Variational Quantum Linear Solver

Bravo-Prieto, Carlos; LaRose, Ryan; Cerezo, M.; Subasi, Yigit; Cincio, Lukasz; Coles, Patrick J.

doi:10.22331/q-2023-11-22-1188

Variational Quantum Linear Solver

Journal Article · Wed Nov 22 00:00:00 EST 2023 · Quantum

DOI:https://doi.org/10.22331/q-2023-11-22-1188· OSTI ID:2367476

Bravo-Prieto, Carlos ^[1]; LaRose, Ryan ^[2]; Cerezo, M. ^[3]; Subasi, Yigit ^[3]; Cincio, Lukasz ^[3]; Coles, Patrick J. ^[3]

Los Alamos National Laboratory (LANL), Los Alamos, NM (United States); Barcelona Supercomputing Center (Spain); Univ. de Barcelona (Spain)
Michigan State Univ., East Lansing, MI (United States)
Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)

Previously proposed quantum algorithms for solving linear systems of equations cannot be implemented in the near term due to the re quired 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$$\rangle$$ ∝ |b$$\rangle$$. We derive an operationally meaningful termination condition for VQLS that allows one to guarantee that a desired solution precision ϵ is achieved. Specifically, we prove that C $⩾$ ϵ²/κ², where C is the VQLS cost function and κ 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 success fully implement VQLS up to a problem size of 1024 × 1024. Finally, we numerically solve nontrivial problems of size up to 2⁵⁰ × 2⁵⁰. For the specific examples that we consider, we heuristically find that the time complexity of VQLS scales efficiently in ϵ, κ, and the system size N.

View Accepted Manuscript (DOE)

Research Organization:: Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)

Sponsoring Organization:: USDOE Laboratory Directed Research and Development (LDRD) Program; USDOE National Nuclear Security Administration (NNSA); USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)

Grant/Contract Number:: 89233218CNA000001

OSTI ID:: 2367476

Report Number(s):: LA-UR-19-29101; TRN: US2411384

Journal Information:: Quantum, Vol. 7; ISSN 2521-327X

Publisher:: Quantum Science Open CommunityCopyright Statement

Country of Publication:: United States

Language:: English

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