Quantum and classical query complexities of functions of matrices

TL;DR

This paper investigates the quantum and classical query complexities of approximating matrix functions of sparse Hermitian matrices, establishing bounds that demonstrate an exponential separation and the optimality of quantum singular value transformation.

Contribution

It provides tight bounds on quantum and classical query complexities for matrix function approximation, revealing exponential separation and confirming the optimality of quantum methods.

Findings

Quantum query complexity lower bounded by the approximate degree of the function.

Classical query complexity exhibits exponential lower bounds in terms of the approximate degree.

Entry estimation problem is BQP-complete for functions with large approximate degree.

Abstract

Let $A$ be an $s$-sparse Hermitian matrix, $f(x)$ be a univariate function, and $i, j$ be two indices. In this work, we investigate the query complexity of approximating $\bra{i} f(A) \ket{j}$. We show that for any continuous function $f(x):[-1,1]\rightarrow [-1,1]$, the quantum query complexity of computing $\bra{i} f(A) \ket{j}\pm \varepsilon/4$ is lower bounded by $\Omega(\widetilde{\deg}_\varepsilon(f))$. The upper bound is at most quadratic in $\widetilde{\deg}_\varepsilon(f)$ and is linear in $\widetilde{\deg}_\varepsilon(f)$ under certain mild assumptions on $A$. Here the approximate degree $\widetilde{\deg}_\varepsilon(f)$ is the minimum degree such that there is a polynomial of that degree approximating $f$ up to additive error $\varepsilon$ in the interval $[-1,1]$. We also show that the classical query complexity is lower bounded by $\widetilde{\Omega}((s/2)^{(\widetilde{\deg}_{2\varepsilon}(f)-1)/6})$ for any $s\geq 4$. Our results show that the quantum and classical separation is exponential for any continuous function of sparse Hermitian matrices, and also imply the optimality of implementing smooth functions of sparse Hermitian matrices by quantum singular value transformation. As another hardness result, we show that entry estimation problem (i.e., deciding $\bra{i} f(A) \ket{j}\geq \varepsilon$ or $\bra{i} f(A) \ket{j}\leq -\varepsilon$) is BQP-complete for any continuous function $f(x)$ as long as its approximate degree is large enough. The main techniques we used are the dual polynomial method for functions over the reals, linear semi-infinite programming, and tridiagonal matrices.