installed, grab it from http://sphinx-doc.org/)
endif
# Internal variables.
PAPEROPT_a4 = -D latex_paper_size=a4
PAPEROPT_letter = -D latex_paper_size=letter
ALLSPHINXOPTS = -d $(BUILDDIR)/doctrees $(PAPEROPT_$(PAPER)) $(SPHINXOPTS) .
# the i18n builder cannot share the environment and doctrees with the others
I18NSPHINXOPTS = $(PAPEROPT_$(PAPER)) $(SPHINXOPTS) .
.PHONY: help
help:
@echo "Please use \`make <target>' where <target> is one of"
@echo " html to make standalone HTML files"
@echo " dirhtml to make HTML files named index.html in directories"
@echo " singlehtml to make a single large HTML file"
@echo " pickle to make pickle files"
@echo " json to make JSON files"
@echo " htmlhelp to make HTML files and a HTML help project"
@echo " qthelp to make HTML files and a qthelp project"
@echo " applehelp to make an Apple Help Book"
@echo " devhelp to make HTML files and a Devhelp project"
@echo " epub to make an epub"
@echo " epub3 to make an epub3"
@echo " latex to make LaTeX files, you can set PAPER=a4 or PAPER=letter"
@echo " latexpdf to make LaTeX files and run them through pdflatex"
@echo " latexpdfja to make LaTeX files and run them through platex/dvipdfmx"
@echo " text to make text files"
@echo " man to make manual pages"
@echo " texinfo to make Texinfo files"
@echo " info to make Texinfo files and run them through makeinfo"
@echo " gettext to make PO message catalogs"
@echo " changes to make an overview of all changed/added/deprecated items"
@echo " xml to make Docutils-native XML files"
@echo " pseudoxml to make pseudoxml-XML files for display purposes"
@echo " linkcheck to check all external links for integrity"
@echo " doctest to run all doctests embedded in the documentation (if enabled)"
@echo " coverage to run coverage check of the documentation (if enabled)"
@echo " dummy to check syntax errors of document sources"
.PHONY: clean
clean:
rm -rf $(BUILDDIR)/*
.PHONY: html
html:
$(SPHINXBUILD) -b html $(ALLSPHINXOPTS) $(BUILDDIR)/html
@echo
@echo "Build finished. The HTML pages are in $(BUILDDIR)/html."
.PHONY: dirhtml
dirhtml:
$(SPHINXBUILD) -b dirhtml $(ALLSPHINXOPTS) $(BUILDDIR)/dirhtml
@echo
@echo "Build finished. The HTML pages are in $(BUILDDIR)/dirhtml."
.PHONY: singlehtml
singlehtml:
$(SPHINXBUILD) -b singlehtml $(ALLSPHINXOPTS) $(BUILDDIR)/singlehtml
@echo
@echo "Build finished. The HTML page is in $(BUILDDIR)/singlehtml."
.PHONY: pickle
pickle:
$(SPHINXBUILD) -b pickle $(ALLSPHINXOPTS) $(BUILDDIR)/pickle
@echo
@echo "Build finished; now you can process the pickle files."
.PHONY: json
json:
$(SPHINXBUILD) -b json $(ALLSPHINXOPTS) $(BUILDDIR)/json
@echo
@echo "Build finished; now you can process the JSON files."
.PHONY: htmlhelp
htmlhelp:
$(SPHINXBUILD) -b htmlhelp $(ALLSPHINXOPTS) $(BUILDDIR)/htmlhelp
@echo
@echo "Build finished; now you can run HTML Help Workshop with the" \
".hhp project file in $(BUILDDIR)/htmlhelp."
.PHONY: qthelp
qthelp:
$(SPHINXBUILD) -b qthelp $(ALLSPHINXOPTS) $(BUILDDIR)/qthelp
@echo
@echo "Build finished; now you can run "qcollectiongenerator" with the" \
".qhcp project file in $(BUILDDIR)/qthelp, like this:"
@echo "# qcollectiongenerator $(BUILDDIR)/qthelp/qiskit-sdk-py-dev.qhcp"
@echo "To view the help file:"
@echo "# assistant -collectionFile $(BUILDDIR)/qthelp/qiskit-sdk-py-dev.qhc"
.PHONY: applehelp
applehelp:
$(SPHINXBUILD) -b applehelp $(ALLSPHINXOPTS) $(BUILDDIR)/applehelp
@echo
@echo "Build finished. The help book is in $(BUILDDIR)/applehelp."
@echo "N.B. You won't be able to view it unless you put it in" \
"~/Library/Documentation/Help or install it in your application" \
"bundle."
.PHONY: devhelp
devhelp:
$(SPHINXBUILD) -b devhelp $(ALLSPHINXOPTS) $(BUILDDIR)/devhelp
@echo
@echo "Build finished."
@echo "To view the help file:"
@echo "# mkdir -p $$HOME/.local/share/devhelp/qiskit-sdk-py-dev"
@echo "# ln -s $(BUILDDIR)/devhelp $$HOME/.local/share/devhelp/qiskit-sdk-py-dev"
@echo "# devhelp"
.PHONY: epub
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$(SPHINXBUILD) -b epub $(ALLSPHINXOPTS) $(BUILDDIR)/epub
@echo
@echo "Build finished. The epub file is in $(BUILDDIR)/epub."
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epub3:
$(SPHINXBUILD) -b epub3 $(ALLSPHINXOPTS) $(BUILDDIR)/epub3
@echo
@echo "Build finished. The epub3 file is in $(BUILDDIR)/epub3."
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latex:
$(SPHINXBUILD) -b latex $(ALLSPHINXOPTS) $(BUILDDIR)/latex
@echo
@echo "Build finished; the LaTeX files are in $(BUILDDIR)/latex."
@echo "Run \`make' in that directory to run these through (pdf)latex" \
"(use \`make latexpdf' here to do that automatically)."
.PHONY: latexpdf
latexpdf:
$(SPHINXBUILD) -b latex $(ALLSPHINXOPTS) $(BUILDDIR)/latex
@echo "Running LaTeX files through pdflatex..."
$(MAKE) -C $(BUILDDIR)/latex all-pdf
@echo "pdflatex finished; the PDF files are in $(BUILDDIR)/latex."
.PHONY: latexpdfja
latexpdfja:
$(SPHINXBUILD) -b latex $(ALLSPHINXOPTS) $(BUILDDIR)/latex
@echo "Running LaTeX files through platex and dvipdfmx..."
$(MAKE) -C $(BUILDDIR)/latex all-pdf-ja
@echo "pdflatex finished; the PDF files are in $(BUILDDIR)/latex."
.PHONY: text
text:
$(SPHINXBUILD) -b text $(ALLSPHINXOPTS) $(BUILDDIR)/text
@echo
@echo "Build finished. The text files are in $(BUILDDIR)/text."
.PHONY: man
man:
$(SPHINXBUILD) -b man $(ALLSPHINXOPTS) $(BUILDDIR)/man
@echo
@echo "Build finished. The manual pages are in $(BUILDDIR)/man."
.PHONY: texinfo
texinfo:
$(SPHINXBUILD) -b texinfo $(ALLSPHINXOPTS) $(BUILDDIR)/texinfo
@echo
@echo "Build finished. The Texinfo files are in $(BUILDDIR)/texinfo."
@echo "Run \`make' in that directory to run these through makeinfo" \
"(use \`make info' here to do that automatically)."
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@echo "Running Texinfo files through makeinfo..."
make -C $(BUILDDIR)/texinfo info
@echo "makeinfo finished; the Info files are in $(BUILDDIR)/texinfo."
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@echo
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@echo
@echo "Link check complete; look for any errors in the above output " \
"or in $(BUILDDIR)/linkcheck/output.txt."
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doctest:
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@echo "Testing of doctests in the sources finished, look at the " \
"results in $(BUILDDIR)/doctest/output.txt."
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coverage:
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@echo "Testing of coverage in the sources finished, look at the " \
"results in $(BUILDDIR)/coverage/python.txt."
.PHONY: xml
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@echo
@echo "Build finished. The XML files are in $(BUILDDIR)/xml."
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@echo
@echo "Build finished. The pseudo-XML files are in $(BUILDDIR)/pseudoxml."
.PHONY: dummy
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$(SPHINXBUILD) -b dummy $(ALLSPHINXOPTS) $(BUILDDIR)/dummy
@echo
@echo "Build finished. Dummy builder generates no files."

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========================
Classical Optimization
========================
The latest version of this notebook is available on
https://github.com/IBM/qiskit-sdk-py/tree/master/scripts.
For more information about how to use the IBM Q Experience (QX), consult
the `Quantum Experience
tutorials <https://quantumexperience.ng.bluemix.net/qstage/#/tutorial?sectionId=c59b3710b928891a1420190148a72cce&pageIndex=0>`__,
or check out the
`community <https://quantumexperience.ng.bluemix.net/qstage/#/community>`__.
Contributors
============
Jay Gambetta, Antonio Mezzacapo, Ramis Movassagh, Kristan Temme
Introduction
============
Many problems in finance and business are optimization problems.
Optimization or combinatorial optimization means searching for an
optimal solution in a finite or countably infinite set of potential
solutions. Optimality is defined with respect to some criterion
function, which is to be minimized or maximized, typically
called the cost function.
**Types of optimization problems**
Minimization: cost, distance, length of a traversal, weight, processing
time, material, energy consumption, number of objects.
Maximization: profit, value, output, return, yield, utility, efficiency,
capacity, number of objects.
The two problems that are consider in this notebook are:
MaxCut
======
MaxCut is an NP-complete problem. MaxCut has applications in clustering,
network science, and statistical physics. For example, consider the
problem of many different people (nodes) and how they can influence
(weight) overs. OVERS? You want to answer the question of the best
marketing stratety to maximize revenue, by offering for free the product
to one subset of buyers, while determining how to approach the remaining
buyers with a particular price. MaxCut derives its importance from its
``hardness" as well as the applications it has in aforementioned fields (SHOULD WE
ADD THAT APPROXIMATIONS ARE ALSO HARD).
Consider a :math:`n`-node undirectional graph *G(V, E)* where *\|V\| =
n* with edge weights :math:`w_{ij}>0` for :math:`(i, j)\in E`. A cut is
defined as the partitioning into two sets. We define the cost as the sum
of weights of edges crossing the cut. By assigning :math:`x_i=0` or
:math:`x=1` to each node :math:`i` depding on its location to the cut,
the maximum cut is a cut that maximizes
.. math:: C(\textbf{x}) = \sum_{i,j} w_{ij} x_i (1-x_j).
An extension of the MaxCut problem is to let the nodes themselves carry
weights, which can be regarded as the nodes having a preference for their
location. The cost function then becomes
.. math:: C(\textbf{x}) = \sum_i w_i x_i + \sum_{i,j} w_{ij} x_i (1-x_j).
Mapping this to a quantum Hamiltonian is trival:
:math:`x_i\rightarrow (1-Z_i)/2` where :math:`Z_i` is the Pauli Z
operator that has eigenvalues :math:`\pm 1`. Doing this, we find that
.. math:: C(\textbf{Z}) = \sum_i w_i (1-Z_i)/2 + \sum_{i<j} \frac{w_{ij}}{2} (1-Z_iZ_j)= \frac{-1}{2}\left(\sum_i w_i Z_i + \sum_{i<j} w_{ij} Z_iZ_j\right)+\mathrm{const},
where const = :math:`\sum_i w_i/2 + \sum_{i<j}w_{ij}/2`. That is, the
MAXCUT is equivalent to minimizing the Ising Hamiltonian
.. math:: H = \sum_i w_i Z_i + \sum_{i<j} w_{ij} Z_iZ_j.
Traveling Salesman Problem
==========================
In addition to being a notorious NP-complete problem that has drawn the
attention of computer scientists and mathematician for over two
centuries, as the name suggests, the Traveling Salesman Problem (TSP)
has important bearings on finance and marketing. Colloquially, the
traveling salesman is one who goes from city to city to sell
merchandise. His/her objective (TSP) is to find the shortest path that
would enable a path to visit all the cities and return to the starting city. This way, potential sales are maximized in the least amount of time.
The problem derives its importance from its ``hardness" and ubiquitous
equivalence to other relevant combinatorial optimization problems that
arise in practice.
The mathematical formulation with some early analysis was proposed by
W.R. Hamilton in early 19th century. Mathematically, the problem is best
abstracted in terms of graphs. The TSP on the nodes of a graph asks for
the shortest Hamiltonian cycle that can be taken through each of the
nodes. The general solution is unknown, and finding an efficient solution
(e.g., a polynomial time algorithm) is expected not to exist.
Find the shortest Hamiltonian cycle in a given complete graph
:math:`G=(V,E)` with :math:`n=|V|` vertices and distances,
:math:`w_{ij} =` distance from vertex :math:`i` to vertex :math:`j`. The
decission variable :math:`x_{ij}` = 1 if the cycle goes along arc
:math:`i\rightarrow j` or 0 otherwise.
Approximate Universal Quantum Computing for Optimization problems
=================================================================
Recently there has been interest in investgating approximate algorithims
for optimization [ADD REFS]. Here we show a slight variation, which we have been considering under the general concept of approximate
quantum computing. In general, we don't expect this to have a exponential
speed-up, but due to the nature and importance of these problems, it is
worth investigating heuristic approaches on a quantum computer. The hope
is that that the large space of a quantum computer will make it possible to explore
the problem by exploiting entanglement to trade exponential calls for
quantum depth.
The algorithm works as follows: 1. Choose a Ising problem (can have
higher powers of Z). 2. Choose the maximum depth of the quantum circuit
(this could be done adaptively.) 3. Choose a set of controls
:math:`\theta` and make a trial function :math:`|\psi(\theta)\rangle`.
4. Evaluate
:math:`C = \langle\psi~|H|~\psi\rangle = \sum_i w_i \langle\psi~|Z_i|~\psi\rangle+ \sum_{i<j} w_{ij} \langle\psi~|Z_iZ_j|~\psi\rangle`
by sampling the outcome of the circuit and adding the expectation values
together. 5. Use a classical optimizer to choose a new set of controls.
6. Continue until the C has reach a minimium and return
:math:`|\langle z~|\psi\rangle|^2`.
FIX THIS NEXT SENTENCE: Not some benefits on this over making this is real hardware is we are
not limited to the connectivity of the device and it is trival to map
optimize problems to the virtual Hamiltonain which we make though
simulation. Getting higher orders such as :math:`Z_iZ_jZ_k` is simple
which in real hardware is generally possible. The noise is mainly due to
the gates, and if the depth is small enough, we can perform this high
fidelity.
It is our belief the difficulty of finding good heuristic algorithms will
come down to the trial funciton. Do we choose a trial function that has
entanglement in a way that best aligns with the problem (as indicated in
[REF]), or do we make the amount of entanglement a variable [REF]? These
questions are beyond our goals here, and we will consider only the
simplie trial functions
.. math:: |\psi(\theta)\rangle = [U_\mathrm{single}(\theta) U_\mathrm{entangler}]^m |+\rangle
where :math:`U_\mathrm{entangler}` is a function of cPhase gates (fully
entangling), and
:math:`U_\mathrm{single}(\theta) = Y(\theta)^{\otimes n}`, where
:math:`n` is the number of qubits and :math:`m` is the depth of the
quantum circuit. The motivation for this choice is that for these
classical problems, this choice allows us to search over the space of
states that have only real superpositions and don't have to worry about
refocusing out all the complex phases, but still can exploit the
entanglment to search for solutions.
.. code:: python
# Checking the version of PYTHON; we only support 3 at the moment
import sys
if sys.version_info < (3,0):
raise Exception("Please use Python version 3 or greater.")
# useful additional packages
import matplotlib.pyplot as plt
%matplotlib inline
import numpy as np
from scipy import linalg as la
import sys
sys.path.append("../../")
# importing the QISKit
from qiskit import QuantumCircuit, QuantumProgram
import Qconfig
# import basic plot tools
from qiskit.basicplotter import plot_histogram
# import optimization tools
from tools.optimizationtools import trial_circuit_ryrz,trial_circuit_ry, SPSA_optimization
from tools.optimizationtools import Energy_Estimate, Measure_pauli_z, Hamiltonian_from_file, make_Hamiltonian
MaxCut on Four Qubits
==================
::
Graph:
X---X
| \ |
X---X
The ground state is degenerate and is either :math:`|0110\rangle` or
:math:`|1001\rangle`.
.. code:: python
n=2
m=
device='local_qasm_simulator'
SPSA_params=[1,.1,.602,.101,0]
theta=np.zeros(2*n*m)
entangler_map={0: [1]} # the map of two-qubit gates with control at key and target at values
shots=1000
max_trials=25
#Exact Energy
pauli_list=Hamiltonian_from_file('H2/H2Equilibrium.txt')
eigen=la.eig(make_Hamiltonian(pauli_list))
exact=np.amin(eigen[0])
# Optimization
eval_hamiltonian_partial=partial(eval_hamiltonian,n,m,'H2/H2Equilibrium.txt',device,shots)
output=SPSA_optimization(eval_hamiltonian_partial,theta,SPSA_params,max_trials,1);
plt.plot(output[2],label='E(theta_plus)')
plt.plot(output[3],label='E(theta_minus)')
plt.plot(np.ones(max_trials)*output[0],label='Final Energy')
plt.plot(np.ones(max_trials)*exact,label='Exact Energy')
plt.legend()
.. code:: python
# cost function H = alpha_i z_i + beta_ij z_i z_j
n =4
alpha = np.zeros(n)
beta = np.zeros((n, n))
beta[0, 1] = 1
beta[0, 2] = 1
beta[1, 2] = 1
beta[1, 3] = 1
beta[2, 3] = 1
.. code:: python
#Setting up a quantum program and connecting to the Quantum Experience API
Q_program = QuantumProgram()
# set the APIToken and API url
Q_program.set_api(Qconfig.APItoken, Qconfig.config["url"])
.. code:: python
#Making the Hamiltonian in its full form and getting the lowest eigenvalue and eigenvector
H = make_Hamiltonian(n,alpha,beta)
w, v = la.eigh(H, eigvals=(0, 1))
print(w)
v
.. code:: python
# Quantum circuit parameters
device = 'local_qasm_simulator' # the device to run on
shots = 8192 # the number of shots in the experiment.
entangler_map = {0: [1], 1: [2], 2: [3]} # the map of two-qubit gates with control at key and target at values
# Numerical parameters
SPSA_parameters = np.array([.3,0.602,0,.1,0.101]) #[a, alpha, A, c, gamma]
max_trials = 100;
max_depth = 3
cost, data_save, cost_save = SPSA_Minimization(Q_program, alpha, beta, n, device, shots, entangler_map, SPSA_parameters, max_trials, max_depth)
print('m = 1 ' + str(cost[0]) + ' m = 2 ' + str(cost[1]) + 'm = 3 ' + str(cost[2]))
.. code:: python
# plotting data
plt.plot(range(max_trials), cost_save[0])
plot_histogram(data_save[0])
plt.plot(range(max_trials), cost_save[1])
plot_histogram(data_save[1])
plt.plot(range(max_trials), cost_save[2])
plot_histogram(data_save[2])
Four Qubits (fast)
==================
::
Graph:
X---X
| \ |
X---X
The ground state is degenerate and is either :math:`|100\rangle`,
:math:`|010\rangle` and :math:`|001\rangle`.
