Complexity of Quantum State Verification in the Quantum Linear Systems Problem

Open Access

Complexity of Quantum State Verification in the Quantum Linear Systems Problem

Rolando D. Somma and Yiğit Subaşı

PRX Quantum 2, 010315 – Published 27 January 2021

Abstract

We analyze the complexity of quantum state verification in the context of solving systems of linear equations of the form $A \vec{x} = \vec{b}$ . We show that any quantum operation that verifies whether a given quantum state is within a constant distance from the solution of the quantum linear systems problem requires $q = Ω (κ)$ uses of a unitary that prepares a quantum state $| b ⟩$ , proportional to $\vec{b}$ , and its inverse in the worst case. Here, $κ$ is the condition number of matrix $A$ . For typical instances, we show that $q = Ω (\sqrt{κ})$ with high probability. These lower bounds are almost achieved if quantum state verification is performed using known quantum algorithms for the quantum linear systems problem. We also analyze the number of copies of $| b ⟩$ required by verification procedures of the prepare-and-measure type. In this case, the lower bounds are quadratically worse, being $Ω (κ^{2})$ in the worst case and $Ω (κ)$ in typical instances with high probability. We discuss the implications of our results to known variational and related approaches to this problem, where state preparation, gate, and measurement errors will need to decrease rapidly with $κ$ for worst-case and typical instances if error correction is not used, and present some open problems.

Received 14 August 2020
Accepted 16 December 2020

DOI:https://doi.org/10.1103/PRXQuantum.2.010315

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Quantum algorithms

Quantum Information, Science & Technology

Authors & Affiliations

Rolando D. Somma ^1,* and Yiğit Subaşı ^2,†

¹Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
²Computer, Computational, and Statistical Sciences Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA

^*somma@lanl.gov
^†ysubasi@lanl.gov

Popular Summary

Quantum computers may one day solve problems related to systems of linear equations exponentially faster than classical computers. There is extensive work on quantum algorithms for this purpose, and most are expected to necessitate a large-scale and fault-tolerant quantum computer for implementation. However, today’s quantum devices are relatively small (less than 100 qubits), fairly noisy, and do not allow for quantum error correction. Significant recent research aims at using these so-called noisy intermediate-scale quantum (NISQ) devices to solve practical problems that classical computers cannot. Can these devices be used for solving systems of linear equations? An answer to this question is essential to understand if quantum advantage is possible for a paramount problem that has applications in linear algebra, fluid dynamics, physics, and beyond.

Quantum state verification is key to test if a quantum computation has been correctly implemented. Our work demonstrates that a surprisingly large number of resources (gates and measurements) are needed to verify the solution to a linear systems problem with a quantum device. It suggests that, under fairly general conditions, NISQ devices will not be able to show quantum advantage for this problem in general.

We investigate strong limits placed by our results on so-called variational quantum algorithms, which have been specially devised for NISQ devices. As these algorithms use a form of quantum state verification, they will require many resources and very low error levels for their implementation, which are currently prohibitive. We also discuss variants of the quantum linear systems problem where the limits imposed by our results do not apply, opening new questions that may be of interest to the quantum computing community.

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