Character Complexity: A Novel Measure for Quantum Circuit Analysis

Abstract

In the rapidly evolving field of quantum computing, quantifying circuit complexity remains a critical challenge. This paper introduces \textit{character complexity}, a novel measure that bridges group-theoretic concepts with practical quantum computing concerns. By leveraging tools from representation theory, I prove several key properties of character complexity and establish a surprising connection to the classical simulability of quantum circuits. This new measure offers a fresh perspective on the complexity landscape of quantum algorithms, potentially reshaping our understanding of quantum-classical computational boundaries. I present innovative visualization methods for character complexity, providing intuitive insights into the structure of quantum circuits. The empirical results reveal intriguing scaling behaviors with respect to qubit and gate counts, opening new avenues for quantum algorithm design and optimization. This work not only contributes to the theoretical foundations of quantum complexity but also offers practical tools for the quantum computing community. As quantum hardware continues to advance, character complexity could play a crucial role in developing more efficient quantum algorithms and in exploring the fundamental limits of quantum computation.