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MPF Intro and getting started (#2367)
Closes #2109 and closes #2111 --------- Co-authored-by: abbycross <[email protected]>
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| { | ||
| "cells": [ | ||
| { | ||
| "cell_type": "markdown", | ||
| "id": "67e54fee-28ed-4ebe-bcb8-715aa06721a1", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "# Multi-product formulas (MPF)\n", | ||
| "\n", | ||
| "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", | ||
| "\n", | ||
| "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." | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "id": "1a2f529c-4983-447d-9923-a423b34e7215", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "## Install the MPF package\n", | ||
| "\n", | ||
| "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", | ||
| "\n", | ||
| "### Install from PyPI\n", | ||
| "\n", | ||
| "The most straightforward way to install the `qiskit-addon-mpf` package is via PyPI.\n", | ||
| "```bash\n", | ||
| "pip install qiskit-addon-mpf\n", | ||
| "```\n", | ||
| "\n", | ||
| "### Build from source\n", | ||
| "Users who wish to develop in the repository or who want to install it manually may do so by first cloning the repository:\n", | ||
| "```bash\n", | ||
| "git clone [email protected]:Qiskit/qiskit-addon-mpf.git\n", | ||
| "```\n", | ||
| "\n", | ||
| "and install the package via `pip`. The repository also contains a number of optional dependencies that enable certain features.\n", | ||
| "\n", | ||
| "Adjust the options to suit your needs.\n", | ||
| "```bash\n", | ||
| "pip install tox notebook -e '.[notebook-dependencies,dev]'\n", | ||
| "```" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "id": "7ac19afa-954c-4c1d-b3b1-7871db85e8b4", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "## Theoretical background\n", | ||
| "\n", | ||
| "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", | ||
| "\n", | ||
| "$$ \\mu(t) = \\sum_j x_j\\rho_j^{k_j}\\left(\\frac{t}{k_j}\\right) + \\text{some remaining Trotter error}, $$\n", | ||
| "\n", | ||
| "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", | ||
| "\n", | ||
| "\n", | ||
| "\n", | ||
| "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", | ||
| "\n", | ||
| "$$ \\langle \\mathcal{O}(t) \\rangle = \\sum_j x_j(t) \\langle \\mathcal{O}_{k_j}(t) \\rangle. $$\n", | ||
| "\n", | ||
| "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", | ||
| "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", | ||
| "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", | ||
| "\n", | ||
| "\n", | ||
| "\n", | ||
| "## Determine MPF coefficients\n", | ||
| "\n", | ||
| "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", | ||
| "\n", | ||
| "1. The **largest** $k_j$ value is bounded by the highest depth of circuit that can be reliably executed\n", | ||
| "2. The **smallest** $k_j$ should satisfy $dt = t/k_j < 1$ since that is where the Trotter error is most well behaved\n", | ||
| "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", | ||
| "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", | ||
| "5. Lastly, the norm of the coefficients $x_j$ obtained should be small, as this indicates a well-conditioned MPF [1](#references)\n", | ||
| "\n", | ||
| "## Next steps\n", | ||
| "\n", | ||
| "<Admonition type=\"tip\" title=\"Recommendations\">\n", | ||
| " - Read the page on [getting started with MPF](/guides/qiskit-addons-mpf-get-started).\n", | ||
| "</Admonition>\n", | ||
| "\n", | ||
| "## References\n", | ||
| "\n", | ||
| "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", | ||
| "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", | ||
| "3. N. Robertson, et al. [\"Tensor Network enhanced Dynamic Multiproduct Formulas\"](https://arxiv.org/abs/2407.17405v2), arXiv:2407.17405v2 [quant-ph]." | ||
| ] | ||
| } | ||
| ], | ||
| "metadata": { | ||
| "description": "Overview page of the qiskit addon that generates multi-product formulas (MPF)", | ||
| "kernelspec": { | ||
| "display_name": "Python 3", | ||
| "language": "python", | ||
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| "nbconvert_exporter": "python", | ||
| "pygments_lexer": "ipython3", | ||
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| }, | ||
| "title": "Multi-product formulas (MPF)" | ||
| }, | ||
| "nbformat": 4, | ||
| "nbformat_minor": 2 | ||
| } |
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