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  • Open Access

Quantum Computational Complexity of Matrix Functions

Santiago Cifuentes1, Samson Wang2,*, Thais L. Silva3, Mario Berta4,5, and Leandro Aolita3

  • *Contact author: samsonwang@outlook.com
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PRX Quantum 7, 020364 – Published 18 June, 2026

DOI: https://doi.org/10.1103/g5x4-jcsz

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, we compute a matrix element of f(A) or a local measurement on f(A)|0⟩⊗n, with |0⟩⊗n being 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 and 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 poly(n), both problems can be efficiently classically simulated when A has O(logn) nonzero 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(logn). Our work provides a catalog of efficient quantum and classical algorithms for fundamental linear-algebra tasks.

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Popular Summary

Many candidate applications for quantum algorithms can be framed as a matrix function problem. This paper provides a sweeping assessment of the amenability of matrix function problems to quantum advantage, by characterizing regimes which are easy for quantum computers and hard for classical one or, conversely, easy for classical computers. Different building-block functions that are popular and prominent in the literature are studied, as well as a spectrum of many parameter regimes. Along the way the paper gives a simple-to-test formula for assessing advantage and also demonstrates that the format in which data is provided in can have a strong effect on computational complexity.

Article Text

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PRX Quantum 7, 020364– Published 18 June, 2026
  • Received 8 September 2025
  • Revised 25 November 2025
  • Accepted 2 March 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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