Efficiently simulating quantum circuits on classical computers is a fundamental challenge in quantum computing. This paper presents a novel theoretical approach that achieves exponential speedups (polynomial runtime) 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.

翻译：在经典计算机上高效模拟量子电路是量子计算领域的一项基础性挑战。本文提出了一种新颖的理论方法，针对广泛类别的量子电路，相比现有模拟器实现了指数级加速（多项式运行时间）。该技术利用先进的群论与对称性考量，将量子电路映射至适于高效经典模拟的等价形式。本文证明了若干奠定该方法数学基础的基本定理，包括一个广义的Gottesman-Knill定理。通过理论分析和初步基准测试，展示了该方法的潜力。本工作有助于理解经典计算与量子计算的边界，为量子电路分析与优化提供了新工具，并开辟了群论与量子计算交叉领域的进一步研究方向。这些发现可能对量子算法设计、纠错以及更高效量子模拟器的开发产生影响。