Variational Quantum Linear Solver

TL;DR

The paper introduces a hybrid variational quantum algorithm, VQLS, capable of solving large linear systems on near-term quantum computers, with proven guarantees on solution accuracy and demonstrated implementation on Rigetti's hardware.

Contribution

It proposes a new variational quantum algorithm for linear systems, with a guaranteed termination condition and efficient circuit design, suitable for near-term quantum devices.

Findings

Successfully implemented VQLS on Rigetti's quantum computer for 1024x1024 systems.

Numerically solved large problems up to 2^50 dimensions.

Heuristically observed efficient scaling with system size, condition number, and precision.

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 $|x\rangle$ such that $A|x\rangle\propto|b\rangle$. 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 $C \geq \epsilon^2 / \kappa^2$, 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 $1024\times1024$. 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 $N$.