Classical simulations of quantum circuits are essential for verifying and benchmarking quantum algorithms, particularly for large circuits, where computational demands increase exponentially with the number of qubits. Among available methods, the classical simulation of quantum circuits inspired by density functional theory -- the so-called QC-DFT method, shows promise for large circuit simulations as it approximates the quantum circuits using single-qubit reduced density matrices to model multi-qubit systems. However, the QC-DFT method performs very poorly when dealing with multi-qubit gates. In this work, we introduce a novel CNOT "functional" that leverages neural networks to generate unitary transformations, effectively mitigating the simulation errors observed in the original QC-DFT method. For random circuit simulations, our modified QC-DFT enables efficient computation of single-qubit marginal measurement probabilities, or single-qubit probability (SQPs), and achieves lower SQP errors and higher fidelities than the original QC-DFT method. Despite limitations in capturing full entanglement and joint probability distributions, we find potential applications of SQPs in simulating Shor's and Grover's algorithms for specific solution classes. These findings advance the capabilities of classical simulations for some quantum problems and provide insights into managing entanglement and gate errors in practical quantum computing.

翻译：量子线路的经典模拟对于验证和基准测试量子算法至关重要，尤其对于大规模线路，其计算需求随量子比特数量呈指数增长。在现有方法中，受密度泛函理论启发的量子线路经典模拟——即所谓的QC-DFT方法，通过使用单量子比特约化密度矩阵来建模多量子比特系统，为大规模线路模拟提供了前景。然而，QC-DFT方法在处理多量子比特门时表现极差。本工作中，我们引入了一种新颖的CNOT“泛函”，它利用神经网络生成幺正变换，有效缓解了原始QC-DFT方法中观察到的模拟误差。对于随机线路模拟，我们改进的QC-DFT方法能够高效计算单量子比特边际测量概率（或称单量子比特概率，SQPs），并且相比原始QC-DFT方法实现了更低的SQP误差和更高的保真度。尽管在捕捉完整纠缠态和联合概率分布方面存在局限，我们发现SQPs在模拟特定解类别的Shor算法和Grover算法中具有潜在应用价值。这些发现提升了经典模拟处理某些量子问题的能力，并为管理实际量子计算中的纠缠和门误差提供了见解。