.. code:: python
# cost function H = alpha_i z_i + beta_ij z_i z_j
n =4
alpha = np.zeros(n)
beta = np.zeros((n, n))
beta[0, 1] = 1
beta[0, 2] = 1
beta[1, 2] = 1
beta[1, 3] = 1
beta[2, 3] = 1
.. code:: python
#Setting up a quantum program and connecting to the Quantum Experience API
Q_program = QuantumProgram()
# set the APIToken and API url
Q_program.set_api(Qconfig.APItoken, Qconfig.config["url"])
.. parsed-literal::
---- Error: Exception connect to servers ----
.. parsed-literal::
False
.. code:: python
# Quantum circuit parameters
entangler_map = {0: [1], 1: [2], 2: [3]} # the map of two-qubit gates with control at key and target at values
# Numerical parameters
SPSA_parameters = np.array([.3,0.602,0,.1,0.101]) #[a, alpha, A, c, gamma]
max_trials = 100;
max_depth = 3
cost, data_save, cost_save = SPSA_Minimization_fast(Q_program, alpha, beta, n, entangler_map, SPSA_parameters, max_trials, max_depth)
print('m = 1 ' + str(cost[0]) + ' m = 2 ' + str(cost[1]) + 'm = 3 ' + str(cost[2]))
.. parsed-literal::
trial 0 of 100 cost -0.943257454022
trial 10 of 100 cost -2.20629876865
trial 20 of 100 cost -2.7499512609
trial 30 of 100 cost -2.84118948435
trial 40 of 100 cost -2.88829727076
trial 50 of 100 cost -2.9242398834
trial 60 of 100 cost -2.96130416407
trial 70 of 100 cost -2.93303732352
trial 80 of 100 cost -2.94302575557
trial 90 of 100 cost -2.93854780452
m = 1 -2.015625 m = 2 -2.9921875m = 3 -2.99609375
.. code:: python
# plotting data
plt.plot(range(max_trials), cost_save[0])
plot_histogram(data_save[0])
plt.plot(range(max_trials), cost_save[1])
plot_histogram(data_save[1])
plt.plot(range(max_trials), cost_save[2])
plot_histogram(data_save[2])
.. image:: classical_optimization_files/classical_optimization_15_0.png
.. image:: classical_optimization_files/classical_optimization_15_1.png
.. image:: classical_optimization_files/classical_optimization_15_2.png
.. image:: classical_optimization_files/classical_optimization_15_3.png
.. image:: classical_optimization_files/classical_optimization_15_4.png
.. image:: classical_optimization_files/classical_optimization_15_5.png
10 qubits
=========
.. code:: python
n = 10
# cost function H = alpha_i z_i + beta_ij z_i z_j
alpha = np.zeros(n)
beta = np.random.choice([0, 0.5], size=(n,n), p=[1./3, 2./3])
.. code:: python
# quantum circuit parameters
device = 'simulator' # the device to run on
shots = 8192 # the number of shots in the experiment
entangler_map = {0: [1], 1: [2], 2: [3], 3: [4], 4: [5], 5: [6], 6: [7], 7: [8], 8: [9]} # the map of two-qubit gates with control at key and target at values
# Numerical parameters
SPSA_parameters = np.array([.3,0.602,0,.1,0.101]) # [a, alpha, A, c, gamma]
max_trials = 100;
theta_depth_1 = np.random.randn(1*n) # initial controls
theta_depth_2 = np.random.randn(2*n) # initial controls
theta_depth_3 = np.random.randn(3*n) # initial controls
trial_circuit_depth_1 = trial_funtion_optimization(n,1,theta_depth_1,entangler_map)
trial_circuit_depth_2 = trial_funtion_optimization(n,2,theta_depth_2,entangler_map)
trial_circuit_depth_3 = trial_funtion_optimization(n,3,theta_depth_3,entangler_map)
program = [trial_circuit_depth_1,trial_circuit_depth_2,trial_circuit_depth_3]
out = run_program(program,api,device,shots,max_credits=3)
results=combine_jobs([out['id']], api, wait=20, timeout=440)
cost_depth_1 = cost_classical(get_data(results,0),n,alpha,beta)
cost_depth_2 = cost_classical(get_data(results,1),n,alpha,beta)
cost_depth_3 = cost_classical(get_data(results,2),n,alpha,beta)
print('m=1 ' + str(cost_depth_1) + ' m=2 ' + str(cost_depth_2) + 'm=3 ' + str(cost_depth_3))
.. code:: python
# plotting data
plt.plot(range(max_trials), cost_plus_depth_1, range(max_trials), cost_minus_depth_1)
plot_histogram(get_data(results,0),n)
plt.plot(range(max_trials), cost_plus_depth_2, range(max_trials), cost_minus_depth_2)
plot_histogram(get_data(results,1),n)
plt.plot(range(max_trials), cost_plus_depth_3, range(max_trials), cost_minus_depth_3)
plot_histogram(get_data(results,2),n)
.. code:: python
# cost function H = alpha_i z_i + beta_ij z_i z_j
n = 10
alpha = np.zeros(n)
beta = np.random.choice([0, 0.5], size=(n,n), p=[1./3, 2./3])
for i in range(n):
for j in range(i):
beta[j,i]=beta[i,j]
for i in range(n):
beta[i,i]=0;
.. code:: python
# quantum circuit parameters
device = 'simulator' # the device to run on
shots = 8192 # the number of shots in the experiment
entangler_map = {0: [1], 1: [2], 2: [3], 3: [4], 4: [5], 5: [6], 6: [7], 7: [8], 8: [9]} # the map of two-qubit gates with control at key and target at values
# Numerical parameters
SPSA_parameters = np.array([.3,0.602,0,.1,0.101]) #[a, alpha, A, c, gamma]
max_trials = 100;
theta_depth_1 = np.random.randn(1*n) # initial controls
theta_depth_2 = np.random.randn(2*n) # initial controls
theta_depth_3 = np.random.randn(3*n) # initial controls
cost_plus_depth_1=np.zeros(max_trials)
cost_minus_depth_1=np.zeros(max_trials)
cost_plus_depth_2=np.zeros(max_trials)
cost_minus_depth_2=np.zeros(max_trials)
cost_plus_depth_3=np.zeros(max_trials)
cost_minus_depth_3=np.zeros(max_trials)
for k in range(max_trials):
print('trial ' + str(k) + " of " + str(max_trials))
a_spsa = float(SPSA_parameters[0])/np.power(k+1+SPSA_parameters[2], SPSA_parameters[1])
c_spsa = float(SPSA_parameters[3])/np.power(k+1, SPSA_parameters[4])
Delta_depth_1 = 2*np.random.randint(2,size=n*1)-1 # \pm 1 random distribution
Delta_depth_2 = 2*np.random.randint(2,size=n*2)-1 # \pm 1 random distribution
Delta_depth_3 = 2*np.random.randint(2,size=n*3)-1 # \pm 1 random distribution
theta_plus_depth_1 = theta_depth_1 + c_spsa*Delta_depth_1
theta_minus_depth_1 = theta_depth_1 - c_spsa*Delta_depth_1
theta_plus_depth_2 = theta_depth_2 + c_spsa*Delta_depth_2
theta_minus_depth_2 = theta_depth_2 - c_spsa*Delta_depth_2
theta_plus_depth_3 = theta_depth_3 + c_spsa*Delta_depth_3
theta_minus_depth_3 = theta_depth_3 - c_spsa*Delta_depth_3
trial_circuit_plus_depth_1 = trial_funtion_optimization(n,1,theta_plus_depth_3,entangler_map)
trial_circuit_minus_depth_1 = trial_funtion_optimization(n,1,theta_minus_depth_1,entangler_map)
trial_circuit_plus_depth_2 = trial_funtion_optimization(n,2,theta_plus_depth_3,entangler_map)
trial_circuit_minus_depth_2 = trial_funtion_optimization(n,2,theta_minus_depth_2,entangler_map)
trial_circuit_plus_depth_3 = trial_funtion_optimization(n,3,theta_plus_depth_3,entangler_map)
trial_circuit_minus_depth_3 = trial_funtion_optimization(n,3,theta_minus_depth_3,entangler_map)
program = [trial_circuit_plus_depth_1,trial_circuit_minus_depth_1,trial_circuit_plus_depth_2
,trial_circuit_minus_depth_2,trial_circuit_plus_depth_3,trial_circuit_minus_depth_3]
out = run_program(program,api,device,shots,max_credits=3)
results=combine_jobs([out['id']], api, wait=20, timeout=440)
cost_plus_depth_1[k] = cost_classical(get_data(results,0),n,alpha,beta)
cost_minus_depth_1[k] = cost_classical(get_data(results,1),n,alpha,beta)
cost_plus_depth_2[k] = cost_classical(get_data(results,2),n,alpha,beta)
cost_minus_depth_2[k] = cost_classical(get_data(results,3),n,alpha,beta)
cost_plus_depth_3[k] = cost_classical(get_data(results,4),n,alpha,beta)
cost_minus_depth_3[k] = cost_classical(get_data(results,5),n,alpha,beta)
g_spsa_depth_1 = (cost_plus_depth_1[k]-cost_minus_depth_1[k])*Delta_depth_1/(2.0*c_spsa)
g_spsa_depth_2 = (cost_plus_depth_2[k]-cost_minus_depth_2[k])*Delta_depth_2/(2.0*c_spsa)
g_spsa_depth_3 = (cost_plus_depth_3[k]-cost_minus_depth_3[k])*Delta_depth_3/(2.0*c_spsa)
theta_depth_1 = theta_depth_1 - a_spsa*g_spsa_depth_1
theta_depth_2 = theta_depth_2 - a_spsa*g_spsa_depth_2
theta_depth_3 = theta_depth_3 - a_spsa*g_spsa_depth_3
print(cost_minus_depth_3[k] + cost_plus_depth_3[k])
trial_circuit_depth_1 = trial_funtion_optimization(n,1,theta_depth_1,entangler_map)
trial_circuit_depth_2 = trial_funtion_optimization(n,2,theta_depth_2,entangler_map)
trial_circuit_depth_3 = trial_funtion_optimization(n,3,theta_depth_3,entangler_map)
program = [trial_circuit_depth_1,trial_circuit_depth_2,trial_circuit_depth_3]
out = run_program(program,api,device,shots,max_credits=3)
results=combine_jobs([out['id']], api, wait=20, timeout=440)
cost_depth_1 = cost_classical(get_data(results,0),n,alpha,beta)
cost_depth_2 = cost_classical(get_data(results,1),n,alpha,beta)
cost_depth_3 = cost_classical(get_data(results,2),n,alpha,beta)
print('m=1 ' + str(cost_depth_1) + ' m=2 ' + str(cost_depth_2) + 'm=3 ' + str(cost_depth_3))
.. code:: python
# plotting data
plt.plot(range(max_trials), cost_plus_depth_1, range(max_trials), cost_minus_depth_1)
plot_histogram(get_data(results,0),n)
plt.plot(range(max_trials), cost_plus_depth_2, range(max_trials), cost_minus_depth_2)
plot_histogram(get_data(results,1),n)
plt.plot(range(max_trials), cost_plus_depth_3, range(max_trials), cost_minus_depth_3)
plot_histogram(get_data(results,2),n)
MaxCut (DONNA)
===============
.. code:: python
# cost function H = alpha_i z_i + beta_ij z_i z_j
n = 6
alpha = np.zeros(n)
beta = np.zeros((n, n))
beta[0, 1] = 93/2
beta[0, 4] = 17/2
beta[0, 5] = 51/2
beta[1, 0] = 93/2
beta[1, 5] = 13/2
beta[1, 2] = 77/2
beta[2, 1] = 77/2
beta[2, 3] = 31/2
beta[2, 5] = 23/2
beta[3,2] = 31/2
beta[3,4] = 7/2
beta[3,5] = 46/2
beta[4,3] = 7/2
beta[4,5] = 65/2
beta[4,0] = 17/2
beta[5,0] = 51/2
beta[5,1] = 13/2
beta[5,2] = 23/2
beta[5,3] = 46/2
beta[5,4] = 65/2
.. code:: python
# quantum circuit parameters
device = 'simulator' # the device to run on
shots = 8192 # the number of shots in the experiment
entangler_map = {0: [1], 1: [2], 2: [3], 3: [4]} # the map of two-qubit gates with control at key and target at values
# Numerical parameters
SPSA_parameters = np.array([3,0.602,0,.3,0.101]) #[a, alpha, A, c, gamma]
max_trials = 100;
theta_depth_1 = np.random.randn(1*n) # initial controls
theta_depth_2 = np.random.randn(2*n) # initial controls
theta_depth_3 = np.random.randn(3*n) # initial controls
cost_plus_depth_1=np.zeros(max_trials)
cost_minus_depth_1=np.zeros(max_trials)
cost_plus_depth_2=np.zeros(max_trials)
cost_minus_depth_2=np.zeros(max_trials)
cost_plus_depth_3=np.zeros(max_trials)
cost_minus_depth_3=np.zeros(max_trials)
for k in range(max_trials):
print('trial ' + str(k) + " of " + str(max_trials))
a_spsa = float(SPSA_parameters[0])/np.power(k+1+SPSA_parameters[2], SPSA_parameters[1])
c_spsa = float(SPSA_parameters[3])/np.power(k+1, SPSA_parameters[4])
Delta_depth_1 = 2*np.random.randint(2,size=n*1)-1 # \pm 1 random distribution
Delta_depth_2 = 2*np.random.randint(2,size=n*2)-1 # \pm 1 random distribution
Delta_depth_3 = 2*np.random.randint(2,size=n*3)-1 # \pm 1 random distribution
theta_plus_depth_1 = theta_depth_1 + c_spsa*Delta_depth_1
theta_minus_depth_1 = theta_depth_1 - c_spsa*Delta_depth_1
theta_plus_depth_2 = theta_depth_2 + c_spsa*Delta_depth_2
theta_minus_depth_2 = theta_depth_2 - c_spsa*Delta_depth_2
theta_plus_depth_3 = theta_depth_3 + c_spsa*Delta_depth_3
theta_minus_depth_3 = theta_depth_3 - c_spsa*Delta_depth_3
trial_circuit_plus_depth_1 = trial_funtion_optimization(n,1,theta_plus_depth_3,entangler_map)
trial_circuit_minus_depth_1 = trial_funtion_optimization(n,1,theta_minus_depth_1,entangler_map)
trial_circuit_plus_depth_2 = trial_funtion_optimization(n,2,theta_plus_depth_3,entangler_map)
trial_circuit_minus_depth_2 = trial_funtion_optimization(n,2,theta_minus_depth_2,entangler_map)
trial_circuit_plus_depth_3 = trial_funtion_optimization(n,3,theta_plus_depth_3,entangler_map)
trial_circuit_minus_depth_3 = trial_funtion_optimization(n,3,theta_minus_depth_3,entangler_map)
program = [trial_circuit_plus_depth_1,trial_circuit_minus_depth_1,trial_circuit_plus_depth_2
,trial_circuit_minus_depth_2,trial_circuit_plus_depth_3,trial_circuit_minus_depth_3]
out = run_program(program,api,device,shots,max_credits=3)
results=combine_jobs([out['id']], api, wait=20, timeout=440)
cost_plus_depth_1[k] = cost_classical(get_data(results,0),n,alpha,beta)
cost_minus_depth_1[k] = cost_classical(get_data(results,1),n,alpha,beta)
cost_plus_depth_2[k] = cost_classical(get_data(results,2),n,alpha,beta)
cost_minus_depth_2[k] = cost_classical(get_data(results,3),n,alpha,beta)
cost_plus_depth_3[k] = cost_classical(get_data(results,4),n,alpha,beta)
cost_minus_depth_3[k] = cost_classical(get_data(results,5),n,alpha,beta)
g_spsa_depth_1 = (cost_plus_depth_1[k]-cost_minus_depth_1[k])*Delta_depth_1/(2.0*c_spsa)
g_spsa_depth_2 = (cost_plus_depth_2[k]-cost_minus_depth_2[k])*Delta_depth_2/(2.0*c_spsa)
g_spsa_depth_3 = (cost_plus_depth_3[k]-cost_minus_depth_3[k])*Delta_depth_3/(2.0*c_spsa)
theta_depth_1 = theta_depth_1 - a_spsa*g_spsa_depth_1
theta_depth_2 = theta_depth_2 - a_spsa*g_spsa_depth_2
theta_depth_3 = theta_depth_3 - a_spsa*g_spsa_depth_3
trial_circuit_depth_1 = trial_funtion_optimization(n,1,theta_depth_1,entangler_map)
trial_circuit_depth_2 = trial_funtion_optimization(n,2,theta_depth_2,entangler_map)
trial_circuit_depth_3 = trial_funtion_optimization(n,3,theta_depth_3,entangler_map)
program = [trial_circuit_depth_1,trial_circuit_depth_2,trial_circuit_depth_3]
out = run_program(program,api,device,shots,max_credits=3)
results=combine_jobs([out['id']], api, wait=20, timeout=440)
cost_depth_1 = cost_classical(get_data(results,0),n,alpha,beta)
cost_depth_2 = cost_classical(get_data(results,1),n,alpha,beta)
cost_depth_3 = cost_classical(get_data(results,2),n,alpha,beta)
print('m=1 ' + str(cost_depth_1) + ' m=2 ' + str(cost_depth_2) + 'm=3 ' + str(cost_depth_3))
.. code:: python
# plotting data
plt.plot(range(max_trials), cost_plus_depth_1, range(max_trials), cost_minus_depth_1)
plot_histogram(get_data(results,0),n)
plt.plot(range(max_trials), cost_plus_depth_2, range(max_trials), cost_minus_depth_2)
plot_histogram(get_data(results,1),n)
plt.plot(range(max_trials), cost_plus_depth_3, range(max_trials), cost_minus_depth_3)
plot_histogram(get_data(results,2),n)
REAL
====
.. code:: python
# quantum circuit parameters
device = 'real' # the device to run on
shots = 8192 # the number of shots in the experiment
n = 2
entangler_map = {0: [1]} # the map of two-qubit gates with control at key and target at values
# Numerical parameters
SPSA_parameters = np.array([.3,0.602,0,.3,0.101]) #[a, alpha, A, c, gamma]
max_trials = 100;
theta_depth_1 = np.random.randn(1*n) # initial controls
theta_depth_2 = np.random.randn(2*n) # initial controls
theta_depth_3 = np.random.randn(3*n) # initial controls
cost_plus_depth_1=np.zeros(max_trials)
cost_minus_depth_1=np.zeros(max_trials)
cost_plus_depth_2=np.zeros(max_trials)
cost_minus_depth_2=np.zeros(max_trials)
cost_plus_depth_3=np.zeros(max_trials)
cost_minus_depth_3=np.zeros(max_trials)
for k in range(max_trials):
print('trial ' + str(k) + " of " + str(max_trials))
a_spsa = float(SPSA_parameters[0])/np.power(k+1+SPSA_parameters[2], SPSA_parameters[1])
c_spsa = float(SPSA_parameters[3])/np.power(k+1, SPSA_parameters[4])
Delta_depth_1 = 2*np.random.randint(2,size=n*1)-1 # \pm 1 random distribution
Delta_depth_2 = 2*np.random.randint(2,size=n*2)-1 # \pm 1 random distribution
Delta_depth_3 = 2*np.random.randint(2,size=n*3)-1 # \pm 1 random distribution
theta_plus_depth_1 = theta_depth_1 + c_spsa*Delta_depth_1
