A Note on Fine-Grained Quantum Reductions for Linear Algebraic Problems

2025-09-23

AI Summary

This work investigates the fine-grained complexity equivalence between fundamental linear algebra problems—including determinant computation, the matrix triple trace $mathrm{tr}(ABC)$, and the trace of the matrix inverse—and matrix multiplication in the quantum computing model. Method: We introduce the first direct quantum reduction based on the Bernstein–Vazirani algorithm, reducing matrix multiplication to $mathrm{tr}(ABC)$, and systematically extend it to multiple linear algebraic problems. Contribution/Results: We prove that all target problems admit quantum reductions to $n imes n$ matrix multiplication in time $O(T(n)) + ilde{O}(n^2)$, where $T(n)$ denotes the quantum time complexity of matrix multiplication. This establishes a tight complexity-theoretic linkage: any quantum speedup for any of these problems implies an equivalent speedup for matrix multiplication. Our result provides a unified complexity framework for quantum linear algebra and introduces a novel paradigm for designing quantum algorithms via fine-grained reductions.

Technology Category

• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Computer Vision: Other Foundations of Computer Vision

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Applications of cryptography
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We observe that any $T(n)$ time algorithm (quantum or classical) for several central linear algebraic problems, such as computing $det(A)$, $tr(A^3)$, or $tr(A^{-1})$ for an $n imes n$ integer matrix $A$, yields a $O(T(n)) + ilde O(n^2)$ time extit{quantum algorithm} for $n imes n$ matrix-matrix multiplication. That is, on quantum computers, the complexity of these problems is essentially equivalent to that of matrix multiplication. Our results follow by first observing that the Bernstein-Vazirani algorithm gives a direct quantum reduction from matrix multiplication to computing $tr(ABC)$ for $n imes n$ inputs $A,B,C$. We can then reduce $tr(ABC)$ to each of our problems of interest. For the above problems, and many others in linear algebra, their fastest known algorithms require $Θ(n^ω)$ time, where $ωapprox 2.37$ is the current exponent of fast matrix multiplication. Our finding shows that any improvements beyond this barrier would lead to faster quantum algorithms for matrix multiplication. Our results complement existing reductions from matrix multiplication in algebraic circuits [BCS13], and reductions that work for standard classical algorithms, but are not tight -- i.e., which roughly show that an $O(n^{3-δ})$ time algorithm for the problem yields an $O(n^{3-δ/3})$ matrix multiplication algorithm [WW10].

Problem

• Establishing quantum equivalence between matrix multiplication and linear algebraic problems
• Reducing matrix multiplication to trace computations using Bernstein-Vazirani algorithm
• Showing quantum algorithmic improvements require breaking matrix multiplication barrier

Innovation

• Quantum reduction from matrix multiplication to trace computation
• Bernstein-Vazirani algorithm enables efficient trace calculations
• Linear algebra problems equivalent to matrix multiplication complexity