Quantum computational complexity of matrix functions

2024-10-17

arXiv.org

AI Summary

This work systematically characterizes the computational boundaries between classical and quantum algorithms for estimating properties of matrix functions $f(A)$ of a Hermitian matrix $A$, including matrix elements and local measurements on $f(A)|0 angle^{otimes n}$. Method: We analyze four fundamental function classes—monomials, Chebyshev polynomials, time evolution $e^{iAt}$, and matrix inversion $A^{-1}$—under distinct input models (e.g., Pauli sparsity vs. sparse-access oracles) and parameter regimes, employing Hamiltonian simulation, polynomial approximation, Pauli expansion analysis, and complexity-theoretic reductions. Contribution/Results: We establish rigorous complexity hierarchies, revealing that for $O(log n)$-Pauli-sparse $A$, polynomial functions admit efficient classical simulation, whereas under the sparse-access model the same problem remains BQP-complete—demonstrating the critical role of input representation. This is the first comprehensive complexity mapping for these four function classes, providing a foundational quantum/classical solvability classification for basic linear-algebraic tasks.

Technology Category

• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Responsible Web: Algorithmic accountability and transparency on the web

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 $f(A)|0 angle^{otimes n}$, with $|0 angle^{otimes n}$ 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.

Problem

• Distinguish classical and quantum power in matrix function estimation
• Study complexity of computing matrix elements and local measurements
• Identify BQP-complete and classically easy regimes for functions

Innovation

• Quantum algorithms for matrix function estimation
• BQP-complete problems in sparse access
• Classical simulation for low Pauli sparsity