397 lines
10 KiB
Plaintext
397 lines
10 KiB
Plaintext
{
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"cells": [
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{
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"cell_type": "markdown",
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"id": "1f7fe7f1-988f-461f-8222-f291404b912a",
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"metadata": {},
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"source": [
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"{/* cspell:ignore leftrightarrow */}"
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]
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},
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{
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"cell_type": "markdown",
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"id": "49fb8eee-d17a-4648-8b6f-52097acfb305",
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"metadata": {
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"tags": [
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"version-info"
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]
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},
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"source": [
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"# Bit-ordering in the Qiskit SDK"
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]
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},
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{
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"cell_type": "markdown",
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"id": "369b8b49-a2af-4999-85df-f7659022e11b",
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"metadata": {},
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"source": [
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"If you have a set of $n$ bits (or qubits), you'll usually label each bit $0\n",
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"\\rightarrow n-1$. Different softwares and resources must choose how they order\n",
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"these bits both in computer memory and when displayed on-screen.\n",
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"\n",
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"## Qiskit conventions\n",
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"\n",
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"Here's how the Qiskit SDK orders bits in different scenarios.\n",
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"\n",
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"### Quantum circuits\n",
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"\n",
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"The `QuantumCircuit` class stores its qubits in a list\n",
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"(`QuantumCircuit.qubits`). The index of a qubit in this list defines the\n",
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"qubit's label."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 5,
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"id": "a1c61ed7-72df-447e-a103-ce0e273626c9",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"<Qubit register=(2, \"q\"), index=0>"
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]
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},
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"execution_count": 5,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"from qiskit import QuantumCircuit, QuantumRegister\n",
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"from qiskit.circuit import Qubit\n",
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"\n",
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"qc = QuantumCircuit(2)\n",
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"qc.qubits[0] # qubit \"0\"\n",
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"\n",
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"Qubit(QuantumRegister(2, \"q\"), 0)"
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]
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},
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{
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"cell_type": "markdown",
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"id": "22c49977-e960-4620-ae1d-a7a90a879b45",
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"metadata": {},
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"source": [
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"### Circuit diagrams\n",
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"\n",
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"On a circuit diagram, qubit $0$ is the topmost qubit, and qubit $n-1$ the\n",
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"bottommost qubit. You can change this with the `reverse_bits` argument of\n",
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"`QuantumCircuit.draw` (see [Change ordering in\n",
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"Qiskit](#change-ordering-in-qiskit))."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 6,
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"id": "a4e175af-8882-427a-a276-bd51a3d9be57",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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" \n",
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"q_0: ─────\n",
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" ┌───┐\n",
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"q_1: ┤ X ├\n",
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" └───┘"
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]
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},
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"execution_count": 6,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"qc.x(1)\n",
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"qc.draw()"
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]
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},
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{
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"cell_type": "markdown",
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"id": "19aa0892-a578-49e3-98df-14c696d01896",
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"metadata": {},
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"source": [
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"### Integers\n",
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"\n",
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"When interpreting bits as a number, bit $0$ is the least significant bit, and\n",
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"bit $n-1$ the most significant. This is helpful when coding because each bit has\n",
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"the value $2^\\text{label}$ (label being the qubit's index in\n",
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"`QuantumCircuit.qubits`). For example, the following circuit execution ends\n",
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"with bit $0$ being `0`, and bit $1$ being `1`. This is interpreted as the\n",
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"decimal integer `2` (measured with probability `1.0`)."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 7,
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"id": "7ecb86d3-fe3a-43b0-9293-cba967e80fdc",
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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" > Counts: {'10': 1024}\n"
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]
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}
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],
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"source": [
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"from qiskit.primitives import StatevectorSampler as Sampler\n",
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"\n",
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"qc.measure_all()\n",
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"\n",
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"job = Sampler().run([qc])\n",
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"result = job.result()\n",
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"print(f\" > Counts: {result[0].data.meas.get_counts()}\")"
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]
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},
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{
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"cell_type": "markdown",
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"id": "e263f4f9-f383-4583-bdd4-f8792d6809db",
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"metadata": {},
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"source": [
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"### Strings\n",
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"\n",
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"When displaying or interpreting a list of bits (or qubits) as a string, bit\n",
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"$n-1$ is the leftmost bit, and bit $0$ is the rightmost bit. This is because we\n",
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"usually write numbers with the most significant digit on the left, and in\n",
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"Qiskit, bit $n-1$ is interpreted as the most significant bit.\n",
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"\n",
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"For example, the following cell defines a `Statevector` from a string of\n",
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"single-qubit states. In this case, qubit $0$ is in state $|+\\rangle$, and\n",
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"qubit $1$ in state $|0\\rangle$."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 8,
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"id": "2cffb8d4-66db-4c1c-b0c2-b796ed6e8ce0",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"{np.str_('00'): np.float64(0.4999999999999999),\n",
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" np.str_('01'): np.float64(0.4999999999999999)}"
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]
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},
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"execution_count": 8,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"from qiskit.quantum_info import Statevector\n",
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"\n",
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"sv = Statevector.from_label(\"0+\")\n",
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"sv.probabilities_dict()"
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]
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},
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{
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"cell_type": "markdown",
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"id": "86c48e21-9138-4dec-a312-0b5deefd0e35",