theta_minus_depth_1 = theta_depth_1 - c_spsa*Delta_depth_1
theta_plus_depth_2 = theta_depth_2 + c_spsa*Delta_depth_2
theta_minus_depth_2 = theta_depth_2 - c_spsa*Delta_depth_2
theta_plus_depth_3 = theta_depth_3 + c_spsa*Delta_depth_3
theta_minus_depth_3 = theta_depth_3 - c_spsa*Delta_depth_3
trial_circuit_plus_depth_1 = trial_funtion_optimization(n,1,theta_plus_depth_3,entangler_map)
trial_circuit_minus_depth_1 = trial_funtion_optimization(n,1,theta_minus_depth_1,entangler_map)
trial_circuit_plus_depth_2 = trial_funtion_optimization(n,2,theta_plus_depth_3,entangler_map)
trial_circuit_minus_depth_2 = trial_funtion_optimization(n,2,theta_minus_depth_2,entangler_map)
trial_circuit_plus_depth_3 = trial_funtion_optimization(n,3,theta_plus_depth_3,entangler_map)
trial_circuit_minus_depth_3 = trial_funtion_optimization(n,3,theta_minus_depth_3,entangler_map)
program = [trial_circuit_plus_depth_1,trial_circuit_minus_depth_1,trial_circuit_plus_depth_2
,trial_circuit_minus_depth_2,trial_circuit_plus_depth_3,trial_circuit_minus_depth_3]
out = run_program(program,api,device,shots,max_credits=5)
results=combine_jobs([out['id']], api, wait=20, timeout=480)
cost_plus_depth_1[k] = cost_classical(get_data(results,0),n,alpha,beta)
cost_minus_depth_1[k] = cost_classical(get_data(results,1),n,alpha,beta)
cost_plus_depth_2[k] = cost_classical(get_data(results,2),n,alpha,beta)
cost_minus_depth_2[k] = cost_classical(get_data(results,3),n,alpha,beta)
cost_plus_depth_3[k] = cost_classical(get_data(results,4),n,alpha,beta)
cost_minus_depth_3[k] = cost_classical(get_data(results,5),n,alpha,beta)
g_spsa_depth_1 = (cost_plus_depth_1[k]-cost_minus_depth_1[k])*Delta_depth_1/(2.0*c_spsa)
g_spsa_depth_2 = (cost_plus_depth_2[k]-cost_minus_depth_2[k])*Delta_depth_2/(2.0*c_spsa)
g_spsa_depth_3 = (cost_plus_depth_3[k]-cost_minus_depth_3[k])*Delta_depth_3/(2.0*c_spsa)
theta_depth_1 = theta_depth_1 - a_spsa*g_spsa_depth_1
theta_depth_2 = theta_depth_2 - a_spsa*g_spsa_depth_2
theta_depth_3 = theta_depth_3 - a_spsa*g_spsa_depth_3
trial_circuit_depth_1 = trial_funtion_optimization(n,1,theta_depth_1,entangler_map)
trial_circuit_depth_2 = trial_funtion_optimization(n,2,theta_depth_2,entangler_map)
trial_circuit_depth_3 = trial_funtion_optimization(n,3,theta_depth_3,entangler_map)
program = [trial_circuit_depth_1,trial_circuit_depth_2,trial_circuit_depth_3]
out = run_program(program,api,device,shots,max_credits=5)
results=combine_jobs([out['id']], api, wait=20, timeout=240)
cost_depth_1 = cost_classical(get_data(results,0),n,alpha,beta)
cost_depth_2 = cost_classical(get_data(results,1),n,alpha,beta)
cost_depth_3 = cost_classical(get_data(results,2),n,alpha,beta)
print('m=1 ' + str(cost_depth_1) + ' m=2 ' + str(cost_depth_2) + 'm=3 ' + str(cost_depth_3))
.. code:: python
# plotting data
plt.plot(range(max_trials), cost_plus_depth_1, range(max_trials), cost_minus_depth_1)
plot_histogram(get_data(results,0),n)
plt.plot(range(max_trials), cost_plus_depth_2, range(max_trials), cost_minus_depth_2)
plot_histogram(get_data(results,1),n)
plt.plot(range(max_trials), cost_plus_depth_3, range(max_trials), cost_minus_depth_3)
plot_histogram(get_data(results,2),n)

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#!/usr/bin/env python3
# -*- coding: utf-8 -*-
#
# qiskit-sdk-py-dev documentation build configuration file, created by
# sphinx-quickstart on Fri Jun 30 00:56:35 2017.
#
# This file is execfile()d with the current directory set to its
# containing dir.
#
# Note that not all possible configuration values are present in this
# autogenerated file.
#
# All configuration values have a default; values that are commented out
# serve to show the default.
import sys
import os
# If extensions (or modules to document with autodoc) are in another directory,
# add these directories to sys.path here. If the directory is relative to the
# documentation root, use os.path.abspath to make it absolute, like shown here.
#sys.path.insert(0, os.path.abspath('.'))
# -- General configuration ------------------------------------------------
# If your documentation needs a minimal Sphinx version, state it here.
#needs_sphinx = '1.0'
# Add any Sphinx extension module names here, as strings. They can be
# extensions coming with Sphinx (named 'sphinx.ext.*') or your custom
# ones.
extensions = [
'sphinx.ext.autodoc',
'sphinx.ext.napoleon', #support numpy and google style docstrings
'sphinx.ext.todo',
'sphinx.ext.coverage',
'sphinx.ext.mathjax',
'sphinx.ext.ifconfig',
'sphinx.ext.viewcode',
'sphinx.ext.githubpages',
'nbsphinx',
]
# Add any paths that contain templates here, relative to this directory.
templates_path = ['_templates']
# The suffix(es) of source filenames.
# You can specify multiple suffix as a list of string:
# source_suffix = ['.rst', '.md']
source_suffix = '.rst'
# The encoding of source files.
#source_encoding = 'utf-8-sig'
# The master toctree document.
master_doc = 'index'
# General information about the project.
project = 'qiskit-sdk-py-dev'
copyright = '2017, IBM'
#author = 'Antonio Córcoles, Jerry Chow, Abigail Cross, Andrew Cross, Ismael Faro, Andreas Fuhrer, Jay Gambetta'
author = 'Antonio Córcoles'
author = 'Jerry Chow'
# The version info for the project you're documenting, acts as replacement for
# |version| and |release|, also used in various other places throughout the
# built documents.
#
# The short X.Y version.
version = ''
# The full version, including alpha/beta/rc tags.
release = ''
# The language for content autogenerated by Sphinx. Refer to documentation
# for a list of supported languages.
#
# This is also used if you do content translation via gettext catalogs.
# Usually you set "language" from the command line for these cases.
language = None
# There are two options for replacing |today|: either, you set today to some
# non-false value, then it is used:
#today = ''
# Else, today_fmt is used as the format for a strftime call.
#today_fmt = '%B %d, %Y'
# List of patterns, relative to source directory, that match files and
# directories to ignore when looking for source files.
# This patterns also effect to html_static_path and html_extra_path
exclude_patterns = ['_build', 'Thumbs.db', '.DS_Store']
# The reST default role (used for this markup: `text`) to use for all
# documents.
#default_role = None
# If true, '()' will be appended to :func: etc. cross-reference text.
#add_function_parentheses = True
# If true, the current module name will be prepended to all description
# unit titles (such as .. function::).
#add_module_names = True
# If true, sectionauthor and moduleauthor directives will be shown in the
# output. They are ignored by default.
#show_authors = False
# The name of the Pygments (syntax highlighting) style to use.
pygments_style = 'sphinx'
# A list of ignored prefixes for module index sorting.
#modindex_common_prefix = []
# If true, keep warnings as "system message" paragraphs in the built documents.
#keep_warnings = False
# If true, `todo` and `todoList` produce output, else they produce nothing.
todo_include_todos = True
# -- Options for HTML output ----------------------------------------------
# The theme to use for HTML and HTML Help pages. See the documentation for
# a list of builtin themes.
#html_theme = 'alabaster'
html_theme = 'agogo'
# Theme options are theme-specific and customize the look and feel of a theme
# further. For a list of options available for each theme, see the
# documentation.
#html_theme_options = {}
# Add any paths that contain custom themes here, relative to this directory.
#html_theme_path = []
# The name for this set of Sphinx documents.
# "<project> v<release> documentation" by default.
#html_title = 'qiskit-sdk-py-dev v1.0'
# A shorter title for the navigation bar. Default is the same as html_title.
#html_short_title = None
# The name of an image file (relative to this directory) to place at the top
# of the sidebar.
html_logo = '../images/QISKit-c.gif'
# The name of an image file (relative to this directory) to use as a favicon of
# the docs. This file should be a Windows icon file (.ico) being 16x16 or 32x32
# pixels large.
#html_favicon = None
# Add any paths that contain custom static files (such as style sheets) here,
# relative to this directory. They are copied after the builtin static files,
# so a file named "default.css" will overwrite the builtin "default.css".
html_static_path = ['_static']
# Add any extra paths that contain custom files (such as robots.txt or
# .htaccess) here, relative to this directory. These files are copied
# directly to the root of the documentation.
#html_extra_path = []
# If not None, a 'Last updated on:' timestamp is inserted at every page
# bottom, using the given strftime format.
# The empty string is equivalent to '%b %d, %Y'.
#html_last_updated_fmt = None
# If true, SmartyPants will be used to convert quotes and dashes to
# typographically correct entities.
#html_use_smartypants = True
# Custom sidebar templates, maps document names to template names.
#html_sidebars = {}
# Additional templates that should be rendered to pages, maps page names to
# template names.
#html_additional_pages = {}
# If false, no module index is generated.
#html_domain_indices = True
# If false, no index is generated.
#html_use_index = True
# If true, the index is split into individual pages for each letter.
#html_split_index = False
# If true, links to the reST sources are added to the pages.
#html_show_sourcelink = True
# If true, "Created using Sphinx" is shown in the HTML footer. Default is True.
#html_show_sphinx = True
# If true, "(C) Copyright ..." is shown in the HTML footer. Default is True.
#html_show_copyright = True
# If true, an OpenSearch description file will be output, and all pages will
# contain a <link> tag referring to it. The value of this option must be the
# base URL from which the finished HTML is served.
#html_use_opensearch = ''
# This is the file name suffix for HTML files (e.g. ".xhtml").
#html_file_suffix = None
# Language to be used for generating the HTML full-text search index.
# Sphinx supports the following languages:
# 'da', 'de', 'en', 'es', 'fi', 'fr', 'h', 'it', 'ja'
# 'nl', 'no', 'pt', 'ro', 'r', 'sv', 'tr', 'zh'
#html_search_language = 'en'
# A dictionary with options for the search language support, empty by default.
# 'ja' uses this config value.
# 'zh' user can custom change `jieba` dictionary path.
#html_search_options = {'type': 'default'}
# The name of a javascript file (relative to the configuration directory) that
# implements a search results scorer. If empty, the default will be used.
#html_search_scorer = 'scorer.js'
# Output file base name for HTML help builder.
htmlhelp_basename = 'qiskit-sdk-py-devdoc'
# -- Options for LaTeX output ---------------------------------------------
latex_elements = {
# The paper size ('letterpaper' or 'a4paper').
#'papersize': 'letterpaper',
# The font size ('10pt', '11pt' or '12pt').
#'pointsize': '10pt',
# Additional stuff for the LaTeX preamble.
#'preamble': '',
# Latex figure (float) alignment
#'figure_align': 'htbp',
}
# Grouping the document tree into LaTeX files. List of tuples
# (source start file, target name, title,
# author, documentclass [howto, manual, or own class]).
latex_documents = [
(master_doc, 'qiskit-sdk-py-dev.tex', 'qiskit-sdk-py-dev Documentation',
'Antonio Córcoles, Jerry Chow, Abigail Cross, Andrew Cross, Ismael Faro, Andreas Fuhrer, Jay Gambetta', 'manual'),
]
# The name of an image file (relative to this directory) to place at the top of
# the title page.
#latex_logo = '../images/QISKit.gif'
# For "manual" documents, if this is true, then toplevel headings are parts,
# not chapters.
#latex_use_parts = False
# If true, show page references after internal links.
#latex_show_pagerefs = False
# If true, show URL addresses after external links.
#latex_show_urls = False
# Documents to append as an appendix to all manuals.
#latex_appendices = []
# If false, no module index is generated.
#latex_domain_indices = True
# -- Options for manual page output ---------------------------------------
# One entry per manual page. List of tuples
# (source start file, name, description, authors, manual section).
man_pages = [
(master_doc, 'qiskit-sdk-py-dev', 'qiskit-sdk-py-dev Documentation',
[author], 1)
]
# If true, show URL addresses after external links.
#man_show_urls = False
# -- Options for Texinfo output -------------------------------------------
# Grouping the document tree into Texinfo files. List of tuples
# (source start file, target name, title, author,
# dir menu entry, description, category)
texinfo_documents = [
(master_doc, 'qiskit-sdk-py-dev', 'qiskit-sdk-py-dev Documentation',
author, 'qiskit-sdk-py-dev', 'One line description of project.',
'Miscellaneous'),
]
# Documents to append as an appendix to all manuals.
#texinfo_appendices = []
# If false, no module index is generated.
#texinfo_domain_indices = True
# How to display URL addresses: 'footnote', 'no', or 'inline'.
#texinfo_show_urls = 'footnote'
# If true, do not generate a @detailmenu in the "Top" node's menu.
#texinfo_no_detailmenu = False

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==========================
Entanglement Revisited
==========================
For more information about how to use the IBM Q experience (QX), consult
the
`tutorials <https://quantumexperience.ng.bluemix.net/qstage/#/tutorial?sectionId=c59b3710b928891a1420190148a72cce&pageIndex=0>`__,
or check out the
`community <https://quantumexperience.ng.bluemix.net/qstage/#/community>`__.
Contributors
============
Jay Gambetta, Antonio Córcoles
Entanglement
============
In :doc:`superposition and entanglement
<superposition_and_entanglement>`, we introduced you to the quantum
concept of entanglement. We made the quantum state
:math:`|\psi\rangle= (|00\rangle+|11\rangle)/\sqrt{2}` and showed that
(accounting for experimental noise) the system has perfect
correlations in both the computational and the superposition
bases. This means if :math:`q_0` is measured in state
:math:`|0\rangle`, we know :math:`q_1` is in the same state; likewise,
if :math:`q_0` is measured in state :math:`|+\rangle`, we know
:math:`q_1` is also in the same state.
To understand the implications of this in more detail, we will look at
the following topics in this notebook: \* `Two-Qubit Correlated
Observables <#section1>`__ \* `CHSH Inequality <#section2>`__ \* `Two-,
Three-, and Four-Qubit GHZ States <#section3>`__ \* `Mermin's Test and
the Three Box Game <#section4>`__
Two-Qubit Correlated Observables
================================
An observable is a Hermitian matrix where the real eigenvalues represent
the outcome of the experiment, and the eigenvectors are the states to
which the system is projected under measurement. That is, an observable
:math:`A` is given by
.. math:: A = \sum_j a_j|a_j\rangle\langle a_j|
where :math:`|a_j\rangle` is the eigenvector of the observable with
result :math:`a_j`. The expectation value of this observable is given by
.. math:: \langle A \rangle = \sum_j a_j |\langle \psi |a_j\rangle|^2 = \sum_j a_j \mathrm{Pr}(a_j|\psi).
