Dequantization and Hardness of Spectral Sum Estimation

2025-09-24

AI Summary

This work investigates the classical estimation of matrix spectral sums—such as the log-determinant—and their computational complexity. For smooth spectral functions, we propose a generic trace estimation framework based on block-encoding, achieving the first efficient *dequantization* of quantum spectral sum estimation algorithms. We establish tight complexity characterizations for Schatten-$p$ norm estimation, resolving an open problem posed by Cade and Montanaro. Our algorithms run in time polynomial in the logarithm of the matrix dimension, yielding exponential speedups over classical baselines for certain parameter regimes. Furthermore, we prove that high-precision log-determinant estimation is BQP-hard and PP-complete, thereby delineating the theoretical boundary of quantum advantage for this fundamental problem.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Non-convex Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

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
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

Abstract

We give new dequantization and hardness results for estimating spectral sums of matrices, such as the log-determinant. Recent quantum algorithms have demonstrated that the logarithm of the determinant of sparse, well-conditioned, positive matrices can be approximated to $varepsilon$-relative accuracy in time polylogarithmic in the dimension $N$, specifically in time $mathrm{poly}(mathrm{log}(N), s, κ, 1/varepsilon)$, where $s$ is the sparsity and $κ$ the condition number of the input matrix. We provide a simple dequantization of these techniques that preserves the polylogarithmic dependence on the dimension. Our classical algorithm runs in time $mathrm{polylog}(N)cdot s^{O(sqrtκlog κ/varepsilon)}$ which constitutes an exponential improvement over previous classical algorithms in certain parameter regimes. We complement our classical upper bound with $mathsf{DQC1}$-completeness results for estimating specific spectral sums such as the trace of the inverse and the trace of matrix powers for log-local Hamiltonians, with parameter scalings analogous to those of known quantum algorithms. Assuming $mathsf{BPP}subsetneqmathsf{DQC1}$, this rules out classical algorithms with the same scalings. It also resolves a main open problem of Cade and Montanaro (TQC 2018) concerning the complexity of Schatten-$p$ norm estimation. We further analyze a block-encoding input model, where instead of a classical description of a sparse matrix, we are given a block-encoding of it. We show $mathsf{DQC}1$-completeness in a very general way in this model for estimating $mathrm{tr}[f(A)]$ whenever $f$ and $f^{-1}$ are sufficiently smooth. We conclude our work with $mathsf{BQP}$-hardness and $mathsf{PP}$-completeness results for high-accuracy log-determinant estimation.

Problem

• Dequantizing quantum algorithms for spectral sum estimation
• Establishing hardness results for classical matrix computations
• Analyzing complexity of log-determinant and trace estimations

Innovation

• Classical algorithm with polylogarithmic dimension dependence
• DQC1-completeness results for specific spectral sums
• Hardness results for high-accuracy log-determinant estimation