Quantum computational complexity of matrix functions

TL;DR

This paper explores the boundary between classical and quantum computational capabilities in estimating matrix function properties, identifying regimes where problems are BQP-hard or classically simulable, and providing a comprehensive complexity landscape.

Contribution

It characterizes the quantum and classical complexity of matrix function estimation problems across various regimes, revealing a hierarchy of hardness and efficient simulation conditions.

Findings

BQP-complete for certain matrix functions in specific regimes

Classical efficient algorithms exist for polynomial functions with sparse Pauli representations

Quantum hardness persists for non-polynomial functions even when classical algorithms are efficient

Abstract

We investigate the dividing line between classical and quantum computational power in estimating properties of matrix functions. More precisely, we study the computational complexity of two primitive problems: given a function $f$ and a Hermitian matrix $A$, compute a matrix element of $f(A)$ or compute a local measurement on $f(A)|0\rangle^{\otimes n}$, with $|0\rangle^{\otimes n}$ an $n$-qubit reference state vector, in both cases up to additive approximation error. We consider four functions -- monomials, Chebyshev polynomials, the time evolution function, and the inverse function -- and probe the complexity across a broad landscape covering different problem input regimes. Namely, we consider two types of matrix inputs (sparse and Pauli access), matrix properties (norm, sparsity), the approximation error, and function-specific parameters. We identify BQP-complete forms of both problems for each function and then toggle the problem parameters to easier regimes to see where hardness remains, or where the problem becomes classically easy. As part of our results, we make concrete a hierarchy of hardness across the functions; in parameter regimes where we have classically efficient algorithms for monomials, all three other functions remain robustly BQP-hard, or hard under usual computational complexity assumptions. In identifying classically easy regimes, among others, we show that for any polynomial of degree $\mathrm{poly}(n)$ both problems can be efficiently classically simulated when $A$ has $O(\log n)$ non-zero coefficients in the Pauli basis. This contrasts with the fact that the problems are BQP-complete in the sparse access model even for constant row sparsity, whereas the stated Pauli access efficiently constructs sparse access with row sparsity $O(\log n)$. Our work provides a catalog of efficient quantum and classical algorithms for fundamental linear-algebra tasks.