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.

💡 Research Summary

The paper proposes a new theoretical framework for classically simulating quantum circuits by exploiting group‑theoretic structure. The central idea is to view a quantum circuit as an element of a finite group G under matrix multiplication and to express any such element as a linear combination of irreducible representations weighted by their characters. The authors first prove a “Character Function Decomposition” theorem (Theorem 1), showing that any element u in the complex group algebra ℂ