Tight Bound for Estimating Expectation Values from a System of Linear Equations

TL;DR

This paper establishes a tight quantum query complexity lower bound for estimating expectation values in linear systems, showing that the complexity scales linearly with the inverse of the desired accuracy when using block encoding.

Contribution

It provides a matching quantum algorithm that saturates the lower bound, confirming the optimality of the linear dependence on accuracy for the problem.

Findings

Quantum query complexity for SLEP is Θ(α/ε).

Lower bounds are proven via reduction from mean estimation.

Polynomial improvement over linear accuracy dependence is impossible with block encoding.

Abstract

The System of Linear Equations Problem (SLEP) is specified by a complex invertible matrix $A$, the condition number $\kappa$ of $A$, a vector $b$, a Hermitian matrix $M$ and an accuracy $\epsilon$, and the task is to estimate $x^\dagger Mx$, where $x$ is the solution vector to the equation $Ax = b$. We aim to establish a lower bound on the complexity of the end-to-end quantum algorithms for SLEP with respect to $\epsilon$, and devise a quantum algorithm that saturates this bound. To make lower bounds attainable, we consider query complexity in the setting in which a block encoding of $M$ is given, i.e., a unitary black box $U_M$ that contains $M/\alpha$ as a block for some $\alpha \in \mathbb R^+$. We show that the quantum query complexity for SLEP in this setting is $\Theta(\alpha/\epsilon)$. Our lower bound is established by reducing the problem of estimating the mean of a black box function to SLEP. Our $\Theta(\alpha/\epsilon)$ result tightens and proves the common assertion of polynomial accuracy dependence (poly$(1/\epsilon)$) for SLEP, and shows that improvement beyond linear dependence on accuracy is not possible if $M$ is provided via block encoding.