Computational Complexity Analysis of Quantum Algorithms using Random Matrix Theory

Jincheng Zhang

Published Sep 1, 2026• 0 citations

Abstract

This paper investigates the application of Random Matrix Theory (RMT) to the analytical complexity assessment of quantum algorithms. Traditional methods for estimating the computational complexity of quantum algorithms often rely on heuristics and approximations, frequently leading to overly pessimistic or overly optimistic results. We propose a novel approach utilizing RMT to model the statistical behavior of quantum circuits. Specifically, we leverage the eigenvalues and eigenvectors of random matrices to characterize the evolution of quantum states under arbitrary quantum circuits. This allows for a more precise estimation of the number of quantum gates (operations) required to achieve a desired level of accuracy in solving a given problem. We demonstrate, through illustrative examples, how RMT can provide tighter and more accurate bounds on the computational complexity compared to conventional techniques. The core claim of this work is that RMT offers a fundamentally new perspective on complexity analysis, promising to reveal deeper insights into the performance characteristics of quantum algorithms and potentially guide the design of more efficient quantum solutions.

DOI: 10.5281/zenodo.22220217

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