118 lines
7.4 KiB
Plaintext
118 lines
7.4 KiB
Plaintext
{
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"cells": [
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{
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"cell_type": "markdown",
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"id": "67e54fee-28ed-4ebe-bcb8-715aa06721a1",
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"metadata": {},
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"source": [
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"# Multi-product formulas (MPF)\n",
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"\n",
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"Multi-product formulas (MPF) can be used to more accurately simulate the dynamics of a quantum system, at the cost of increased circuit executions. This is a post-processing technique that mitigates the error of expectation values for time-evolved states.\n",
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"\n",
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"Classical computing is used to solve a system of linear equations that provides coefficients to a weighted combination of several circuit executions. Using this weighted combination can reduce the error associated with simulating time evolution, given a good selection of Trotter steps. The MPF tool will ingest a selection of data --- including the number of Trotter steps and the order of the Trotter approximation --- to prepare and solve (or approximate the solution of) the associated system of linear equations, which you can then use to post-process expectation value measurements of a time-evolved state."
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]
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},
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{
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"cell_type": "markdown",
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"id": "1a2f529c-4983-447d-9923-a423b34e7215",
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"metadata": {},
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"source": [
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"## Install the MPF package\n",
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"\n",
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"There are two ways to install the MPF package: via PyPI and building from source. It is recommended to install in a [virtual environment](https://docs.python.org/3.10/tutorial/venv.html) to ensure separation between package dependencies.\n",
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"\n",
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"### Install from PyPI\n",
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"\n",
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"The most straightforward way to install the `qiskit-addon-mpf` package is via PyPI.\n",
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"```bash\n",
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"pip install qiskit-addon-mpf\n",
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"```\n",
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"\n",
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"### Build from source\n",
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"Users who wish to develop in the repository or who want to install it manually may do so by first cloning the repository:\n",
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"```bash\n",
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"git clone git@github.com:Qiskit/qiskit-addon-mpf.git\n",
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"```\n",
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"\n",
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"and install the package via `pip`. The repository also contains a number of optional dependencies that enable certain features.\n",
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"\n",
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"Adjust the options to suit your needs.\n",
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"```bash\n",
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"pip install tox notebook -e '.[notebook-dependencies,dev]'\n",
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"```"
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]
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},
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{
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"cell_type": "markdown",
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"id": "7ac19afa-954c-4c1d-b3b1-7871db85e8b4",
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"metadata": {},
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"source": [
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"## Theoretical background\n",
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"\n",
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"MPFs can reduce the Trotter approximation error associated with simulating the dynamics of quantum systems through a weighted combination of several circuit executions. This weighted sum is defined as:\n",
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"\n",
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"$$ \\mu(t) = \\sum_j x_j\\rho_j^{k_j}\\left(\\frac{t}{k_j}\\right) + \\text{some remaining Trotter error}, $$\n",
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"\n",
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"where $x_j$ are the weighting coefficients, $\\rho_j^{k_j}$ is the density matrix that corresponds to the pure state obtained by evolving the initial state via a product formula $S^{k_j}$ approximating the time-evolution operator using $k_j$ Trotter steps, and $j$ indexes each product formula used in the sum.\n",
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"\n",
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"\n",
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"\n",
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"Generally, however, the goal of simulating quantum dynamics is to measure some observable $\\mathcal{O}(t)$, which is a function of time. When using MPFs, multiple circuits -- each using $k_j$ Trotter steps -- are executed to obtain several measurements of the target observable $\\mathcal{O}_{k_j}(t)$. The measurement of the target observable is then obtained by computing:\n",
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"\n",
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"$$ \\langle \\mathcal{O}(t) \\rangle = \\sum_j x_j(t) \\langle \\mathcal{O}_{k_j}(t) \\rangle. $$\n",
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"\n",
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"In essence, you can reduce the overall Trotter error by approximating the time-evolution operator using several product formulas with a variable number of Trotter steps instead of a single product formula. You construct a circuit for each term in the weighted sum, which evolves the system according to each of the $k_j$ number of Trotter steps. Each circuit is then executed separately on a QPU to reconstruct the results in a post-processing step. The utility of this technique can be viewed from two perspectives:\n",
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"1. For a fixed number of Trotter steps that you execute, you can obtain results with a Trotter error that is smaller in total.\n",
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"2. For a number of Trotter steps that result in deep circuits, you can use MPF to find several shorter-depth circuits to run, which results in a similar Trotter approximation error.\n",
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"\n",
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"\n",
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"\n",
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"## Determine MPF coefficients\n",
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"\n",
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"The core functionality of the `qiskit-addon-mpf` package lies in determining the MPF coefficients $x_j(t)$ (which may be time-dependent). The process to obtain each $x_j(t)$ involves solving a system of linear equations $Ax=b$ where $x$ is the vector of coefficients to be determined, $A$ is a matrix that depends on each of the $k_j$ and the product formula used $S$ (as in, the approximation order and number of Trotter steps), and $b$ is a vector of constraints. This system of equations can be solved either exactly or with an approximate model that minimizes the 1-norm of the coefficients. Additionally, the choice for each $k_j$ is a heuristic process, but can be bounded by the following constraints:\n",
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"\n",
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"1. The **largest** $k_j$ value is bounded by the highest depth of circuit that can be reliably executed\n",
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"2. The **smallest** $k_j$ should satisfy $dt = t/k_j < 1$ since that is where the Trotter error is most well behaved\n",
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"3. None of the coefficients $x_j$ should be close to $0$ since this would imply they don't contribute much to the MPF\n",
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"4. Similarly, the coefficient associated with the largest $k_j$ value should not dominate, since this implies that you are using a single product formula\n",
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"5. Lastly, the norm of the coefficients $x_j$ obtained should be small, as this indicates a well-conditioned MPF [1](#references)\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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" - Read the page on [getting started with MPF](/guides/qiskit-addons-mpf-get-started).\n",
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"</Admonition>\n",
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"\n",
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"## References\n",
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"\n",
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"1. A. Carrera Vazquez, D. J. Egger, D. Ochsner, and S. Wörner, [\"Well-conditioned multi-product formulas for hardware-friendly Hamiltonian simulation\"](https://quantum-journal.org/papers/q-2023-07-25-1067/), Quantum 7, 1067 (2023).\n",
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"2. S. Zhuk, N. Robertson, and S. Bravyi, [\"Trotter error bounds and dynamic multi-product formulas for Hamiltonian simulation\"](https://journals.aps.org/prresearch/abstract/10.1103/PhysRevResearch.6.033309), Phys. Rev. Research 6, 033309 (2024).\n",
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"3. N. Robertson, et al. [\"Tensor Network enhanced Dynamic Multiproduct Formulas\"](https://arxiv.org/abs/2407.17405v2), arXiv:2407.17405v2 [quant-ph]."
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}
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