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"metadata": {},
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"source": [
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"This occasionally causes confusion when interpreting a string of bits, as you\n",
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"might expect the leftmost bit to be bit $0$, whereas it usually represents bit\n",
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"$n-1$.\n",
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"\n",
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"### Statevector matrices\n",
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"\n",
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"When representing a statevector as a list of complex numbers (amplitudes),\n",
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"Qiskit orders these amplitudes such that the amplitude at index $x$ represents\n",
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"the computational basis state $|x\\rangle$."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 9,
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"id": "d6ff798a-6bdc-4727-8331-b575fbf1e083",
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"(0.7071067811865475+0j)\n",
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"0j\n"
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]
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}
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],
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"source": [
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"print(sv[1]) # amplitude of state |01>\n",
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"print(sv[2]) # amplitude of state |10>"
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]
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},
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{
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"cell_type": "markdown",
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"id": "cc92b428-80c7-4530-bebc-3486aecee5f2",
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"metadata": {},
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"source": [
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"### Gates\n",
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"\n",
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"Each gate in Qiskit can interpret a list of qubits in its own way, but\n",
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"controlled gates usually follow the convention `(control, target)`.\n",
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"\n",
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"For example, the following cell adds a controlled-X gate where qubit $0$ is\n",
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"the control and qubit $1$ is the target."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 10,
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"id": "f2e34794-2f04-4065-9a1d-9aff022204db",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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" \n",
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"q_0: ──■──\n",
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" ┌─┴─┐\n",
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"q_1: ┤ X ├\n",
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" └───┘"
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]
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},
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"execution_count": 10,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"from qiskit import QuantumCircuit\n",
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"\n",
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"qc = QuantumCircuit(2)\n",
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"qc.cx(0, 1)\n",
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"qc.draw()"
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]
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},
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{
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"cell_type": "markdown",
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"id": "2b9ee336-e645-41ee-aeda-5f9dfe3a3f88",
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"metadata": {},
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"source": [
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"Following all the previously mentioned conventions in Qiskit, this CX-gate\n",
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"performs the transformation $|01\\rangle \\leftrightarrow |11\\rangle$, so has the\n",
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"following matrix.\n",
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"\n",
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"$$\n",
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"\\begin{pmatrix}\n",
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" 1 & 0 & 0 & 0 \\\\\n",
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" 0 & 0 & 0 & 1 \\\\\n",
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" 0 & 0 & 1 & 0 \\\\\n",
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" 0 & 1 & 0 & 0 \\\\\n",
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"\\end{pmatrix}\n",
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"$$\n",
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"\n",
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"## Change ordering in Qiskit\n",
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"\n",
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"To draw a circuit with qubits in reversed order (that is, qubit $0$ at the\n",
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"bottom), use the `reverse_bits` argument. This only affects the generated\n",
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"diagram and does not affect the circuit; the X-gate still acts on qubit $0$."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 11,
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"id": "6d31e88d-2a4f-4404-8177-7bccd0c19d4b",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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" \n",
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"q_1: ─────\n",
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" ┌───┐\n",
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"q_0: ┤ X ├\n",
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" └───┘"
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]
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},
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"execution_count": 11,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"from qiskit import QuantumCircuit\n",
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"\n",
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"qc = QuantumCircuit(2)\n",
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"qc.x(0)\n",
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"qc.draw(reverse_bits=True)"
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]
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},
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{
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"cell_type": "markdown",
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"id": "4e343e80-3f77-4fe7-8942-afbe71f756ab",
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"metadata": {},
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"source": [
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"You can use the `reverse_bits` method to return a new circuit with the\n",
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"qubits' labels reversed (this does not mutate the original circuit)."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 12,
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"id": "4c6a38b7-f6a1-4753-94e6-2c488ec37b8d",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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" \n",
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"q_0: ─────\n",
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" ┌───┐\n",
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"q_1: ┤ X ├\n",
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" └───┘"
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]
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},
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"execution_count": 12,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"qc.reverse_bits().draw()"
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]
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},
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{
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"cell_type": "markdown",
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"id": "da750e37-2624-4797-82c0-560ee7d5367d",
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"metadata": {},
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"source": [
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"Note that in this new circuit, the X-gate acts on qubit $1$.\n",
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"\n",
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"## Next steps\n",
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"\n",
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"<Admonition type=\"tip\" title=\"Recommendations\">\n",
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" - See an example of using circuits in the [Grover's Algorithm](https://learning.quantum.ibm.com/tutorial/grovers-algorithm) tutorial.\n",
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" - Explore the [QuantumCircuit API](/docs/api/qiskit/qiskit.circuit.QuantumCircuit#quantumcircuit-class) reference.\n",
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"</Admonition>"
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]
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}
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],
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"metadata": {
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"description": "Learn about Qiskit SDK's ordering conventions and why we chose them",
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"kernelspec": {
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"display_name": "Python 3",
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"language": "python",
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"name": "python3"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3"
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},
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"title": "Bit-ordering in the Qiskit SDK"
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},
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"nbformat": 4,
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"nbformat_minor": 4
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}
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