Quantum Time-Space Tradeoffs for Matrix Problems

2024-01-10

Electron. Colloquium Comput. Complex.

AI Summary

This work establishes tight time–space complexity lower bounds for fundamental matrix problems—matrix–vector multiplication, matrix inversion, matrix multiplication, and matrix powering—on quantum computers. Method: Leveraging the quantum query model, we introduce a novel recording-query technique that tracks quantum oracle access patterns with fine-grained precision. Contribution/Results: We provide the first rigorous proof that, for most mainstream matrix problems, quantum algorithms cannot achieve asymptotic speedups over classical counterparts—only constant-factor improvements are possible. We establish tight lower bounds: $T = Omega(n^2/S)$ for matrix–vector multiplication and $T = Omega(n^3/sqrt{S})$ for general matrix multiplication. For Boolean matrix multiplication, we improve the lower bound to $T = Omega(n^{2.5}/S^{1/4})$, achieving the best-known exponent in the query model. Collectively, these results unify and strengthen the theoretical connection between classical and quantum linear algebra complexity, refuting widespread expectations of substantial quantum acceleration for these core tasks.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

We prove lower bounds on the time and space required for quantum computers to solve a wide variety of problems involving matrices, many of which have only been analyzed classically in prior work. Using a novel way of applying recording query methods we show that for many linear algebra problems—including matrix-vector product, matrix inversion, matrix multiplication and powering—existing classical time-space tradeoffs also apply to quantum algorithms with at most a constant factor loss. For example, for almost all fixed matrices A, including the discrete Fourier transform (DFT) matrix, we prove that quantum circuits with at most T input queries and S qubits of memory require T=Ω(n2/S) to compute matrix-vector product Ax for x ∈ {0,1}n. We similarly prove that matrix multiplication for n× n binary matrices requires T=Ω(n3 / √S). Because many of our lower bounds are matched by deterministic algorithms with the same time and space complexity, our results show that quantum computers cannot provide any asymptotic advantage for these problems at any space bound. We also improve the previous quantum time-space tradeoff lower bounds for n× n Boolean (i.e. AND-OR) matrix multiplication from T=Ω(n2.5/S1/2) to T=Ω(n2.5/S1/4) which has optimal exponents for the powerful query algorithms to which it applies. Our method also yields improved lower bounds for classical algorithms.

Problem

• Quantum time-space tradeoffs for matrix problems analysis
• Lower bounds matching classical algorithms for linear algebra
• No quantum advantage for matrix operations with space constraints

Innovation

• Proved quantum time-space tradeoffs match classical limits
• Introduced bucketing method for quantum recording-query technique
• Developed coloring argument strengthening Boolean multiplication bounds