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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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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[205] Oxana Shaya, "When could NISQ algorithms start to create value in discrete manufacturing ?", arXiv:2209.09650, (2022).

[206] Willie Aboumrad and Dominic Widdows, "Mod2VQLS: a Variational Quantum Algorithm for Solving Systems of Linear Equations Modulo 2", arXiv:2311.12771, (2023).

[207] Nicolas PD Sawaya and Joonsuk Huh, "Improved resource-tunable near-term quantum algorithms for transition probabilities, with applications in physics and variational quantum linear algebra", arXiv:2206.14213, (2022).

[208] Minati Rath and Hema Date, "Quantum-Assisted Simulation: A Framework for Designing Machine Learning Models in the Quantum Computing Domain", arXiv:2311.10363, (2023).

[209] Yoshiyuki Saito, Xinwei Lee, Dongsheng Cai, and Nobuyoshi Asai, "Quantum Multi-Resolution Measurement with application to Quantum Linear Solver", arXiv:2304.05960, (2023).

[210] Po-Wei Huang, Xiufan Li, Kelvin Koor, and Patrick Rebentrost, "Hybrid quantum-classical and quantum-inspired classical algorithms for solving banded circulant linear systems", arXiv:2309.11451, (2023).

[211] Dingjie Lu, Zhao Wang, Jun Liu, Yangfan Li, Wei-Bin Ewe, and Zhuangjian Liu, "From Ad-Hoc to Systematic: A Strategy for Imposing General Boundary Conditions in Discretized PDEs in variational quantum algorithm", arXiv:2310.11764, (2023).

[212] Sanjay Suresh and Krishnan Suresh, "Computing a Sparse Approximate Inverse on Quantum Annealing Machines", arXiv:2310.02388, (2023).

The above citations are from Crossref's cited-by service (last updated successfully 2024-02-27 21:24:13) and SAO/NASA ADS (last updated successfully 2024-02-27 21:24:15). The list may be incomplete as not all publishers provide suitable and complete citation data.