Variational Quantum Linear Solver

Carlos Bravo-Prieto1,2,3, Ryan LaRose4, M. Cerezo1,5, Yigit Subasi6, Lukasz Cincio1, and Patrick J. Coles1

1Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA.
2Barcelona Supercomputing Center, Barcelona, Spain.
3Institut de Ciències del Cosmos, Universitat de Barcelona, Barcelona, Spain.
4Department of Computational Mathematics, Science, and Engineering & Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48823, USA.
5Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, NM, USA
6Computer, Computational and Statistical Sciences Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA

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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 $|x\rangle$ 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 $C \geqslant \epsilon^2 / \kappa^2$, 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 $1024\times1024$. 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 $N$.

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