[Paper Review] Quantum algorithms for spectral sums

Alessandro Luongo, Changpeng Shao|arXiv (Cornell University)|Nov 12, 2020

Quantum Computing Algorithms and Architecture62 references4 citations

TL;DR

This paper presents new quantum algorithms for estimating spectral sums of symmetric positive definite matrices, such as the log-determinant, von Neumann entropy, trace of inverse, and Schatten-p norms. By leveraging quantum singular value transformation and quantum trace estimation, the algorithms achieve sub-linear runtime in matrix size with polynomial dependence on condition number and error, outperforming classical randomized methods that scale linearly with matrix non-zero entries.

ABSTRACT

We propose new quantum algorithms for estimating spectral sums of positive semi-definite (PSD) matrices. The spectral sum of an PSD matrix $A$, for a function $f$, is defined as $ ext{Tr}[f(A)] = \sum_j f(λ_j)$, where $λ_j$ are the eigenvalues of $A$. Typical examples of spectral sums are the von Neumann entropy, the trace of $A^{-1}$, the log-determinant, and the Schatten $p$-norm, where the latter does not require the matrix to be PSD. The current best classical randomized algorithms estimating these quantities have a runtime that is at least linearly in the number of nonzero entries of the matrix and quadratic in the estimation error. Assuming access to a block-encoding of a matrix, our algorithms are sub-linear in the matrix size, and depend at most quadratically on other parameters, like the condition number and the approximation error, and thus can compete with most of the randomized and distributed classical algorithms proposed in the literature, and polynomially improve the runtime of other quantum algorithms proposed for the same problems. We show how the algorithms and techniques used in this work can be applied to three problems in spectral graph theory: approximating the number of triangles, the effective resistance, and the number of spanning trees within a graph.

Motivation & Objective

• To develop quantum algorithms that efficiently estimate spectral sums of symmetric positive definite matrices, which are computationally expensive in classical settings.
• To reduce the runtime dependence on matrix size from linear (classical) to sub-linear (quantum), leveraging quantum access to matrix data.
• To enable faster estimation of key spectral quantities like log-determinant, trace of inverse, von Neumann entropy, and Schatten-p norms for use in machine learning, statistics, and computational chemistry.
• To generalize quantum algorithms beyond SPD matrices by extending to singular values and non-SPD cases, such as Schatten-p norms.
• To identify limitations of current quantum approaches, such as challenges in estimating the Estrada index due to negative eigenvalues, and propose future directions.

Proposed method

• The algorithms use quantum block-encoding to represent the matrix A, enabling efficient manipulation of its spectrum via quantum singular value transformation (SVT).
• A key subroutine is quantum trace estimation, which estimates the trace of a function of the matrix using amplitude estimation and controlled unitaries.
• Polynomial approximation techniques, such as Chebyshev or Taylor expansions, are used to approximate functions like log(λ) or 1/λ applied to eigenvalues.
• The framework assumes access to a (μ, log n, 0)-block-encoding of A/μ, which allows efficient implementation of f(A) up to desired precision.
• Error bounds are derived by combining approximation error from polynomial expansion and estimation error from trace estimation, leading to relative error guarantees.
• For the trace of inverse, the method constructs a block-encoding of a rational approximation to 1/λ, then applies trace estimation with controlled error.

Experimental results

Research questions

• RQ1 Can quantum algorithms achieve sub-linear runtime in matrix size for estimating spectral sums like log-determinant and von Neumann entropy?
• RQ2 How does the quantum approach scale with respect to condition number, error tolerance, and matrix sparsity compared to classical randomized algorithms?
• RQ3 What are the limitations of current quantum algorithms when estimating spectral sums involving negative eigenvalues, such as the Estrada index?
• RQ4 Can quantum singular value transformation be adapted to handle non-SPD matrices and functions defined on eigenvalues rather than singular values?
• RQ5 What are the practical applications of these quantum spectral sum estimators in machine learning, statistics, and network analysis?

Key findings

• The proposed quantum algorithms achieve a runtime of Õ(μ²κ²‖A‖/ε) for estimating the trace of inverse, where μ is the scaling factor, κ the condition number, and ε the error tolerance.
• For spectral sums like log-determinant and von Neumann entropy, the algorithms scale poly-logarithmically in n (matrix size), polynomially in condition number and error, enabling sub-linear speedups over classical methods.
• The framework generalizes to non-SPD matrices via singular values, enabling estimation of Schatten-p norms with similar complexity bounds.
• The algorithm for the trace of inverse achieves relative error ε with complexity Õ(μ²κ²‖A‖/ε), improving upon classical methods that scale linearly with the number of non-zero entries.
• The Estrada index cannot be efficiently estimated using current SVT-based methods due to negative eigenvalues; alternative approaches using quantum signal processing or spectral shifting are suggested.
• The work establishes a foundation for quantum subroutines in applications such as covariance matrix log-determinant estimation, triangle counting, and spanning tree enumeration in graphs.