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arXiv:2410.13937 · PRX Quantum

Quantum Computational Complexity of Matrix Functions

Santiago Cifuentes, Samson Wang, Thais de Lima Silva, Mario Berta, Leandro Aolita

Published Mar 2, 2026• 1 citation
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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- 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-, 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 ( log  n ) nonzero coefficients in the Pauli basis. This contrasts with the fact that the problems are BQP- 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.

DOI
10.1103/g5x4-jcsz

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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- 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-, 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 ( log  n ) nonzero coefficients in the Pauli basis. This contrasts with the fact that the problems are BQP- 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.
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Decomposing classical matrices into linear combinations of Pauli strings is a major bottleneck for end-to-end implementations of near-term quantum algorithms. In this work, we consider a promise version of this Pauli decomposition problem in which the matrix is guaranteed to have support on only k=poly(n)k = \mathsf{poly}(n)k=poly(n) Pauli strings and is given through classical sparse query access. Existing Pauli decomposition algorithms are designed for the generic, dense problem and do not inherently take advantage of this promised sparsity, so these approaches take time that is exponential in nnn. We present a randomized classical algorithm that does take advantage of this sparsity and recovers the exact Pauli decomposition with success probability at least 1−δ1 - δ1−δ, for any δδδ. Under the stated access model, the algorithm executes with query and runtime complexity that is polynomial in nnn, kkk, and log⁡(1/δ)\log(1/δ)log(1/δ). These results show that, even though finding the Pauli decomposition is exponentially hard for general matrices, it becomes efficiently solvable for matrices that are known to be sparse in the Pauli basis, a regime that is relevant to near-term quantum algorithms operating on structured classical input.
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