Deep Learning-Integrated Pairwise-Qubit Subsystems for Highly Efficient Quantum Circuit Simulation

Classically simulating large quantum circuits is well-known to be difficult due to its exponential scaling O(2n) in computations and memory with respect to the number of qubits (n). We propose a novel classical simulation framework for quantum circuits, combining pairwise-qubit subsystems (PQS) with deep learning, thus called PQS simulation, to achieve scalable and highly efficient circuit evaluation. By representing quantum states as sequences of reduced density matrices and modeling two-qubit interactions through learnable unitary operators, this representation yields quadratic space and linear time per elementary gate, i.e., O(n2) memory and O(n) time per-gate update. A custom multi-hidden long short-term memory architecture is trained to learn and predict the evolution of pairwise entangled qubit subsystems. We demonstrate the applicability of the method to Shor period-finding circuits in the special case where the period is a power-of-two, for which the ideal circuit yields a unique outcome and the period can be inferred from single-qubit measurement probabilities. Despite being trained on smaller circuits, the model generalizes to higher qubit counts than those required during training, achieving success probabilities up to 35.48% on the 18-qubit circuit, while simultaneously offering significant run-time advantages over exact simulation (up to 66 times faster on the 24-qubit circuit). Because Shor’s classical post-processing is cheap, we can wrap PQS simulation in an error-tolerant loop for these circuit instances, which results in an expected runtime of O(n4/(1 − ε)), where 1 − ε is the probability that PQS simulation returns the correct period. For comparisons, we provide FLOP/s-normalized projections: our measured 36-qubit Quantum Fourier Transform runtime projects to 0.158 s on a single Argonne Theta node (versus a reported 78.98 h), and 40-qubit Shor projects to 19.3 s on a 2,048-A100 cluster (versus ∼ 200 s). We also analyze failure modes via the distribution of single-qubit measurement probabilities and find errors concentrated in the lowest index qubit, suggesting that targeted mitigation on only a few qubits may suffice. The PQS simulation exhibits a promising direction for approximate, flexible, and scalable simulation of quantum algorithms on classical hardware.