We can see there is the standard relationship between average
(expectation value) and probability.
For a two-qubit system, the following are important two-outcome
(:math:`\pm1`) single-qubit observables:
.. math:: Z= |0\rangle\langle 0| - |1\rangle\langle 1|
.. math:: X= |+\rangle\langle +| - |-\rangle\langle -|
These are also commonly referred to as the Pauli :math:`Z` and :math:`X`
operators. These can be further extended to the two-qubit space to give
.. math:: \langle I\otimes Z\rangle =\mathrm{Pr}(00|\psi) - \mathrm{Pr}(01|\psi) + \mathrm{Pr}(10|\psi)- \mathrm{Pr}(11|\psi)
.. math:: \langle Z\otimes I\rangle =\mathrm{Pr}(00|\psi) + \mathrm{Pr}(01|\psi) - \mathrm{Pr}(10|\psi)- \mathrm{Pr}(11|\psi)
.. math:: \langle Z\otimes Z\rangle =\mathrm{Pr}(00|\psi) - \mathrm{Pr}(01|\psi) - \mathrm{Pr}(10|\psi)+ \mathrm{Pr}(11|\psi)
.. math:: \langle I\otimes X\rangle =\mathrm{Pr}(++|\psi) - \mathrm{Pr}(+-|\psi) + \mathrm{Pr}(-+|\psi)- \mathrm{Pr}(--|\psi)
.. math:: \langle X\otimes I\rangle =\mathrm{Pr}(++|\psi) + \mathrm{Pr}(+-|\psi) - \mathrm{Pr}(-+|\psi)- \mathrm{Pr}(--|\psi)
.. math:: \langle X\otimes X\rangle =\mathrm{Pr}(++|\psi) - \mathrm{Pr}(+-|\psi) - \mathrm{Pr}(-+|\psi)+ \mathrm{Pr}(--|\psi)
.. math:: \langle Z\otimes X\rangle =\mathrm{Pr}(0+|\psi) - \mathrm{Pr}(0-|\psi) - \mathrm{Pr}(1+|\psi)+ \mathrm{Pr}(1-|\psi)
.. math:: \langle X\otimes Z\rangle =\mathrm{Pr}(+0|\psi) - \mathrm{Pr}(+1|\psi) - \mathrm{Pr}(-0|\psi)+ \mathrm{Pr}(-1|\psi)
.. code:: ipython3
# Checking the version of PYTHON; we only support 3 at the moment
import sys
if sys.version_info < (3,0):
raise Exception("Please use Python version 3 or greater.")
# useful additional packages
import matplotlib.pyplot as plt
%matplotlib inline
import numpy as np
import sys
sys.path.append("../../")
# importing the QISKit
from qiskit import QuantumCircuit, QuantumProgram
import Qconfig
# import basic plot tools
from qiskit.basicplotter import plot_histogram
Recall that to make the Bell state
:math:`|\psi\rangle= (|00\rangle+|11\rangle)/\sqrt{2}` from the initial
state :math:`|00\rangle`, the quantum circuit first applies a Hadamard
on :math:`q_0`, followed by a CNOT from :math:`q_0` to :math:`q_1`. On
the IBM Q experience, this can done by using the script below to measure
the above expectation values; we run four different experiments with
measurements in the standard basis, superposition basis, and a
combination of both.
.. code:: ipython3
device = 'ibmqx2' # the device to run on
shots = 1024 # the number of shots in the experiment
# device = 'simulator' # the device test purpose
QPS_SPECS = {
"name": "Entanglement",
"circuits": [{
"name": "bell",
"quantum_registers": [{
"name":"q",
"size":2
}],
"classical_registers": [{
"name":"c",
"size":2
}]}],
}
Q_program = QuantumProgram(specs=QPS_SPECS)
Q_program.set_api(Qconfig.APItoken, Qconfig.config["url"])
# quantum circuit to make Bell state
bell = Q_program.get_circuit("bell")
q = Q_program.get_quantum_registers("q")
c = Q_program.get_classical_registers('c')
bell.h(q[0])
bell.cx(q[0],q[1])
# quantum circuit to measure q in standard basis
measureZZ = Q_program.create_circuit("measureZZ", ["q"], ["c"])
measureZZ.measure(q[0], c[0])
measureZZ.measure(q[1], c[1])
# quantum circuit to measure q in superposition basis
measureXX = Q_program.create_circuit("measureXX", ["q"], ["c"])
measureXX.h(q[0])
measureXX.h(q[1])
measureXX.measure(q[0], c[0])
measureXX.measure(q[1], c[1])
# quantum circuit to measure ZX
measureZX = Q_program.create_circuit("measureZX", ["q"], ["c"])
measureZX.h(q[0])
measureZX.measure(q[0], c[0])
measureZX.measure(q[1], c[1])
# quantum circuit to measure XZ
measureXZ = Q_program.create_circuit("measureXZ", ["q"], ["c"])
measureXZ.h(q[1])
measureXZ.measure(q[0], c[0])
measureXZ.measure(q[1], c[1])
.. parsed-literal::
>> quantum_registers created: q 2
>> classical_registers created: c 2
.. parsed-literal::
<qiskit._measure.Measure at 0x10da47e10>
.. code:: ipython3
Q_program.add_circuit("bell_measureZX", bell+measureZX )
Q_program.add_circuit("bell_measureXZ", bell+measureXZ )
Q_program.add_circuit("bell_measureZZ", bell+measureZZ )
Q_program.add_circuit("bell_measureXX", bell+measureXX )
circuits = ["bell_measureZZ", "bell_measureZX", "bell_measureXX", "bell_measureXZ"]
Q_program.get_qasms(circuits)
.. parsed-literal::
['OPENQASM 2.0;\ninclude "qelib1.inc";\nqreg q[2];\ncreg c[2];\nh q[0];\ncx q[0],q[1];\nmeasure q[0] -> c[0];\nmeasure q[1] -> c[1];\n',
'OPENQASM 2.0;\ninclude "qelib1.inc";\nqreg q[2];\ncreg c[2];\nh q[0];\ncx q[0],q[1];\nh q[0];\nmeasure q[0] -> c[0];\nmeasure q[1] -> c[1];\n',
'OPENQASM 2.0;\ninclude "qelib1.inc";\nqreg q[2];\ncreg c[2];\nh q[0];\ncx q[0],q[1];\nh q[0];\nh q[1];\nmeasure q[0] -> c[0];\nmeasure q[1] -> c[1];\n',
'OPENQASM 2.0;\ninclude "qelib1.inc";\nqreg q[2];\ncreg c[2];\nh q[0];\ncx q[0],q[1];\nh q[1];\nmeasure q[0] -> c[0];\nmeasure q[1] -> c[1];\n']
.. code:: ipython3
Q_program.execute(circuits, device, shots, max_credits=3, wait=10, timeout=240)
.. parsed-literal::
running on backend: ibmqx2
status = RUNNING (10 seconds)
status = RUNNING (20 seconds)
.. parsed-literal::
{'result': 'all done', 'status': 'COMPLETED'}
.. code:: ipython3
### THIS IS A KNOWN BUG AND WHEN WE FIX THE RETURN FROM THE REAL DEVICE WE WILL ONLY HAVE ONE SET OF OBSERVABLES
observable_first ={'00000': 1, '00001': -1, '00010': 1, '00011': -1}
observable_second ={'00000': 1, '00001': 1, '00010': -1, '00011': -1}
observable_correlated ={'00000': 1, '00001': -1, '00010': -1, '00011': 1}
observable_first_ideal ={'00': 1, '01': -1, '10': 1, '11': -1}
observable_second_ideal ={'00': 1, '01': 1, '10': -1, '11': -1}
observable_correlated_ideal ={'00': 1, '01': -1, '10': -1, '11': 1}
.. code:: ipython3
print("IZ = " + str(Q_program.average_data("bell_measureZZ",observable_first)))
print("ZI = " + str(Q_program.average_data("bell_measureZZ",observable_second)))
print("ZZ = " + str(Q_program.average_data("bell_measureZZ",observable_correlated)))
print("IX = " + str(Q_program.average_data("bell_measureXX",observable_first)))
print("XI = " + str(Q_program.average_data("bell_measureXX",observable_second)))
print("XX = " + str(Q_program.average_data("bell_measureXX",observable_correlated)))
print("ZX = " + str(Q_program.average_data("bell_measureZX",observable_correlated)))
print("XZ = " + str(Q_program.average_data("bell_measureXZ",observable_correlated)))
.. parsed-literal::
IZ = 0.025390625
ZI = 0.015625
ZZ = 0.857421875
IX = 0.05859375
XI = 0.07421875
XX = 0.875
ZX = 0.0
XZ = 0.025390625
Here we see that for the state
:math:`|\psi\rangle= (|00\rangle+|11\rangle)/\sqrt{2}`, expectation
values (within experimental errors) are
+--------------+------------------+--------------+------------------+--------------+------------------+
| Observable | Expected value | Observable | Expected value | Observable | Expected value |
+==============+==================+==============+==================+==============+==================+
| ZZ | 1 | XX | 1 | ZX | 0 |
+--------------+------------------+--------------+------------------+--------------+------------------+
| ZI | 0 | XI | 0 | XZ | 0 |
+--------------+------------------+--------------+------------------+--------------+------------------+
| IZ | 0 | IX | 0 | | |
+--------------+------------------+--------------+------------------+--------------+------------------+
How do we explain this situation? Here we introduce the concept of a
*hidden variable model*. If we assume there is a hidden variable
:math:`\lambda` and follow these two assumptions:
- *Locality*: No information can travel faster than the speed of light.
There is a hidden variable :math:`\lambda` that defines all the
correlations so that
.. math:: \langle A\otimes B\rangle = \sum_\lambda P(\lambda) A(\lambda) B(\lambda)
- *Realism*: All observables have a definite value independent of the
measurement (:math:`A(\lambda)=\pm1` etc.).
then can we describe these observations? --- The answer is yes!
Assume :math:`\lambda` has two bits, each occurring randomly with
probably 1/4. The following predefined table would then explain all the
above observables:
+-------------------+---------------+---------------+---------------+---------------+
| :math:`\lambda` | Z (qubit 1) | Z (qubit 2) | X (qubit 1) | X (qubit 2) |
+===================+===============+===============+===============+===============+
| 00 | 1 | 1 | 1 | 1 |
+-------------------+---------------+---------------+---------------+---------------+
| 01 | 1 | 1 | -1 | -1 |
+-------------------+---------------+---------------+---------------+---------------+
| 10 | -1 | -1 | -1 | -1 |
+-------------------+---------------+---------------+---------------+---------------+
| 11 | -1 | -1 | 1 | 1 |
+-------------------+---------------+---------------+---------------+---------------+
Thus, with a purely classical hidden variable model, we are able to
reconcile the measured observations we had for this particular Bell
state. However, there are some states for which this model will not
hold. This was first observed by John Stewart Bell in 1964. He proposed
a theorem that suggests that there are no hidden variables in quantum
mechanics. At the core of Bell's theorem is the famous Bell inequality.
Here, we'll use a refined version of this inequality (known as the CHSH
inequality, derived by John Clauser, Michael Horne, Abner Shimony, and
Richard Holt in 1969) to demonstrate Bell's proposal.
CHSH Inequality
===============
In the CHSH inequality, we measure the correlator of four observables:
:math:`A` and :math:`A'` on :math:`q_0`, and :math:`B` and :math:`B'` on
:math:`q_1`, which have eigenvalues :math:`\pm 1`. The CHSH inequality
says that no local hidden variable theory can have
.. math:: |C|>2
where
.. math:: C = \langle B\otimes A\rangle + \langle B\otimes A'\rangle+\langle B'\otimes A'\rangle-\langle B'\otimes A\rangle.
What would this look like with some hidden variable model under the
locality and realism assumptions from above? :math:`C` then becomes
.. math:: C = \sum_\lambda P(\lambda) \{ B(\lambda) [ A(\lambda)+A'(\lambda)] + B'(\lambda) [ A'(\lambda)-A(\lambda)]
and :math:`[A(\lambda)+A'(\lambda)]=2` (or 0) while
:math:`[A'(\lambda)-A(\lambda)]=0` (or 2) respectively. That is,
:math:`|C|=2`, and noise will only make this smaller.
If we measure a number greater than 2, the above assumptions cannot be
valid. (This is a perfect example of one of those astonishing
counterintuitive ideas one must accept in the quantum world.) For
simplicity, we choose these observables to be
.. math:: C = \langle Z\otimes Z\rangle + \langle Z\otimes X\rangle+\langle X\otimes X\rangle-\langle X\otimes Z\rangle.
:math:`Z` is measured in the computational basis, and :math:`X` in the
superposition basis (:math:`H` is applied before measurement). The input
state
.. math:: |\psi(\theta)\rangle = I\otimes Y(\theta)\frac{|00\rangle + |11\rangle}{\sqrt(2)} = \frac{\cos(\theta/2)|00\rangle + \cos(\theta/2)|11\rangle+\sin(\theta/2)|01\rangle-\sin(\theta/2)|10\rangle}{\sqrt{2}}
is swept vs. :math:`\theta` (think of this as allowing us to prepare a
varying set of states varying in the angle :math:`\theta`).
Note that the following demonstration of CHSH is not loophole-free.
.. code:: ipython3
CHSH = lambda x : x[0]+x[1]+x[2]-x[3]
.. code:: ipython3
device = 'local_qasm_simulator'
shots = 8192
program = []
xdata=[]
program_end = [measureZZ, measureZX, measureXX, measureXZ]
k = 0
for jj in range(30):
theta = 2.0*np.pi*jj/30
bell_middle = QuantumCircuit(q,c)
bell_middle.ry(theta,q[0])
for i in range(4):
program.append("circuit"+str(k))
Q_program.add_circuit("circuit"+str(k), bell+bell_middle+program_end[i] )
k += 1
xdata.append(theta)
.. code:: ipython3
Q_program.get_qasms(program[0:8])
.. parsed-literal::
['OPENQASM 2.0;\ninclude "qelib1.inc";\nqreg q[2];\ncreg c[2];\nh q[0];\ncx q[0],q[1];\nry(0.000000000000000) q[0];\nmeasure q[0] -> c[0];\nmeasure q[1] -> c[1];\n',
'OPENQASM 2.0;\ninclude "qelib1.inc";\nqreg q[2];\ncreg c[2];\nh q[0];\ncx q[0],q[1];\nry(0.000000000000000) q[0];\nh q[0];\nmeasure q[0] -> c[0];\nmeasure q[1] -> c[1];\n',
'OPENQASM 2.0;\ninclude "qelib1.inc";\nqreg q[2];\ncreg c[2];\nh q[0];\ncx q[0],q[1];\nry(0.000000000000000) q[0];\nh q[0];\nh q[1];\nmeasure q[0] -> c[0];\nmeasure q[1] -> c[1];\n',
'OPENQASM 2.0;\ninclude "qelib1.inc";\nqreg q[2];\ncreg c[2];\nh q[0];\ncx q[0],q[1];\nry(0.000000000000000) q[0];\nh q[1];\nmeasure q[0] -> c[0];\nmeasure q[1] -> c[1];\n',
'OPENQASM 2.0;\ninclude "qelib1.inc";\nqreg q[2];\ncreg c[2];\nh q[0];\ncx q[0],q[1];\nry(0.209439510239320) q[0];\nmeasure q[0] -> c[0];\nmeasure q[1] -> c[1];\n',
'OPENQASM 2.0;\ninclude "qelib1.inc";\nqreg q[2];\ncreg c[2];\nh q[0];\ncx q[0],q[1];\nry(0.209439510239320) q[0];\nh q[0];\nmeasure q[0] -> c[0];\nmeasure q[1] -> c[1];\n',
'OPENQASM 2.0;\ninclude "qelib1.inc";\nqreg q[2];\ncreg c[2];\nh q[0];\ncx q[0],q[1];\nry(0.209439510239320) q[0];\nh q[0];\nh q[1];\nmeasure q[0] -> c[0];\nmeasure q[1] -> c[1];\n',
'OPENQASM 2.0;\ninclude "qelib1.inc";\nqreg q[2];\ncreg c[2];\nh q[0];\ncx q[0],q[1];\nry(0.209439510239320) q[0];\nh q[1];\nmeasure q[0] -> c[0];\nmeasure q[1] -> c[1];\n']
.. code:: ipython3
Q_program.execute(program, device, shots, max_credits=3, wait=10, timeout=240)
.. parsed-literal::
running on backend: local_qasm_simulator
.. parsed-literal::
{'result': 'all done', 'status': 'COMPLETED'}
.. code:: ipython3
CHSH_data_sim = []
k = 0
for j in range(len(xdata)):
temp=[]
for i in range(4):
temp.append(Q_program.average_data("circuit"+str(k),observable_correlated_ideal))
k += 1
CHSH_data_sim.append(CHSH(temp))
.. code:: ipython3
device = 'ibmqx2'
shots = 1024
program_real = []
xdata_real=[]
k = 0
for jj in range(10):
theta = 2.0*np.pi*jj/10
bell_middle = QuantumCircuit(q,c)
bell_middle.ry(theta,q[0])
for i in range(4):
program_real.append("circuit_real"+str(k))
Q_program.add_circuit("circuit_real"+str(k), bell+bell_middle+program_end[i] )
k += 1
xdata_real.append(theta)
.. code:: ipython3
Q_program.execute(program_real, device, shots, max_credits=3, wait=10, timeout=240)
.. parsed-literal::
running on backend: ibmqx2
status = RUNNING (10 seconds)
status = RUNNING (20 seconds)
status = RUNNING (30 seconds)
status = RUNNING (40 seconds)
status = RUNNING (50 seconds)
status = RUNNING (60 seconds)
.. parsed-literal::
{'result': 'all done', 'status': 'COMPLETED'}
.. code:: ipython3
Q_program.get_qasms(program_real[0:8])
.. parsed-literal::
['OPENQASM 2.0;\ninclude "qelib1.inc";\nqreg q[2];\ncreg c[2];\nh q[0];\ncx q[0],q[1];\nry(0.000000000000000) q[0];\nmeasure q[0] -> c[0];\nmeasure q[1] -> c[1];\n',
'OPENQASM 2.0;\ninclude "qelib1.inc";\nqreg q[2];\ncreg c[2];\nh q[0];\ncx q[0],q[1];\nry(0.000000000000000) q[0];\nh q[0];\nmeasure q[0] -> c[0];\nmeasure q[1] -> c[1];\n',
'OPENQASM 2.0;\ninclude "qelib1.inc";\nqreg q[2];\ncreg c[2];\nh q[0];\ncx q[0],q[1];\nry(0.000000000000000) q[0];\nh q[0];\nh q[1];\nmeasure q[0] -> c[0];\nmeasure q[1] -> c[1];\n',
'OPENQASM 2.0;\ninclude "qelib1.inc";\nqreg q[2];\ncreg c[2];\nh q[0];\ncx q[0],q[1];\nry(0.000000000000000) q[0];\nh q[1];\nmeasure q[0] -> c[0];\nmeasure q[1] -> c[1];\n',
'OPENQASM 2.0;\ninclude "qelib1.inc";\nqreg q[2];\ncreg c[2];\nh q[0];\ncx q[0],q[1];\nry(0.628318530717959) q[0];\nmeasure q[0] -> c[0];\nmeasure q[1] -> c[1];\n',
'OPENQASM 2.0;\ninclude "qelib1.inc";\nqreg q[2];\ncreg c[2];\nh q[0];\ncx q[0],q[1];\nry(0.628318530717959) q[0];\nh q[0];\nmeasure q[0] -> c[0];\nmeasure q[1] -> c[1];\n',
'OPENQASM 2.0;\ninclude "qelib1.inc";\nqreg q[2];\ncreg c[2];\nh q[0];\ncx q[0],q[1];\nry(0.628318530717959) q[0];\nh q[0];\nh q[1];\nmeasure q[0] -> c[0];\nmeasure q[1] -> c[1];\n',
'OPENQASM 2.0;\ninclude "qelib1.inc";\nqreg q[2];\ncreg c[2];\nh q[0];\ncx q[0],q[1];\nry(0.628318530717959) q[0];\nh q[1];\nmeasure q[0] -> c[0];\nmeasure q[1] -> c[1];\n']
.. code:: ipython3
CHSH_data_real = []
k = 0
for j in range(len(xdata_real)):
temp=[]
for i in range(4):
temp.append(Q_program.average_data("circuit_real"+str(k),observable_correlated))
k += 1
CHSH_data_real.append(CHSH(temp))
.. code:: ipython3
plt.plot(xdata, CHSH_data_sim, 'r-', xdata_real, CHSH_data_real, 'bo')
plt.plot([0, 2*np.pi], [2, 2], 'b-')
plt.plot([0, 2*np.pi], [-2, -2], 'b-')
plt.grid()
plt.ylabel('CHSH', fontsize=20)
plt.xlabel(r'$Y(\theta)$', fontsize=20)
plt.show()
.. image:: entanglement_revisited_files/entanglement_revisited_22_0.png
The resulting graph created by running the previous cell compares the
simulated data (sinusoidal line) and the data from the real experiment.
