Variational Quantum Linear Solver

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Abstract

Previously proposed quantum algorithms for solving linear systems of equations cannot be implemented in the near term due to the required circuit depth. Here, we propose a hybrid quantum-classical algorithm, called Variational Quantum Linear Solver (VQLS), for solving linear systems on near-term quantum computers. VQLS seeks to variationally prepare $|xrangle$ such that $A|xranglepropto|brangle$. We derive an operationally meaningful termination condition for VQLS that allows one to guarantee that a desired solution precision $epsilon$ is achieved. Specifically, we prove that $C geqslant epsilon^2 / kappa^2$, where $C$ is the VQLS cost function and $kappa$ is the condition number of $A$. We present efficient quantum circuits to estimate $C$, while providing evidence for the classical hardness of its estimation. Using Rigetti’s quantum computer, we successfully implement VQLS up to a problem size of $1024times1024$. Finally, we numerically solve non-trivial problems of size up to $2^{50}times2^{50}$. For the specific examples that we consider, we heuristically find that the time complexity of VQLS scales efficiently in $epsilon$, $kappa$, and the system size $N$.

Featured image: Schematic diagram for the Variational Quantum Linear Solver (VQLS) algorithm. The input to VQLS is a matrix $A$ written as a linear combination of unitaries $A_l$ and a short-depth quantum circuit $U$ which prepares the state $|brangle$. The output of VQLS is a quantum state $|xrangle$ that is approximately proportional to the solution of the linear system $A vec{x} = vec{b}$. Parameters $vec{alpha}$ in the ansatz $V(vec{alpha})$ are adjusted in a hybrid quantum-classical optimization loop until the cost $C(vec{alpha})$ (local or global) is below a user-specified threshold. When this loop terminates, the resulting gate sequence $V(vec{alpha}_text{opt})$ prepares the state $|xrangle = vec{x} / ||vec{x}||_2$, from which observable quantities can be computed. Furthermore, the final value of the cost $C(vec{alpha}_text{opt})$ provides an upper bound on the deviation of observables measured on $|xrangle$ from observables measured on the exact solution.

► BibTeX data

@article{BravoPrieto2023variationalquantum, doi = {10.22331/q-2023-11-22-1188}, url = {https://doi.org/10.22331/q-2023-11-22-1188}, title = {Variational {Q}uantum {L}inear {S}olver}, author = {Bravo-Prieto, Carlos and LaRose, Ryan and Cerezo, M. and Subasi, Yigit and Cincio, Lukasz and Coles, Patrick J.}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{“{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {7}, pages = {1188}, month = nov, year = {2023} }

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[146] Po-Wei Huang and Patrick Rebentrost, “Post-variational quantum neural networks”, arXiv:2307.10560, (2023).

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[148] Óscar Amaro and Diogo Cruz, “A Living Review of Quantum Computing for Plasma Physics”, arXiv:2302.00001, (2023).

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On Crossref’s cited-by service no data on citing works was found (last attempt 2023-11-24 23:22:08).

This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.

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Source: https://quantum-journal.org/papers/q-2023-11-22-1188/

Time Stamp: November 22, 2023

Time Stamp: Jul 7, 2022

Variational Quantum Linear Solver

Republished By Plato

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