Daksh Shami | Searcharxiv

arXiv2024

Character Complexity: A Novel Measure for Quantum Circuit Analysis

In the rapidly evolving field of quantum computing, quantifying circuit complexity remains a critical challenge. This paper introduces 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.

quant-ph↗

arXiv2024

Bridging Classical and Quantum: Group-Theoretic Approach to Quantum Circuit Simulation

Efficiently simulating quantum circuits on classical computers is a fundamental challenge in quantum computing. This paper presents a novel theoretical approach that achieves substantial speedups over existing simulators for a wide class of quantum circuits. The technique leverages advanced group theory and symmetry considerations to map quantum circuits to equivalent forms amenable to efficient classical simulation. Several fundamental theorems are proven that establish the mathematical foundations of this approach, including a generalized Gottesman-Knill theorem. The potential of this method is demonstrated through theoretical analysis and preliminary benchmarks. This work contributes to the understanding of the boundary between classical and quantum computation, provides new tools for quantum circuit analysis and optimization, and opens up avenues for further research at the intersection of group theory and quantum computation. The findings may have implications for quantum algorithm design, error correction, and the development of more efficient quantum simulators.

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