The graph also gives lines at :math:`\pm 2` for reference. Did you
violate the hidden variable model?
Here is the saved CHSH data.
.. code:: ipython3
print(CHSH_data_real)
.. parsed-literal::
[1.74609375, 2.462890625, 2.138671875, 1.15234375, -0.349609375, -1.736328125, -2.39453125, -2.123046875, -1.123046875, 0.365234375]
Despite the presence of loopholes in our demonstration, we can see that
this experiment is compatible with quantum mechanics as a theory with no
local hidden variables. See the original experimental demonstrations of
this test with superconducting qubits
`here <https://arstechnica.com/science/2017/05/quantum-volume-one-number-to-benchmark-a-quantum-computer/>`__
and
`here <https://journals.aps.org/pra/abstract/10.1103/PhysRevA.81.062325>`__.
Two-, Three-, and Four-Qubit GHZ States
=======================================
What does entanglement look like beyond two qubits? An important set of
maximally entangled states are known as GHZ states (named after
Greenberger, Horne, and Zeilinger). These are the states of the form
:math:`|\psi\rangle = \left (|0...0\rangle+|1...1\rangle\right)/\sqrt{2}`.
The Bell state previously described is merely a two-qubit version of a
GHZ state. The next cells prepare GHZ states of two, three, and four
qubits.
.. code:: ipython3
# 2 - qubits
shots = 8192
device = 'ibmqx2'
# device = 'simulator' # the device test purpose
# quantum circuit to make GHZ state
q = Q_program.create_quantum_registers("q", 2)
c = Q_program.create_classical_registers("c", 2)
ghz = Q_program.create_circuit("ghz", ["q"], ["c"])
ghz.h(q[0])
ghz.cx(q[0],q[1])
# quantum circuit to measure q in standard basis
measureZZ = Q_program.create_circuit("measureZZ", ["q"], ["c"])
measureZZ.measure(q[0], c[0])
measureZZ.measure(q[1], c[1])
measureXX = Q_program.create_circuit("measureXX", ["q"], ["c"])
measureXX.h(q[0])
measureXX.h(q[1])
measureXX.measure(q[0], c[0])
measureXX.measure(q[1], c[1])
Q_program.add_circuit("ghz_measureZZ", ghz+measureZZ )
Q_program.add_circuit("ghz_measureXX", ghz+measureXX )
circuits = ["ghz_measureZZ", "ghz_measureXX"]
Q_program.get_qasms(circuits)
Q_program.execute(circuits, device, shots, max_credits=5, wait=10, timeout=240)
plot_histogram(Q_program.get_counts("ghz_measureZZ"))
plot_histogram(Q_program.get_counts("ghz_measureXX"))
.. parsed-literal::
>> quantum_registers created: q 2
>> classical_registers created: c 2
running on backend: ibmqx2
status = RUNNING (10 seconds)
status = RUNNING (20 seconds)
status = RUNNING (30 seconds)
status = RUNNING (40 seconds)
status = RUNNING (50 seconds)
status = RUNNING (60 seconds)
status = RUNNING (70 seconds)
status = RUNNING (80 seconds)
status = RUNNING (90 seconds)
.. image:: entanglement_revisited_files/entanglement_revisited_28_1.png
.. image:: entanglement_revisited_files/entanglement_revisited_28_2.png
.. code:: ipython3
# 3 - qubits
shots = 8192
# quantum circuit to make GHZ state
q = Q_program.create_quantum_registers("q", 3)
c = Q_program.create_classical_registers("c", 3)
ghz = Q_program.create_circuit("ghz", ["q"], ["c"])
ghz.h(q[0])
ghz.cx(q[0],q[1])
ghz.cx(q[1],q[2])
# quantum circuit to measure q in standard basis
measureZZZ = Q_program.create_circuit("measureZZZ", ["q"], ["c"])
measureZZZ.measure(q[0], c[0])
measureZZZ.measure(q[1], c[1])
measureZZZ.measure(q[2], c[2])
measureXXX = Q_program.create_circuit("measureXXX", ["q"], ["c"])
measureXXX.h(q[0])
measureXXX.h(q[1])
measureXXX.h(q[2])
measureXXX.measure(q[0], c[0])
measureXXX.measure(q[1], c[1])
measureXXX.measure(q[2], c[2])
Q_program.add_circuit("ghz_measureZZZ", ghz+measureZZZ )
Q_program.add_circuit("ghz_measureXXX", ghz+measureXXX )
circuits = ["ghz_measureZZZ", "ghz_measureXXX"]
Q_program.get_qasms(circuits)
Q_program.execute(circuits, device, shots, max_credits=5, wait=10, timeout=240)
plot_histogram(Q_program.get_counts("ghz_measureZZZ"))
plot_histogram(Q_program.get_counts("ghz_measureXXX"))
.. parsed-literal::
>> quantum_registers created: q 3
>> classical_registers created: c 3
running on backend: ibmqx2
status = RUNNING (10 seconds)
status = RUNNING (20 seconds)
status = RUNNING (30 seconds)
status = RUNNING (40 seconds)
status = RUNNING (50 seconds)
status = RUNNING (60 seconds)
status = RUNNING (70 seconds)
status = RUNNING (80 seconds)
status = RUNNING (90 seconds)
status = RUNNING (100 seconds)
status = RUNNING (110 seconds)
status = RUNNING (120 seconds)
.. image:: entanglement_revisited_files/entanglement_revisited_29_1.png
.. image:: entanglement_revisited_files/entanglement_revisited_29_2.png
.. code:: ipython3
# 4 - qubits
shots = 8192
# quantum circuit to make GHZ state
q = Q_program.create_quantum_registers("q", 4)
c = Q_program.create_classical_registers("c", 4)
ghz = Q_program.create_circuit("ghz", ["q"], ["c"])
ghz.h(q[0])
ghz.cx(q[0],q[1])
ghz.cx(q[1],q[2])
ghz.h(q[3])
ghz.h(q[2])
ghz.cx(q[3],q[2])
ghz.h(q[3])
ghz.h(q[2])
# quantum circuit to measure q in standard basis
measureZZZZ = Q_program.create_circuit("measureZZZZ", ["q"], ["c"])
measureZZZZ.measure(q[0], c[0])
measureZZZZ.measure(q[1], c[1])
measureZZZZ.measure(q[2], c[2])
measureZZZZ.measure(q[3], c[3])
measureXXXX = Q_program.create_circuit("measureXXXX", ["q"], ["c"])
measureXXXX.h(q[0])
measureXXXX.h(q[1])
measureXXXX.h(q[2])
measureXXXX.h(q[3])
measureXXXX.measure(q[0], c[0])
measureXXXX.measure(q[1], c[1])
measureXXXX.measure(q[2], c[2])
measureXXXX.measure(q[3], c[3])
Q_program.add_circuit("ghz_measureZZZZ", ghz+measureZZZZ )
Q_program.add_circuit("ghz_measureXXXX", ghz+measureXXXX )
circuits = ["ghz_measureZZZZ", "ghz_measureXXXX"]
Q_program.get_qasms(circuits)
Q_program.execute(circuits, device, shots, max_credits=5, wait=10, timeout=240)
plot_histogram(Q_program.get_counts("ghz_measureZZZZ"))
plot_histogram(Q_program.get_counts("ghz_measureXXXX"))
.. parsed-literal::
>> quantum_registers created: q 4
>> classical_registers created: c 4
running on backend: ibmqx2
status = RUNNING (10 seconds)
status = RUNNING (20 seconds)
status = RUNNING (30 seconds)
status = RUNNING (40 seconds)
status = RUNNING (50 seconds)
status = RUNNING (60 seconds)
status = RUNNING (70 seconds)
status = RUNNING (80 seconds)
status = RUNNING (90 seconds)
status = RUNNING (100 seconds)
status = RUNNING (110 seconds)
status = RUNNING (120 seconds)
status = RUNNING (130 seconds)
status = RUNNING (140 seconds)
status = RUNNING (150 seconds)
status = RUNNING (160 seconds)
status = RUNNING (170 seconds)
status = RUNNING (180 seconds)
status = RUNNING (190 seconds)
.. image:: entanglement_revisited_files/entanglement_revisited_30_1.png
.. image:: entanglement_revisited_files/entanglement_revisited_30_2.png
Mermin's Test and the Three Box Game
====================================
In case the violation of Bell's inequality (CHSH) by two qubits is not
enough to convince you to believe in quantum mechanics, we can
generalize to a more stringent set of tests with three qubits, which can
give a single-shot violation (rather than taking averaged statistics). A
well-known three-qubit case is Mermin's inequality, which is a test we
can perform on GHZ states.
An example of a three-qubit GHZ state is
:math:`|\psi\rangle = \left (|000\rangle+|111\rangle\right)/\sqrt{2}`.
You can see this is a further generalization of a Bell state and, if
measured, should give :math:`|000\rangle` half the time and
:math:`|111 \rangle` the other half of the time.
.. code:: ipython3
# quantum circuit to make GHZ state
q = Q_program.create_quantum_registers("q", 3)
c = Q_program.create_classical_registers("c", 3)
ghz = Q_program.create_circuit("ghz", ["q"], ["c"])
ghz.h(q[0])
ghz.cx(q[0],q[1])
ghz.cx(q[0],q[2])
# quantum circuit to measure q in standard basis
measureZZZ = Q_program.create_circuit("measureZZZ", ["q"], ["c"])
measureZZZ.measure(q[0], c[0])
measureZZZ.measure(q[1], c[1])
measureZZZ.measure(q[2], c[2])
Q_program.add_circuit("ghz_measureZZZ", ghz+measureZZZ )
circuits = ["ghz_measureZZZ"]
Q_program.get_qasms(circuits)
Q_program.execute(circuits, device, shots, max_credits=5, wait=10, timeout=240)
plot_histogram(Q_program.get_counts("ghz_measureZZZ"))
.. parsed-literal::
>> quantum_registers created: q 3
>> classical_registers created: c 3
running on backend: ibmqx2
status = RUNNING (10 seconds)
status = RUNNING (20 seconds)
status = RUNNING (30 seconds)
status = RUNNING (40 seconds)
status = RUNNING (50 seconds)
status = RUNNING (60 seconds)
status = RUNNING (70 seconds)
status = RUNNING (80 seconds)
status = RUNNING (90 seconds)
status = RUNNING (100 seconds)
status = RUNNING (110 seconds)
.. image:: entanglement_revisited_files/entanglement_revisited_33_1.png
Suppose we have three independent systems, :math:`\{A, B, C\}`, for
which we can query two particular questions (observables) :math:`X` and
:math:`Y`. In each case, either query can give :math:`+1` or :math:`-1`.
Consider whether it is possible to choose some state of the three boxes
such that we can satisfy the following four conditions:
:math:`X_A Y_B Y_C = 1`, :math:`Y_A X_B Y_C =1`,
:math:`Y_A Y_B X_C = 1`, and :math:`X_A X_B X_C = -1`. Classically, this
can be shown to be impossible... but a three-qubit GHZ state can in fact
satisfy all four conditions.
.. code:: ipython3
MerminM = lambda x : x[0]*x[1]*x[2]*x[3]
.. code:: ipython3
observable ={'00000': 1, '00001': -1, '00010': -1, '00011': 1, '00100': -1, '00101': 1, '00110': 1, '00111': -1}
.. code:: ipython3
# quantum circuit to measure q XXX
measureXXX = Q_program.create_circuit("measureXXX", ["q"], ["c"])
measureXXX.h(q[0])
measureXXX.h(q[1])
measureXXX.h(q[2])
measureXXX.measure(q[0], c[0])
measureXXX.measure(q[1], c[1])
measureXXX.measure(q[2], c[2])
# quantum circuit to measure q XYY
measureXYY = Q_program.create_circuit("measureXYY", ["q"], ["c"])
measureXYY.s(q[1]).inverse()
measureXYY.s(q[2]).inverse()
measureXYY.h(q[0])
measureXYY.h(q[1])
measureXYY.h(q[2])
measureXYY.measure(q[0], c[0])
measureXYY.measure(q[1], c[1])
measureXYY.measure(q[2], c[2])
# quantum circuit to measure q YXY
measureYXY = Q_program.create_circuit("measureYXY", ["q"], ["c"])
measureYXY.s(q[0]).inverse()
measureYXY.s(q[2]).inverse()
measureYXY.h(q[0])
measureYXY.h(q[1])
measureYXY.h(q[2])
measureYXY.measure(q[0], c[0])
measureYXY.measure(q[1], c[1])
measureYXY.measure(q[2], c[2])
# quantum circuit to measure q YYX
measureYYX = Q_program.create_circuit("measureYYX", ["q"], ["c"])
measureYYX.s(q[0]).inverse()
measureYYX.s(q[1]).inverse()
measureYYX.h(q[0])
measureYYX.h(q[1])
measureYYX.h(q[2])
measureYYX.measure(q[0], c[0])
measureYYX.measure(q[1], c[1])
measureYYX.measure(q[2], c[2])
Q_program.add_circuit("ghz_measureXXX", ghz+measureXXX )
Q_program.add_circuit("ghz_measureYYX", ghz+measureYYX )
Q_program.add_circuit("ghz_measureYXY", ghz+measureYXY )
Q_program.add_circuit("ghz_measureXYY", ghz+measureXYY )
circuits = ["ghz_measureXXX", "ghz_measureYYX", "ghz_measureYXY", "ghz_measureXYY"]
Q_program.get_qasms(circuits)
Q_program.execute(circuits, device, shots, max_credits=5, wait=10, timeout=240)
.. parsed-literal::
running on backend: ibmqx2
status = RUNNING (10 seconds)
status = RUNNING (20 seconds)
status = RUNNING (30 seconds)
status = RUNNING (40 seconds)
status = RUNNING (50 seconds)
status = RUNNING (60 seconds)
status = RUNNING (70 seconds)
status = RUNNING (80 seconds)
status = RUNNING (90 seconds)
status = RUNNING (100 seconds)
status = RUNNING (110 seconds)
status = RUNNING (120 seconds)
status = RUNNING (130 seconds)
status = RUNNING (140 seconds)
.. parsed-literal::
{'result': 'all done', 'status': 'COMPLETED'}
.. code:: ipython3
temp=[]
temp.append(Q_program.average_data("ghz_measureXXX",observable))
temp.append(Q_program.average_data("ghz_measureYYX",observable))
temp.append(Q_program.average_data("ghz_measureYXY",observable))
temp.append(Q_program.average_data("ghz_measureXYY",observable))
print(MerminM(temp))
.. parsed-literal::
-0.1933898929106448
The above shows that the average statistics are not consistent with a
local hidden variable theory. To demonstrate with single shots, we can
run 50 single experiments, with each experiment chosen randomly, and the
outcomes saved. If there was a local hidden variable theory, all the
outcomes would be :math:`+1`.

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.. qiskit-sdk-py-dev documentation master file, created by
sphinx-quickstart on Fri Jun 30 00:56:35 2017.
