Variational Quantum Linear Solver

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

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

Variational Quantum Linear Solver

Carlos Bravo-Prieto^1,2,3, Ryan LaRose⁴, M. Cerezo^1,5, 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

Published:	2023-11-22, volume 7, page 1188
Eprint:	arXiv:1909.05820v4
Doi:	https://doi.org/10.22331/q-2023-11-22-1188
Citation:	Quantum 7, 1188 (2023).

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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 $N$.

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 $|b\rangle$. The output of VQLS is a quantum state $|x\rangle$ 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 $|x\rangle = \vec{x} / ||\vec{x}||_2$, 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 $|x\rangle$ 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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