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Quantum computational complexity of matrix functions

Santiago Cifuentes, Samson Wang, Thais L. Silva, Mario Berta, Leandro Aolita

TL;DR

The paper provides a comprehensive complexity map for estimating matrix-function properties, focusing on two problems—matrix-element estimation and localized measurements on $f(A)$—across four function families. By analyzing sparse and Pauli access models, it establishes BQP-completeness for monomials, Chebyshev polynomials, time evolution, and inversion in broad parameter regimes, while also identifying classically easy regimes under stronger norm bounds or structured Pauli representations. It shows that quantum advantage hinges on specific matrix structure and problem parameters, notably revealing that Pauli sparsity can eliminate hardness in some cases, whereas general sparse access often preserves quantum hardness. The results offer a catalog of fundamental quantum and classical algorithms for linear-algebra tasks, clarifying when quantum speedups are plausible and when dequantization-like classical algorithms can match performance. This work thus provides both a theoretical framework and practical guidance for evaluating quantum advantages in matrix-function computations.

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.

Quantum computational complexity of matrix functions

TL;DR

The paper provides a comprehensive complexity map for estimating matrix-function properties, focusing on two problems—matrix-element estimation and localized measurements on —across four function families. By analyzing sparse and Pauli access models, it establishes BQP-completeness for monomials, Chebyshev polynomials, time evolution, and inversion in broad parameter regimes, while also identifying classically easy regimes under stronger norm bounds or structured Pauli representations. It shows that quantum advantage hinges on specific matrix structure and problem parameters, notably revealing that Pauli sparsity can eliminate hardness in some cases, whereas general sparse access often preserves quantum hardness. The results offer a catalog of fundamental quantum and classical algorithms for linear-algebra tasks, clarifying when quantum speedups are plausible and when dequantization-like classical algorithms can match performance. This work thus provides both a theoretical framework and practical guidance for evaluating quantum advantages in matrix-function computations.

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 and a Hermitian matrix , compute a matrix element of or compute a local measurement on , with an -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 both problems can be efficiently classically simulated when has 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 . Our work provides a catalog of efficient quantum and classical algorithms for fundamental linear-algebra tasks.

Paper Structure

This paper contains 22 sections, 50 theorems, 110 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Setting and motivation
  3. Related work
  4. Results
  5. Basic definitions
  6. Summary of results
  7. Discussion
  8. Technical background
  9. BQP and computational complexity.
  10. Hamiltonian simulation for sparse and Pauli access.
  11. Polynomial approximations.
  12. Quantum algorithms for matrix function estimation.
  13. Classical algorithms for matrix functions.
  14. Detailed statements and main proofs
  15. Monomials
  16. ...and 7 more sections

Key Result

Lemma 12

Given sparse access to s-sparse $A \in \mathds{C}^{N \times N}$ it is possible to construct a circuit $U$ such that $\left\lVert e^{-iAt}-U \right\rVert\leq \alpha$ in time polynomial in ($\frac{1}{\alpha}$, $t$, $s$, $\|A\|$, $\log N$) aharononv2007AdiabaticQStateGenerationberry2005EffQAlgSimmSpars

Figures (2)

  • Figure 1: Diagram indicating the complexity of the studied problems. We use the acronyms $\textsc{MM}$ (matrix monomial), $\textsc{CP}$ (Chebyshev polynomials), $\textsc{MI}$ (matrix inversion), $\textsc{TE}$ (time evolution), and $\textsc{ALL}$ (all the previous functions). The superscript indicates the access model: Spar (Sparse), Pauli (Pauli), or Both (the problem belongs to the complexity class for both access models). We denote $\left\lVert A \right\rVert$ as the operator (or spectral) norm of $A$, $\left\lVert A \right \rVert_{1}$ as its induced 1-norm, $\lambda_{A}$ the vector $\ell_1$-norm of its Pauli coefficients, $L$ as the number of Pauli terms, $\#_{\text{nz}}$ its number of non-zero elements in the computational basis, $\kappa_{A}$ its condition number, $t$ as the evolution time, and $\eta>0$ is an $\mathcal{O}(1)$ number. Although the local measurement problem appears, at first sight, a more natural task for quantum algorithms than the matrix element problem, interestingly, we find an almost-complete match between the two problems for almost all settings studied (see Table \ref{['tab:summary']}). The only potentially discrepant cases are $\textsc{CP}$ for inverse polynomially small matrix norms (purple region) and $\textsc{TE}$ for constant time (blue region), for which our hardness proof works only for the matrix element problem. All other results sketched in the figure hold for both problems. When not indicated, it is assumed that $\|A\|\leq 1$ and all other problem parameters (sparsity, inverse precision, problem-specific parameters) scale polynomially in the input size (which is polylogarithmic in the matrix dimension). Hence, for instance, $\textsc{MM}^{\textsc{Spar}}$ indicates both the matrix element and local measurement problems for $f_m(A)=A^m$, for $m$ polynomial in input size , where $A$ is given through the sparse access model.
  • Figure 2: Circuit extension $C'$ of the input circuit $C$ used in janzing2006BQPmixing. Note that the amplitude $\langle \boldsymbol{x} |\langle 0 | C' | \boldsymbol{x} \rangle| 0 \rangle$ is linearly related to the measurement probability that decides the problem. In turn, the estimation of such amplitude reduces to estimating an element of a monomial of a sparse matrix $A$, defined in Eq. \ref{['eq:A']}.

Theorems & Definitions (99)

  • Definition 1: Function of a matrix (eigenvalue transformation)
  • Definition 2: Sparse access
  • Definition 3: Pauli query access and Pauli-sparseness
  • Definition 9: Promise problems and BQP
  • Definition 10: Karp reductions, BQP-hard and BQP-complete
  • Lemma 12: Hamiltonian simulation for Sparse and Pauli access
  • Lemma 13: Polynomial approximation of $\frac{1}{x}$ (childs2015QLinSysExpPrec, Lemmas 17 and 19)
  • Lemma 14: Polynomial approximation of $e^{ixt}$ (gilyen2018QSingValTransf, Lemmas 57 and 59)
  • Lemma 15: Quantum algorithm for entry estimation. (Janzing/Wocjan janzing2006BQPmixing, Lemma 2)
  • Lemma 16: Quantum algorithm for normalized local measurement
  • ...and 89 more