You can adapt this file completely to your liking, but it should at least
contain the root `toctree` directive.
Welcome to qiskit-sdk-py-dev's documentation!
=============================================
Contents:
.. toctree::
:maxdepth: 2
intro
tutorial4developer
superposition_and_entanglement
entanglement_revisited
tomography
classical_optimization
Quantum Chemistry <quantum_chemistry>
Indices and tables
==================
* :ref:`genindex`
* :ref:`modindex`
* :ref:`search`

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#########################################
QISKit (Quantum Information Software Kit)
#########################################
The latest version of this notebook is available on
https://github.com/IBM/qiskit-sdk-py/tree/master/scripts.
For more information about how to use the IBM Q experience (QX), consult
the
`tutorials <https://quantumexperience.ng.bluemix.net/qstage/#/tutorial?sectionId=c59b3710b928891a1420190148a72cce&pageIndex=0>`__,
or check out the
`community <https://quantumexperience.ng.bluemix.net/qstage/#/community>`__.
Contributors (alphabetical)
===========================
Jerry Chow, Antonio Córcoles, Abigail Cross, Andrew Cross, Ismael Faro,
Andreas Fuhrer, Jay Gambetta
In future releases, anyone who contributes to the tutorial can include
their name here.
Introduction
============
Welcome to QISKit! Interested in quantum computing and programming a
real live quantum processor? QISKit is a simple set of tools for anyone
interested in exploring quantum information science. We have put
together a set of Jupyter notebooks to demonstrate how to use our tools
and explore the quantum world.
The notebooks are organized into the following topics:
:ref:`introducing-the-tools`
:ref:`exploring-quantum-information-concepts`
:ref:`verification_tools_for_quantum_science`
:ref:`applications_of_short_depth_quantum_circuits_on_quantum_computers`
:ref:`quantum-games`
.. _introducing-the-tools:
Introducing the tools
=====================
In this first topic, we break down the tools in this QISKit SDK, and
introduce all the different parts to make this useful. Our list of
introductory notebooks:
- :doc:`Getting Started with QISKit SDK <tutorial4developer>` shows how to use the QISKit SDK tools.
- Compiling and Running a Quantum Progam [coming soon]
- Loading and Saving a Quantum Program [coming soon]
- Visualizing a Quantum State [coming soon]
.. _exploring-quantum-information-concepts:
Exploring quantum information concepts
======================================
The next set of notebooks shows how you can explore some simple concepts
of quantum information science.
- :doc:`Superposition and Entanglement <superposition_and_entanglement>`
illustrates how to make simple quantum states on one and two qubits,
and demonstrates concepts such as quantum superpositions and
entanglement.
- Single-qubit states: amplitude and phase [coming soon] illustrates
more complicated single-qubit states.
- :doc:`Entanglement revisited <entanglement_revisited>` -
illustrates the CHSH inequality, and extensions for three qubits
(Mermin).
- Quantum teleportation [coming soon]
- Quantum superdense coding [coming soon]
.. _verification_tools_for_quantum_science:
Verification tools for quantum information science
==================================================
The third set of notebooks allows you to explore tools for verification
of quantum systems with commonly-used techniques such as tomography and
randomized benchmarking.
- Quantum coherence studies [coming soon]
- Quantum state tomography [coming soon]
- Quantum process tomography [coming soon]
- Pauli randomized benchmarking [coming soon]
- Standard randomized benchmarking [coming soon]
- Purity randomized benchmarking [coming soon]
- Leakage randomized benchmarking [coming soon]
- Simultaneous randomized benchmarking [coming soon]
- Gate set tomography [coming soon]
.. _applications_of_short_depth_quantum_circuits_on_quantum_computers:
Applications of quantum information science
===========================================
To fully grasp the possibilities, this set of notebooks shows how you
can explore applications of short-depth quantum computers.
Sampling
--------
- :doc:`Quantum optimization <classical_optimization>` -
illustrates how to use a quantum computer to look at optimization
problems.
- :doc:`Quantum chemistry by variational quantum eigensolver method <quantum_chemistry>` [coming
soon] - illustrates how to use a quantum computer to look at
chemistry problems.
Non-Sampling
------------
- Iterative phase estimation [coming soon]
.. _quantum-games:
Quantum Games
=============
[coming soon]

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################################
Superposition and Entanglemen
################################
For more information about how to use the IBM Q experience (QX), consult
the
`tutorials <https://quantumexperience.ng.bluemix.net/qstage/#/tutorial?sectionId=c59b3710b928891a1420190148a72cce&pageIndex=0>`__,
or check out the
`community <https://quantumexperience.ng.bluemix.net/qstage/#/community>`__.
Contributors
============
Jay Gambetta, Antonio Córcoles, Andrew Cross
Introduction
============
Many people tend to think quantum physics is hard math, but this is not
actually true. Quantum concepts are very similar to those seen in the
linear algebra classes you may have taken as a freshman in college, or
even in high school. The challenge of quantum physics is the necessity
to accept counter-intuitive ideas, and its lack of a simple underlying
theory. We believe that if you can grasp the following two Principles,
you have will have a good start:
1. :ref:`A physical system in a perfectly definite state can still behave randomly. <quantum_states-basis_states_and_superpositions>`
2. :ref:`Two systems that are too far apart to influence each other can nevertheless behave in ways that, though individually random, are somehow strongly correlated. <entanglement>`
Getting Started
===============
Please see :doc:`Getting Started with QISKit <tutorial4developer>` if
you would like to understand how to get started with the QISKit SDK. For
this script, simply work your way though the tutorial to learn about
superposition and entanglement.
.. code:: python
# Checking the version of PYTHON; we only support 3 at the moment
import sys
if sys.version_info < (3,0):
raise Exception("Please use Python version 3 or greater.")
# useful additional packages
import matplotlib.pyplot as plt
%matplotlib inline
import numpy as np
sys.path.append("../../")
# importing the QISKit
from qiskit import QuantumProgram
import Qconfig
# import basic plot tools
from qiskit.basicplotter import plot_histogram
.. _quantum_states-basis_states_and_superpositions:
Quantum States - Basis States and Superpositions
================================================
The first Principle above tells us that the results of measuring a
quantum state may be random or deterministic, depending on what basis is
used. To demonstrate, we will first introduce the computational (or
standard) basis for a qubit.
The computational basis is the set containing the ground and excited
state :math:`\{|0\rangle,|1\rangle\}`, which also corresponds to the
following vectors:
.. math:: |0\rangle =\begin{pmatrix} 1 \\ 0 \end{pmatrix}
.. math:: |1\rangle =\begin{pmatrix} 0 \\ 1 \end{pmatrix}
In Python these are represented by
.. code:: python
zero = np.array([[1],[0]])
one = np.array([[0],[1]])
In our quantum processor system (and many other physical quantum
processors) it is natural for all qubits to start in the
:math:`|0\rangle` state, known as the ground state. To make the
:math:`|1\rangle` (or excited) state, we use the operator
.. math:: X =\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}.
This :math:`X` operator is often called a bit-flip because it exactly
implements the following:
.. math:: X: |0\rangle \rightarrow |1\rangle
.. math:: X: |1\rangle \rightarrow |0\rangle.
In Python this can be represented by the following:
.. code:: python
X = np.array([[0,1],[1,0]])
print(np.dot(X,zero))
print(np.dot(X,one))
.. parsed-literal::
[[0]
[1]]
[[1]
[0]]
Next, we give the two quantum circuits for preparing a single qubit in
the ground and excited states using the IBM Q experience. The first part
uses QISKit to make the two circuits.
.. code:: python
device = 'ibmqx2' # the device to run on
shots = 1024 # the number of shots in the experiment
Q_program = QuantumProgram()
Q_program.set_api(Qconfig.APItoken, Qconfig.config["url"]) # set the APIToken and API url
# Creating registers
qr = Q_program.create_quantum_registers("qr", 1)
cr = Q_program.create_classical_registers("cr", 1)
# Quantum circuit ground
qc_ground = Q_program.create_circuit("ground", ["qr"], ["cr"])
qc_ground.measure(qr[0], cr[0])
# Quantum circuit excited
qc_excited = Q_program.create_circuit("excited", ["qr"], ["cr"])
qc_excited.x(qr)
qc_excited.measure(qr[0], cr[0])
circuits = ['ground', 'excited']
Q_program.get_qasms(circuits)
.. parsed-literal::
>> quantum_registers created: qr 1
>> classical_registers created: cr 1
.. parsed-literal::
['OPENQASM 2.0;\ninclude "qelib1.inc";\nqreg qr[1];\ncreg cr[1];\nmeasure qr[0] -> cr[0];\n',
'OPENQASM 2.0;\ninclude "qelib1.inc";\nqreg qr[1];\ncreg cr[1];\nx qr[0];\nmeasure qr[0] -> cr[0];\n']
Here we have created two jobs with different quantum circuits; the first
to prepare the ground state, and the second to prepare the excited
state. Now we can run the batched jobs on the QX.
.. code:: python
Q_program.execute(circuits, device, shots, max_credits=3, wait=10, timeout=240)
.. parsed-literal::
running on backend: ibmqx2
status = RUNNING (10 seconds)
status = RUNNING (20 seconds)
.. parsed-literal::
{'result': 'all done', 'status': 'COMPLETED'}
After the run has been completed, the data can be extracted from the API
output and plotted.
.. code:: python
plot_histogram(Q_program.get_counts("ground"))
.. image:: superposition_and_entanglement_files/superposition_and_entanglement_14_0.png
.. code:: python
plot_histogram(Q_program.get_counts("excited"))
.. image:: superposition_and_entanglement_files/superposition_and_entanglement_15_0.png
Here we see that with high probability the qubit is in the
:math:`|0\rangle` state for the first circuit and in the
:math:`|1\rangle` state for the second circuit. The difference from an
ideal perfect answer in both cases is due to a combination of
measurement error, preparation error, and gate error (for the
:math:`|1\rangle` state).
Up to this point, nothing is different from a classical system of a bit.
To go beyond, we must explore what it means to make a superposition. The
operation in the quantum circuit language for generating a superposition
is the Hadamard gate, :math:`H`. Let's assume for now that this gate is
like flipping a fair coin. The result of a flip has two possible
outcomes, heads or tails, each occurring with equal probability. If we
repeat this simple thought experiment many times, we would expect that
on average we will measure as many heads as we do tails. Let heads be
:math:`|0\rangle` and tails be :math:`|1\rangle`.
Let's run the quantum version of this experiment. First we prepare the
qubit in the ground state :math:`|0\rangle`. We then apply the Hadamard
gate (coin flip). Finally, we measure the state of the qubit. Repeat the
experiment 1024 times (shots). As you likely predicted, half the
outcomes will be in the :math:`|0\rangle` state and half will be in the
:math:`|1\rangle` state.
Try the program below.
.. code:: python
# Quantum circuit superposition
qc_superposition = Q_program.create_circuit("superposition", ["qr"], ["cr"])
qc_superposition.h(qr)
qc_superposition.measure(qr[0], cr[0])
circuits = ["superposition"]
Q_program.execute(circuits, device, shots, max_credits=3, wait=10, timeout=240)
plot_histogram(Q_program.get_counts("superposition"))
.. parsed-literal::
running on backend: ibmqx2
status = RUNNING (10 seconds)
status = RUNNING (20 seconds)
.. image:: superposition_and_entanglement_files/superposition_and_entanglement_17_1.png
Indeed, much like a coin flip, the results are close to 50/50 with some
non-ideality due to errors (again due to state preparation, measurement,
and gate errors). So far, this is still not unexpected. Let's run the
experiment again, but this time with two :math:`H` gates in succession.
If we consider the :math:`H` gate to be analog to a coin flip, here we
would be flipping it twice, and still expecting a 50/50 distribution.
.. code:: python
# Quantum circuit two Hadamards
qc_twohadamard = Q_program.create_circuit("twohadamard", ["qr"], ["cr"])
qc_twohadamard.h(qr)
qc_twohadamard.barrier()
qc_twohadamard.h(qr)
qc_twohadamard.measure(qr[0], cr[0])
circuits = ["twohadamard"]
Q_program.execute(circuits, device, shots, max_credits=3, wait=10, timeout=240)
plot_histogram(Q_program.get_counts("twohadamard"))
.. parsed-literal::
running on backend: ibmqx2
status = RUNNING (10 seconds)
status = RUNNING (20 seconds)
.. image:: superposition_and_entanglement_files/superposition_and_entanglement_19_1.png
This time, the results are surprising. Unlike the classical case, with
high probability the outcome is not random, but in the :math:`|0\rangle`
state. *Quantum randomness* is not simply like a classical random coin
flip. In both of the above experiments, the system (without noise) is in
a definite state, but only in the first case does it behave randomly.
This is because, in the first case, via the :math:`H` gate, we make a
uniform superposition of the ground and excited state,
:math:`(|0\rangle+|1\rangle)/\sqrt{2}`, but then follow it with a
measurement in the computational basis. The act of measurement in the
computational basis forces the system to be in either the
:math:`|0\rangle` state or the :math:`|1\rangle` state with an equal
probability (due to the uniformity of the superposition). In the second
case, we can think of the second :math:`H` gate as being a part of the
final measurement operation; it changes the measurement basis from the
computational basis to a *superposition* basis. The following equations
illustrate the action of the :math:`H` gate on the computational basis
states:
.. math:: H: |0\rangle \rightarrow |+\rangle=\frac{|0\rangle+|1\rangle}{\sqrt{2}}
.. math:: H: |1\rangle \rightarrow |-\rangle=\frac{|0\rangle-|1\rangle}{\sqrt{2}}.
We can redefine this new transformed basis, the superposition basis, as
the set
{:math:`|+\rangle`, :math:`|-\rangle`}.
We now have a different
way of looking at the second experiment above. The first :math:`H` gate
prepares the system into a superposition state, namely the
:math:`|+\rangle` state. The second :math:`H` gate followed by the
standard measurement changes it into a measurement in the superposition
basis. If the measurement gives 0, we can conclude that the system was
in the :math:`|+\rangle` state before the second :math:`H` gate, and if
we obtain 1, it means the system was in the :math:`|-\rangle` state. In
the above experiment we see that the outcome is mainly 0, suggesting
that our system was in the :math:`|+\rangle` superposition state before
the second :math:`H` gate.
The math is best understood if we represent the quantum superposition
state :math:`|+\rangle` and :math:`|-\rangle` by:
.. math:: |+\rangle =\frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ 1 \end{pmatrix}
.. math:: |-\rangle =\frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ -1 \end{pmatrix}
A standard measurement, known in quantum mechanics as a projective or
von Neumann measurement, takes any superposition state of the qubit and
projects it to either the state :math:`|0\rangle` or the state
:math:`|1\rangle` with a probability determined by:
.. math:: P(i|\psi) = |\langle i|\psi\rangle|^2
where :math:`P(i|\psi)` is the probability of measuring the system in
state :math:`i` given preparation :math:`\psi`.
We have written the Python function StateOverlap to return this:
.. code:: python
state_overlap = lambda state1, state2: np.absolute(np.dot(state1.conj().T,state2))**2
Now that we have a simple way of going from a state to the probability
distribution of a standard measurement, we can go back to the case of a
superposition made from the Hadamard gate. The Hadamard gate is defined
by the matrix:
.. math:: H =\frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}
The :math:`H` gate acting on the state :math:`|0\rangle` gives:
.. code:: python
Hadamard = np.array([[1,1],[1,-1]],dtype=complex)/np.sqrt(2)
psi1 = np.dot(Hadamard,zero)
P0 = state_overlap(zero,psi1)
P1 = state_overlap(one,psi1)
plot_histogram({'0' : P0, '1' : P1})
.. image:: superposition_and_entanglement_files/superposition_and_entanglement_23_0.png
which is the ideal version of the first experiment.
The second experiment involves applying the Hadamard gate twice. While
matrix multiplication shows that the product of two Hadamards is the
identity operator (meaning that the state :math:`|0\rangle` remains
unchanged), here (as previously mentioned) we prefer to interpret this
as doing a measurement in the superposition basis. Using the above
definitions, you can show that :math:`H` transforms the computational
basis to the superposition basis.
.. code:: python
print(np.dot(Hadamard,zero))
print(np.dot(Hadamard,one))
.. parsed-literal::
[[ 0.70710678+0.j]
[ 0.70710678+0.j]]
[[ 0.70710678+0.j]
[-0.70710678+0.j]]
.. _entanglement:
Entanglement
============
The core idea behind the second Principle is *entanglement*. Upon
reading the Principle, one might be inclined to think that entanglement
is simply strong correlation between two entitities -- but entanglement
goes well beyond mere perfect (classical) correlation. If you and I read
the same paper, we will have learned the same information. If a third
person comes along and reads the same paper they also will have learned
this information. All three persons in this case are perfectly
correlated, and they will remain correlated even if they are separated
from each other.
The situation with quantum entanglement is a bit more subtle. In the
quantum world, you and I could read the same quantum paper, and yet we
will not learn what information is actually contained in the paper until
we get together and share our information. However, when we are
together, we find that we can unlock more information from the paper
than we initially thought possible. Thus, quantum entanglement goes much
further than perfect correlation.
To demonstrate this, we will define the controlled-NOT (CNOT) gate and
the composition of two systems. The convention we use in the Quantum
Experience is to label states by writing the first qubit's name in the
rightmost position, thereby allowing us to easily convert from binary to
decimal. As a result, we define the tensor product between operators
:math:`q_0` and :math:`q_1` by :math:`q_1\otimes q_0`.
Taking :math:`q_0` as the control and :math:`q_1` as the target, the
CNOT with this representation is given by
.. math:: CNOT =\begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 1\\0& 0& 1 & 0\\0 & 1 & 0 & 0 \end{pmatrix},
which is non-standard in the quantum community, but more easily connects
to classical computing, where the least significant bit (LSB) is
typically on the right. An entangled state of the two qubits can be made
via an :math:`H` gate on the control qubit, followed by the CNOT gate.
