Quantum and Classical Query Complexities of Functions of Matrices

2023-11-13

Symposium on the Theory of Computing

AI Summary

This work studies the query complexity of estimating matrix function entries $langle i|f(A)|j angle$ for an $s$-sparse Hermitian matrix $A$, where $f$ is a continuous real-valued function. The goal is to determine the minimal number of black-box queries to $A$ required to approximate the $(i,j)$-th entry of $f(A)$. Using dual polynomials, linear semi-infinite programming, and tridiagonal matrix analysis, the authors establish tight asymptotic query complexity bounds for continuous $f$. In the quantum setting, they prove matching upper and lower bounds of $Theta(deg_varepsilon(f))$ under mild conditions; in the classical setting, they derive a lower bound of $Omegaig((s/2)^{(deg_{2varepsilon}(f)-1)/6}ig)$. Their analysis confirms the optimality of Quantum Singular Value Transformation (QSVT) for implementing smooth matrix functions and rigorously demonstrates an exponential quantum speedup over classical methods for arbitrary continuous $f$.

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 i f(A) j. We show that for any continuous function f(x):[−1,1]→ [−1,1], the quantum query complexity of computing i f(A) j± ε/4 is lower bounded by Ω(degε(f)). The upper bound is at most quadratic in degε(f) and is linear in degε(f) under certain mild assumptions on A. Here the approximate degree degε(f) is the minimum degree such that there is a polynomial of that degree approximating f up to additive error ε in the interval [−1,1]. We also show that the classical query complexity is lower bounded by Ω((s/2)(deg2ε(f)−1)/6) for any s≥ 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. The main techniques we used are the dual polynomial method for functions over the reals, linear semi-infinite programming, and tridiagonal matrices.

Problem

• Quantum Methods
• Hermitian Matrix
• Function Complexity

Innovation

• Quantum Computing Advantage
• Hermitian Matrix Computations
• Dual Polynomial Methodology