Complexity of quantum state verification in the quantum linear systems problem

TL;DR

This paper establishes fundamental lower bounds on the complexity of quantum state verification in solving linear systems, revealing that verification costs scale with the condition number and highlighting challenges for practical quantum algorithms.

Contribution

It provides the first rigorous lower bounds on the number of quantum operations needed for state verification in quantum linear systems, relating complexity to the condition number.

Findings

Verification requires at least Ω(κ) unitaries in the worst case.

Typical instances require at least Ω(√κ) unitaries with high probability.

Verification using known algorithms nearly matches these lower bounds.

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 $q=\Omega(\kappa)$ uses of a unitary that prepares a quantum state $\left| b \right>$, proportional to $\vec b$, and its inverse in the worst case. Here, $\kappa$ is the condition number of the matrix $A$. For typical instances, we show that $q=\Omega(\sqrt \kappa)$ 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 $\left| b \right>$ required by verification procedures of the prepare and measure type. In this case, the lower bounds are quadratically worse, being $\Omega(\kappa^2)$ in the worst case and $\Omega(\kappa)$ 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 $\kappa$ for worst-case and typical instances if error correction is not used, and present some open problems.