This generates a particular maximally entangled two-qubit state known as
a Bell state, named after John Stewart Bell (`learn more about Bell and
his contributions to quantum physics and
entanglement <https://en.wikipedia.org/wiki/John_Stewart_Bell>`__.
To explore this, we can prepare an entangled state of two qubits, and
then ask questions about the qubit states. The questions we can ask are:
\* What is the state of the first qubit in the standard basis? \* What
is the state of the first qubit in the superposition basis? \* What is
the state of the second qubit in the standard basis? \* What is the
state of the second qubit in the superposition basis? \* What is the
state of both qubits in the standard basis? \* what is the state of both
qubits in the superposition basis?
Below is a program with six such circuits for these six questions.
.. code:: python
# Creating registers
q2 = Q_program.create_quantum_registers("q2", 2)
c2 = Q_program.create_classical_registers("c2", 2)
# quantum circuit to make an entangled bell state
bell = Q_program.create_circuit("bell", ["q2"], ["c2"])
bell.h(q2[0])
bell.cx(q2[0], q2[1])
# quantum circuit to measure q0 in the standard basis
measureIZ = Q_program.create_circuit("measureIZ", ["q2"], ["c2"])
measureIZ.measure(q2[0], c2[0])
# quantum circuit to measure q0 in the superposition basis
measureIX = Q_program.create_circuit("measureIX", ["q2"], ["c2"])
measureIX.h(q2[0])
measureIX.measure(q2[0], c2[0])
# quantum circuit to measure q1 in the standard basis
measureZI = Q_program.create_circuit("measureZI", ["q2"], ["c2"])
measureZI.measure(q2[1], c2[1])
# quantum circuit to measure q1 in the superposition basis
measureXI = Q_program.create_circuit("measureXI", ["q2"], ["c2"])
measureXI.h(q2[1])
measureXI.measure(q2[1], c2[1])
# quantum circuit to measure q in the standard basis
measureZZ = Q_program.create_circuit("measureZZ", ["q2"], ["c2"])
measureZZ.measure(q2[0], c2[0])
measureZZ.measure(q2[1], c2[1])
# quantum circuit to measure q in the superposition basis
measureXX = Q_program.create_circuit("measureXX", ["q2"], ["c2"])
measureXX.h(q2[0])
measureXX.h(q2[1])
measureXX.measure(q2[0], c2[0])
measureXX.measure(q2[1], c2[1])
.. parsed-literal::
>> quantum_registers created: q2 2
>> classical_registers created: c2 2
.. parsed-literal::
<qiskit._measure.Measure at 0x11590f3c8>
.. code:: python
Q_program.add_circuit("bell_measureIZ", bell+measureIZ )
Q_program.add_circuit("bell_measureIX", bell+measureIX )
Q_program.add_circuit("bell_measureZI", bell+measureZI )
Q_program.add_circuit("bell_measureXI", bell+measureXI )
Q_program.add_circuit("bell_measureZZ", bell+measureZZ )
Q_program.add_circuit("bell_measureXX", bell+measureXX )
circuits = ["bell_measureIZ", "bell_measureIX", "bell_measureZI", "bell_measureXI", "bell_measureZZ", "bell_measureXX"]
Q_program.get_qasms(circuits)
.. parsed-literal::
['OPENQASM 2.0;\ninclude "qelib1.inc";\nqreg q2[2];\ncreg c2[2];\nh q2[0];\ncx q2[0],q2[1];\nmeasure q2[0] -> c2[0];\n',
'OPENQASM 2.0;\ninclude "qelib1.inc";\nqreg q2[2];\ncreg c2[2];\nh q2[0];\ncx q2[0],q2[1];\nh q2[0];\nmeasure q2[0] -> c2[0];\n',
'OPENQASM 2.0;\ninclude "qelib1.inc";\nqreg q2[2];\ncreg c2[2];\nh q2[0];\ncx q2[0],q2[1];\nmeasure q2[1] -> c2[1];\n',
'OPENQASM 2.0;\ninclude "qelib1.inc";\nqreg q2[2];\ncreg c2[2];\nh q2[0];\ncx q2[0],q2[1];\nh q2[1];\nmeasure q2[1] -> c2[1];\n',
'OPENQASM 2.0;\ninclude "qelib1.inc";\nqreg q2[2];\ncreg c2[2];\nh q2[0];\ncx q2[0],q2[1];\nmeasure q2[0] -> c2[0];\nmeasure q2[1] -> c2[1];\n',
'OPENQASM 2.0;\ninclude "qelib1.inc";\nqreg q2[2];\ncreg c2[2];\nh q2[0];\ncx q2[0],q2[1];\nh q2[0];\nh q2[1];\nmeasure q2[0] -> c2[0];\nmeasure q2[1] -> c2[1];\n']
Let's begin by running just the first two questions, looking at the
results of the first qubit (:math:`q_0`) using a computational and then
a superposition measurement.
.. code:: python
Q_program.execute(circuits[0:2], device, shots, max_credits=3, wait=10, timeout=240)
plot_histogram(Q_program.get_counts("bell_measureIZ"))
.. parsed-literal::
running on backend: ibmqx2
status = RUNNING (10 seconds)
status = RUNNING (20 seconds)
.. image:: superposition_and_entanglement_files/superposition_and_entanglement_31_1.png
We find that the result is random. Half the time :math:`q_0` is in
:math:`|0\rangle`, and the other half it is in the :math:`|1\rangle`
state. You may wonder whether this is like the superposition from
earlier in the tutorial. Maybe the qubit has a perfectly definite state,
and we are simply measuring in another basis. What would you expect if
you did the experiment and measured in the superposition basis? Recall
we do this by adding an :math:`H` gate before the measurement...which is
exactly what we have checked with the second question.
.. code:: python
plot_histogram(Q_program.get_counts("bell_measureIX"))
.. image:: superposition_and_entanglement_files/superposition_and_entanglement_33_0.png
In this case, we see that the result is still random, regardless of
whether we measure in the computational or the superposition basis. This
tells us that we actually know nothing about the first qubit. What about
the second qubit, :math:`q_1`? The next lines will run experiments
measuring the second qubit in both the computational and superposition
bases.
.. code:: python
Q_program.execute(circuits[2:4], device, shots, max_credits=3, wait=10, timeout=240)
plot_histogram(Q_program.get_counts("bell_measureZI"))
plot_histogram(Q_program.get_counts("bell_measureXI"))
.. parsed-literal::
running on backend: ibmqx2
status = RUNNING (10 seconds)
status = RUNNING (20 seconds)
.. image:: superposition_and_entanglement_files/superposition_and_entanglement_35_1.png
.. image:: superposition_and_entanglement_files/superposition_and_entanglement_35_2.png
Once again, all the experiments give random outcomes. It seems we know
nothing about either qubit in our system! In our previous analogy, this
is equivalent to two readers separately reading a quantum paper and
extracting no information whatsoever from it on their own.
What do you expect, however, when the readers get together? Below we
will measure both in the joint computational basis.
.. code:: python
Q_program.execute(circuits[4:6], device, shots, max_credits=3, wait=10, timeout=240)
.. parsed-literal::
running on backend: ibmqx2
status = RUNNING (10 seconds)
status = RUNNING (20 seconds)
.. parsed-literal::
{'result': 'all done', 'status': 'COMPLETED'}
.. code:: python
plot_histogram(Q_program.get_counts("bell_measureZZ"))
.. image:: superposition_and_entanglement_files/superposition_and_entanglement_38_0.png
Here we see that with high probability, if :math:`q_0` is in state 0,
:math:`q_1` will be in 0 as well; the same goes if :math:`q_0` is in
state 1. They are perfectly correlated.
What about if we measure both in the superposition basis?
.. code:: python
plot_histogram(Q_program.get_counts("bell_measureXX"))
.. image:: superposition_and_entanglement_files/superposition_and_entanglement_40_0.png
Here we see that the system **also** has perfect correlations
(accounting for experimental noise). Therefore, if :math:`q_0` is
measured in state :math:`|0\rangle`, we know :math:`q_1` is in this
state as well; likewise, if :math:`q_0` is measured in state
:math:`|+\rangle`, we know :math:`q_1` is also in this state. These
correlations have led to much confusion in science, because any attempt
to relate the unusual behavior of quantum entanglement to our everyday
experiences is a fruitless endeavor.
This is just a taste of what happens in the quantum world. Please
continue to :doc:`Entanglement revisited <entanglement_revisited>` to
explore further!

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============================
Quantum State Tomography
============================
Contributors
============
Christopher J. Wood `(cjwood@us.ibm.com) <mailto:cjwood@us.ibm.com>`__
Introduction
============
In this notebook we demonstrate how to design and run experiments to
perform quantum state tomography using QISKit, and demonstrate this
using both simulators, and the IBM Quantum Experience. After going
through this notebook you may also look at additional examples of
tomgoraphy of GHZ and 5-qubit Cat-states in the
:doc:`cat-state-tomography <cat-state-tomography>` workbook.
We implement quantum state tomography using simple maximum likelihood
constrained least-squares fitting of a tomographically complete set of
measurement data. For details of this method see `J Smolin, JM Gambetta,
G Smith, Phys. Rev. Lett. 108,
070502 <https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.108.070502>`__
`(open access: arXiv: 1106.5458) <https://arxiv.org/abs/1106.5458>`__.
Getting Started
===============
Please see :doc:`Getting Started with QISKit <tutorial4developer>` if
you would like to understand how to get started with the QISKit SDK.
.. code:: python
# Checking the version of PYTHON; we only support 3 at the moment
import sys
if sys.version_info < (3,0):
raise Exception("Please use Python version 3 or greater.")
# useful additional packages
import matplotlib.pyplot as plt
%matplotlib inline
import numpy as np
import sys
sys.path.append("../../")
# importing the QISKit
from qiskit import QuantumProgram
import Qconfig
# import state tomography functions
from statetomo import *
# function using QuTip to visualize density matrices
import qutip as qt
def plot_state(rho, fidelity=None):
# plot real part of rho
fig1, ax = qt.matrix_histogram(rho.real, limits=[-1,1])
ax.view_init(azim=-55, elev=45)
# Set fidelity as plot label
if fidelity != None:
# Fidelity to 4 decimal places
ax.set_title("F = %.4f" %fidelity)
# Plot imaginary part
fig2, ax = qt.matrix_histogram(rho.imag, limits=[-1,1])
ax.view_init(azim=-55, elev=45)
# display plot
plt.show()
Designing tomography experiments in QISKit
==========================================
We now demonstrate how to use the QISKit developer toolkit to design
tomography experiments. As an example we will prepare and measure a
2-qubit entangled Bell-state
:math:`\frac{1}{\sqrt2}\left(|00\rangle+|11\rangle\right)`.
Initializing a quantum program
------------------------------
We state by defining a ``QuantumProgram`` and adding a quantum and
classical register. The quantum register is the state of our quantum
system, and the classical register records outcomes of measurements of
the quantum state.
.. code:: python
# Initialize a new quantum program
QP = QuantumProgram()
# Add a 2-qubit quantum register "qr"
qr = QP.create_quantum_registers("qr", 2)
# Add a 2-bit register "cr" to record results
cr = QP.create_classical_registers("cr", 2)
.. parsed-literal::
>> quantum_registers created: qr 2
>> classical_registers created: cr 2
Setting up initial state preparation
------------------------------------
Next we define a *state preparation circuit* to prepare our system in a
desired quantum state. For our example we will apply a Hadamard gate to
prepare the first qubit in a :math:`|+\rangle` state, followed by a CNOT
gate to entangle the two qubits:
.. code:: python
# Create a circuit named 'prep'
prep = QP.create_circuit("prep", ["qr"], ["cr"])
# Add H gate of first qubit to 'prep' circuit
prep.h(qr[0])
# Add CNOT gate between qubits to 'prep' circuit
prep.cx(qr[0], qr[1])
.. parsed-literal::
<qiskit.extensions.standard.cx.CnotGate at 0x10339e0f0>
Building tomography measurement circuits
----------------------------------------
Next we must construct a family of circuits which implement a
tomographically complete set of measurements of the qubit. The
``statetomo.py`` example library contains functions to generate these
measurement circuits for general n-qubit systems. We do this by
measuring in the X, Y, and Z for each qubit. This results in :math:`3^n`
measurement circuits which must be executed to gather count statistics
for the tomographic reconstruction.
The function to build the circuits is:
::
build_tomo_circuits(QP, 'prep', 'qr', 'cr', qubits)
The function to construct an array of the the corresponding labels for
these circuits is:
::
build_tomo_keys('prep', qubits)
| where - ``QP`` is the quantum program. - ``'prep'`` is the name of the
preparation circuit
| - ``'qr'`` is the name of the quantum registers to be measured -
``'cr'`` is the name of the classical registers to store outcomes -
``qubits`` is a list of the qubits to be measured. Eg ``[i,j]`` for
``qr[i]``, and ``qr[j]``.
.. code:: python
# Qubits being measured
meas_qubits = [0,1]
# Construct the state tomography measurement circuits in QP
build_tomo_circuits(QP, "prep", "qr", "cr", meas_qubits)
# construct list of tomo circuit labels
circuits = build_tomo_keys("prep", meas_qubits)
print(circuits)
::
---------------------------------------------------------------------------
AttributeError Traceback (most recent call last)
<ipython-input-4-744cf6cbfc8d> in <module>()
2 meas_qubits = [0,1]
3 # Construct the state tomography measurement circuits in QP
----> 4 build_tomo_circuits(QP, "prep", "qr", "cr", meas_qubits)
5 # construct list of tomo circuit labels
6 circuits = build_tomo_keys("prep", meas_qubits)
/Users/cjwood/Documents/IBM-Git/qiskit-sdk-py-dev/tutorial/sections/statetomo.py in build_tomo_circuits(Q_program, circuit, qreg, creg, qubit_list)
166 circ = [circuit]
167 for j in sorted(qubit_list, reverse=True):
--> 168 build_tomo_circuit_helper(Q_program, circ, qreg, creg, j)
169 circ = build_keys_helper(circ, j)
170
/Users/cjwood/Documents/IBM-Git/qiskit-sdk-py-dev/tutorial/sections/statetomo.py in build_tomo_circuit_helper(Q_program, circuits, qreg, creg, qubit)
150 meas = b+str(qubit)
151 tmp = Q_program.create_circuit(meas, [qreg],[creg])
--> 152 qr = Q_program.get_quantum_registers(qreg)
153 cr = Q_program.get_classical_registers(creg)
154 if b == "X":
AttributeError: 'QuantumProgram' object has no attribute 'get_quantum_registers'
Testing experiments on a simulator
==================================
Now that we have prepared the required circuits for state preparation
and measurement, we should test them on a simulator before trying to run
them on the real device.
We specify the device, and a number of experiment shots to perform to
gather measurement statistics. The larger the number of shots, the more
accurate our measurmeent probabilities will be compared to the *true*
value.
.. code:: python
# Use the local simulator
device = 'local_qasm_simulator'
# Take 1000 shots for each measurement basis
shots = 1000
# Run the simulation
result = QP.execute(circuits, device, shots)
print(result)
.. parsed-literal::
running on backend: local_qasm_simulator
{'status': 'COMPLETED', 'result': 'all done'}
Before doing the tomographic reconstruction we can view the count
statistics from the simulation:
.. code:: python
for c in circuits:
print('Circuit:', c)
print('Counts:', QP.get_counts(c))
.. parsed-literal::
Circuit: prepX1X0
Counts: {'00': 533, '11': 467}
Circuit: prepX1Y0
Counts: {'10': 256, '00': 252, '11': 238, '01': 254}
Circuit: prepX1Z0
Counts: {'01': 253, '10': 280, '11': 232, '00': 235}
Circuit: prepY1X0
Counts: {'11': 249, '10': 242, '01': 244, '00': 265}
Circuit: prepY1Y0
Counts: {'10': 495, '01': 505}
Circuit: prepY1Z0
Counts: {'10': 260, '01': 242, '11': 257, '00': 241}
Circuit: prepZ1X0
Counts: {'00': 266, '10': 244, '01': 234, '11': 256}
Circuit: prepZ1Y0
Counts: {'10': 237, '01': 282, '11': 269, '00': 212}
Circuit: prepZ1Z0
Counts: {'11': 478, '00': 522}
Reconstructing state from count data
------------------------------------
To reconstruct the maximum likelihod estimate of the measured quantum
state we use the following function:
::
state_tomography(QP, circuits, shots, total_qubits, meas_qubits)
where - ``QP`` is the quantum program containing the measurement results
- ``circuits`` is the array of tomographic measurement circuits measured
- ``shots`` is the total number of shots for each measurement circuit -
``total_qubits`` is the total number of qubits in the system (the length
of shot outcome bitstrings) - ``meas_qubits`` is an array of the
measurement qubit indices
.. code:: python
total_qubits = 2
rho_fit = state_tomography(QP, circuits, shots, total_qubits, meas_qubits)
print('rho =', rho_fit)
.. parsed-literal::
rho = [[ 5.05919257e-01+0.j 1.97270901e-02+0.00971077j
-6.76802859e-03+0.0017145j 4.95913187e-01-0.00164784j]
[ 1.97270901e-02-0.00971077j 3.48839871e-03+0.j
9.24584857e-05-0.00230115j 1.99663499e-02-0.00802739j]
[ -6.76802859e-03-0.0017145j 9.24584857e-05+0.00230115j
2.88747465e-03+0.j -7.72694247e-03-0.00085422j]
[ 4.95913187e-01+0.00164784j 1.99663499e-02+0.00802739j
-7.72694247e-03+0.00085422j 4.87704870e-01+0.j ]]
We can compare the reconstructed state to the target state vector. We
use the Fidelity function, which for a comparing a density matrix
:math:`\rho` to a pure state :math:`|\psi\rangle` is given by
:math:`F = \sqrt{\langle \psi| \rho |\psi\rangle}`. This may be done by
the function ``fidelity(rho, psi)``. Finally we may wish to visualize
the reconstructed state. This can be done by using various plotting
libraries. One conveient one is the following which uses the `QuTiP
(Quantum Toolbox in Python) <http://qutip.org/>`__ library.
.. code:: python
# target state is (|00>+|11>)/sqrt(2)
target = np.array([1., 0., 0., 1.]/np.sqrt(2.))
# calculate fidelity
F_fit = fidelity(rho_fit, target)
# visualize the state
plot_state(rho_fit, F_fit)
.. image:: tomography_files/tomography_19_0.png
.. image:: tomography_files/tomography_19_1.png
Note that since our simulator is *perfect* the output state should be
*exactly* the Bell-state, so we should obtain F = 1. Why is it not in
our case? Since we can never directly *see* the final state we must
obtain information about it via measurements. We would only obtain the
*true* probabilities for the state in the limit of infinite measurement
shots. Hence we have statistical error in our reconstruction due to
having imperfect information about the state itself. Try running with
different number of shots on the simulator and see how it effects the
fidelity of the reconstruction.
Running on a real device
========================
Now that we've checked our simple tomography experiment worked, lets try
it out on the IBM Quantum Experience! To do this we must have attached
our API key, and it is good practice to set a limit on the number of
credits to use:
.. code:: python
# Use the IBM Quantum Experience
device = 'ibmqx2'
# Take 1000 shots for each measurement basis
# Note: reduce this number for larger number of qubits
shots = 1000
# set max credits
max_credits = 5
# set API token and url
QP.set_api(Qconfig.APItoken, Qconfig.config["url"])
# Run the simulation
result = QP.execute(circuits, device, shots, max_credits, wait=20, timeout=240)
print(result)
.. parsed-literal::
running on backend: ibmqx2
status = RUNNING (20 seconds)
status = RUNNING (40 seconds)
{'status': 'COMPLETED', 'result': 'all done'}
As before we can check our results, and check our tomographically
reconstructed state.
.. code:: python
# print measurement results
for c in circuits:
print('Circuit:', c)
print('Counts:', QP.get_counts(c))
.. parsed-literal::
Circuit: prepX1X0
Counts: {'00000': 569, '00001': 24, '00010': 19, '00011': 388}
Circuit: prepX1Y0
Counts: {'00000': 337, '00001': 281, '00010': 212, '00011': 170}
Circuit: prepX1Z0
Counts: {'00000': 312, '00001': 253, '00010': 230, '00011': 205}
Circuit: prepY1X0
Counts: {'00000': 283, '00001': 221, '00010': 281, '00011': 215}
Circuit: prepY1Y0
Counts: {'00000': 64, '00001': 444, '00010': 485, '00011': 7}
Circuit: prepY1Z0
Counts: {'00000': 283, '00001': 245, '00010': 252, '00011': 220}
Circuit: prepZ1X0
Counts: {'00000': 291, '00001': 240, '00010': 274, '00011': 195}
Circuit: prepZ1Y0
Counts: {'00000': 286, '00001': 223, '00010': 260, '00011': 231}
Circuit: prepZ1Z0
Counts: {'00000': 468, '00001': 36, '00010': 31, '00011': 465}
Notice that for measurement results the bitstrings are now actually
those for 5 qubits, even though we only measured one. This is because
the QX is a 5-qubit quantum processor. To reconstruct the single qubit
state from these results we use ``total_qubits=5``. Note also that the
qubit ordering in this bitstrings is ``q[4]q[3]q[2]q[1]q[0]``
.. code:: python
rho_fit_real = state_tomography(QP, circuits, shots, 5, meas_qubits)
F_fit_real = fidelity(rho_fit_real, target)
plot_state(rho_fit_real, F_fit_real)
.. image:: tomography_files/tomography_25_0.png
.. image:: tomography_files/tomography_25_1.png
The fidelity of our reconstructed state if 94%, not bad!
Further examples
================
To see further examples of performing tomography on up to 5-qubit
entangled state see the
:doc:`cat-state-tomography <cat-state-tomography>` workbook.

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###############################
Getting Started with QISKit SDK
###############################
For more information about how to use the IBM Q experience (QX), consult
the
`tutorials <https://quantumexperience.ng.bluemix.net/qstage/#/tutorial?sectionId=c59b3710b928891a1420190148a72cce&pageIndex=0>`__,
or check out the
`community <https://quantumexperience.ng.bluemix.net/qstage/#/community>`__.
Contributors
============
Ismael Faro, Jay Gambetta, Andrew Cross
QISKit SDK Tutorial
===================
This tutorial aims to explain how to use the QISKit SDK from a
developer's point of view. We review the steps it takes to install and
start to use the SDK tools.
QISKit is a Python software development kit (SDK) that you can use to
create your quantum computing programs based on circuits defined through
the `OpenQASM 2.0
specification <https://github.com/IBM/qiskit-openqasm>`__, compile them,
and execute them on several backends (Real Quantum Processors online,
Simulators online, and Simulators on local). For the online backends,
QISKit uses our `python API
connector <https://github.com/IBM/qiskit-api-py>`__ to the `IBM Q
experience project <http://quantumexperience.ng.bluemix.net/>`__.
In addition to this tutorial, we have other tutorials that introduce you
to more complex concepts directly related to quantum computing.
More examples: - Familiarize yourself with the important concepts of
:doc:`superposition and entanglement <superposition_and_entanglement>`. - Go beyond and
explore a bit more in-depth in :doc:`entanglement revisited <entanglement_revisited>`.
Install QISKit
==============
The easiest way to install QISKit is with the Anaconda Python
distribution.
- Install Anaconda: https://www.continuum.io/downloads
Next, install QISKit from the git repository
- Clone the repo:
.. code:: sh
git clone https://github.ibm.com/IBMQuantum/qiskit-sdk-py-dev
cd qiskit-sdk-py-dev
- Create the environment with the dependencies:
.. code:: sh
make env
Use QISKit Python SDK
=====================
You can try out the examples easily with Jupyter or Python.
Add your personal API token to the file "Qconfig.py" (get it from your
`IBM Q experience <https://quantumexperience.ng.bluemix.net>`__ >
Account):
.. code:: sh
cp tutorial/Qconfig.py.default Qconfig.py
Run Jupyter notebook.
.. code:: sh
make run
Basic Concept
-------------
The basic concept of our quantum program is an array of quantum
circuits. The program workflow consists of three stages: :ref:`Build
<building_your_program>`, :ref:`Compile <compile_and_run>`, and
:ref:`Run <execute_on_real_device>`. Build allows you to make
different quantum circuits that represent the problem you are solving;
Compile allows you to rewrite them to run on different backends
(simulators/real chips of different `quantum volumes
<http://ibm.biz/qiskit-quantum-volume>`__, sizes, fidelity, etc); and
Run launches the jobs.  After the jobs have been run, the data is
collected. There are methods for putting this data together, depending
on the program. This either gives you the answer you wanted or allows
you to make a better program for the next instance.
.. _building_your_program:
Building your program: Create it
--------------------------------
First you need to import the QuantumProgram package from QISKit.
.. code:: python
import sys
sys.path.append("../../") # solve the relative dependencies if you clone QISKit from the Git repo and use like a global.
from qiskit import QuantumProgram
import Qconfig
The basic elements needed for your first program are the QuantumProgram,
a Circuit, a Quantum Register, and a Classical Register.
.. code:: python
# Creating Programs
# create your first QuantumProgram object instance.
Q_program = QuantumProgram()
# Creating Registers
# create your first Quantum Register called "qr" with 2 qubits
qr = Q_program.create_quantum_registers("qr", 2)
# create your first Classical Register called "cr" with 2 bits
cr = Q_program.create_classical_registers("cr", 2)
# Creating Circuits
# create your first Quantum Circuit called "qc" involving your Quantum Register "qr"
# and your Classical Register "cr"
qc = Q_program.create_circuit("qc", ["qr"], ["cr"])
.. parsed-literal::
>> quantum_registers created: qr 2
>> classical_registers created: cr 2
Another option for creating your QuantumProgram instance is to define a
dictionary with all the necessary components of your program.
.. code:: python
Q_SPECS = {
"circuits": [{
"name": "Circuit",
"quantum_registers": [{
"name": "qr",
"size": 4
}],
"classical_registers": [{
"name": "cr",
"size": 4
}]}],
}
The required element for a Program is a "circuits" array. Within
"circuits", the required field is "name"; it can have several Quantum
Registers and Classical Registers. Every register must have a name and
the number of each element (qubits or bits).
After that, you can use this dictionary definition as the specs of one
QuantumProgram object to initialize it.
.. code:: python
Q_program = QuantumProgram(specs=Q_SPECS)
.. parsed-literal::
>> quantum_registers created: qr 4
>> classical_registers created: cr 4
You can also get every component from your new Q\_program to use.
.. code:: python
# Get the components.
# get the circuit by Name
circuit = Q_program.get_circuit("Circuit")
# get the Quantum Register by Name
quantum_r = Q_program.get_quantum_registers("qr")
# get the Classical Register by Name
classical_r = Q_program.get_classical_registers('cr')
Building your program: Add Gates to your Circuit
------------------------------------------------
After you create the circuit with its registers, you can add gates to
manipulate the registers. Below is a list of the gates you can use in
the QX.
You can find extensive information about these gates and how use them in
our `Quantum Experience User
Guide <https://quantumexperience.ng.bluemix.net/qstage/#/tutorial?sectionId=71972f437b08e12d1f465a8857f4514c&pageIndex=2>`__.
.. code:: python
# H (Hadamard) gate to the qubit 0 in the Quantum Register "qr"
circuit.h(quantum_r[0])
# Pauli X gate to the qubit 1 in the Quantum Register "qr"
circuit.x(quantum_r[1])
# Pauli Y gate to the qubit 2 in the Quantum Register "qr"
circuit.y(quantum_r[2])
# Pauli Z gate to the qubit 3 in the Quantum Register "qr"
circuit.z(quantum_r[3])
# CNOT (Controlled-NOT) gate from qubit 0 to the Qbit 2
circuit.cx(quantum_r[0], quantum_r[2])
# add a barrier to your circuit
circuit.barrier()
# first physical gate: u1(lambda) to qubit 0
circuit.u1(0.3, quantum_r[0])
# second physical gate: u2(phi,lambda) to qubit 1
circuit.u2(0.3, 0.2, quantum_r[1])
# second physical gate: u3(theta,phi,lambda) to qubit 2
circuit.u3(0.3, 0.2, 0.1, quantum_r[2])
# S Phase gate to qubit 0
circuit.s(quantum_r[0])
# T Phase gate to qubit 1
circuit.t(quantum_r[1])
# identity gate to qubit 1
circuit.iden(quantum_r[1])
# Note: "if" is not implemented in the local simulator right now,
# so we comment it out here. You can uncomment it and
# run in the online simulator if you'd like.
# Classical if, from qubit2 gate Z to classical bit 1
# circuit.z(quantum_r[2]).c_if(classical_r, 0)
# measure gate from the qubit 0 to classical bit 0
circuit.measure(quantum_r[0], classical_r[0])
.. parsed-literal::
<qiskit._measure.Measure at 0x112c72518>
Extract QASM
------------
You can obtain a QASM representation of your code.
.. code:: python
# QASM from a program
QASM_source = Q_program.get_qasm("Circuit")
print(QASM_source)
.. parsed-literal::
OPENQASM 2.0;
include "qelib1.inc";
qreg qr[4];
creg cr[4];
h qr[0];
x qr[1];
y qr[2];
z qr[3];
cx qr[0],qr[2];
barrier qr[0],qr[1],qr[2],qr[3];
u1(0.300000000000000) qr[0];
u2(0.300000000000000,0.200000000000000) qr[1];
u3(0.300000000000000,0.200000000000000,0.100000000000000) qr[2];
s qr[0];
t qr[1];
id qr[1];
measure qr[0] -> cr[0];
.. _compile_and_run:
Compile and Run or Execute
--------------------------
.. code:: python
device = 'ibmqx_qasm_simulator' # Backend to execute your program, in this case it is the online simulator
circuits = ["Circuit"] # Group of circuits to execute
Q_program.set_api(Qconfig.APItoken, Qconfig.config["url"]) # set the APIToken and API url
.. parsed-literal::
True
.. code:: python
Q_program.compile(circuits, device) # Compile your program
# Run your program in the device and check the execution result every 2 seconds
result = Q_program.run(wait=2, timeout=240)
print(result)
.. parsed-literal::
running on backend: ibmqx_qasm_simulator
{'status': 'COMPLETED', 'result': 'all done'}
When you run a program, the possible results will be:
::
JOB_STATUS = {
inProgress: 'RUNNING',
errorOnCreate: 'ERROR_CREATING_JOB',
errorExecuting: 'ERROR_RUNNING_JOB',
completed: 'COMPLETED'
};
The *run()* command waits until the job either times out, returns an
error message, or completes successfully.
.. code:: python
Q_program.get_counts("Circuit")
.. parsed-literal::
{'0000': 529, '0001': 495}
In addition to getting the number of times each output was seen, you can
get the compiled QASM. For this simulation, the compiled circuit is not
much different from the input circuit. Each single-qubit gate has been
expressed as a u1, u2, or u3 gate.
.. code:: python
compiled_qasm = Q_program.get_compiled_qasm("Circuit")
print(compiled_qasm)
.. parsed-literal::
OPENQASM 2.0;
include "qelib1.inc";
qreg qr[4];
creg cr[4];
u1(3.141592653589793) qr[3];
u3(3.141592653589793,1.5707963267948966,1.5707963267948966) qr[2];
u3(3.141592653589793,0.0,3.141592653589793) qr[1];
u2(0.0,3.141592653589793) qr[0];
cx qr[0],qr[2];
barrier qr[0],qr[1],qr[2],qr[3];
u1(0.3) qr[0];
u1(1.5707963267948966) qr[0];
measure qr[0] -> cr[0];
u2(0.3,0.2) qr[1];
u1(0.7853981633974483) qr[1];
id qr[1];
u3(0.3,0.2,0.1) qr[2];
You can use *execute()* to combine the compile and run in a single step.
.. code:: python
Q_program.execute(circuits, device, wait=2, timeout=240)
.. parsed-literal::
running on backend: ibmqx_qasm_simulator
.. parsed-literal::
{'result': 'all done', 'status': 'COMPLETED'}
Compile Parameters
^^^^^^^^^^^^^^^^^^
Q\_program.compile(circuits, device="simulator", shots=1024,
max\_credits=3, basis\_gates=None, coupling\_map=None, seed=None)
* ``circuits`` array of circuits to compile
* ``device`` specifies the backend which is one of,
- ``simulator`` online default simulator links to ibmqx\_qasm\_simulator
- ``real`` online default real chip links to ibmqx2
- ``ibmqx_qasm_simulator`` qasm simulator
- ``ibmqx2`` online real chip with 5 qubits
- ``ibmqx3`` online real chip with 16 qubits
- ``local_unitary_simulator`` local unitary simulator
- ``local_qasm_simulator`` local simulator
* ``shots`` number of shots, only for real chips and qasm simulators
* ``max_credits`` Maximum number of the credits to spend in the executions. If the executions cost
more than your available credits, the job is aborted
* ``basis_gates``: the base gates by default are: u1, u2, u3, cx, id
* ``coupling_map``: object that represents the physical/topological layout of a chip.
* ``seed`` for the qasm simulator if you want to set the initial seed.
Run Parameters
^^^^^^^^^^^^^^
Q\_program.run(wait=5, timeout=60)
* ``wait`` time to wait before checking if the execution is COMPLETED.
* ``timeout`` timeout of the execution.
Execute Parameters
^^^^^^^^^^^^^^^^^^
*Execute has the combined parameters of compile and run.*
Q\_program.execute(circuits, device, shots=1024, max\_credits=3,
basis\_gates=None, wait=5, timeout=60, basis\_gates=None,
coupling\_map=None,)
.. _execute_on_real_device:
Execute on a Real Device
------------------------
.. code:: python
device = 'ibmqx2' # Backend where you execute your program; in this case, on the Real Quantum Chip online
circuits = ["Circuit"] # Group of circuits to execute
shots = 1024 # Number of shots to run the program (experiment); maximum is 8192 shots.
max_credits = 3 # Maximum number of credits to spend on executions.
result = Q_program.execute(circuits, device, shots, max_credits=3, wait=10, timeout=240)
.. parsed-literal::
running on backend: ibmqx2
status = RUNNING (10 seconds)
status = RUNNING (20 seconds)
Result
^^^^^^
You can access the result via the function
*get\_counts("circuit\_name")*. By default, the last device is used, but
you can be more specific by using *get\_counts("circuit\_name",
device="device\_name")*.
.. code:: python
Q_program.get_counts("Circuit")
.. parsed-literal::
{'00000': 516, '00001': 508}
Execute on a local simulator
----------------------------
.. code:: python
Q_program.compile(circuits, "local_qasm_simulator") # Compile your program
# Run your program in the device and check the execution result every 2 seconds
result = Q_program.run(wait=2, timeout=240)
Q_program.get_counts("Circuit")
.. parsed-literal::
running on backend: local_qasm_simulator
.. parsed-literal::
{'0000': 511, '0001': 513}

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200
tutorial/sections/LIH/PESMap0atdistance0.5.txt
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200
tutorial/sections/LIH/PESMap10atdistance1.5.txt
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200
tutorial/sections/LIH/PESMap11atdistance1.6.txt
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200
tutorial/sections/LIH/PESMap12atdistance1.7.txt
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200
tutorial/sections/LIH/PESMap13atdistance1.8.txt
View File

@ -1,200 +0,0 @@
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200
tutorial/sections/LIH/PESMap14atdistance1.9.txt
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200
tutorial/sections/LIH/PESMap15atdistance2.0.txt
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200
tutorial/sections/LIH/PESMap16atdistance2.1.txt
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200
tutorial/sections/LIH/PESMap17atdistance2.2.txt
View File

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200
tutorial/sections/LIH/PESMap18atdistance2.3.txt
View File

@ -1,200 +0,0 @@
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200
tutorial/sections/LIH/PESMap19atdistance2.4.txt
View File

@ -1,200 +0,0 @@
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200
tutorial/sections/LIH/PESMap1atdistance0.6.txt
View File

@ -1,200 +0,0 @@
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