Probabilistic Simulation of Deep Quantum Circuits and Applications to Shor’s Algorithm

Abstract

The simulation of deep quantum circuits is traditionally bottlenecked by the exponential scaling of the Hilbert space. This thesis presents a classical simulation framework that circumvents this barrier by mapping quantum states to classical probability distributions using informationally complete positive operator-valued measures (IC-POVMs). By leveraging an autoregressive transformer architecture, this work redefines quantum simulation as a highdimensional generative modeling task, utilizing the self-attention mechanism to capture non-local multi-qubit correlations. To rigorously benchmark the model against volume-law entanglement, the circuits are generated dynamically using uncompiled logical primitives, including the Cuccaro ripple-carry adder routed through the Vedral-Barenco-Ekert (VBE) modular exponentiation architecture. Evaluated via a variance-stabilized mixed Monte Carlo framework, the transformer successfully simulated a 25-qubit quantum Fourier transform (QFT) applied to an entangled 1D cluster state, maintaining a classical fidelity of Fc > 0.99 over 325 gates. Furthermore, the model executed the complete, uncompiled Shor’s algorithm to factor N = 15. Across a depth of 301 gates and 22 qubits, the network sustained a classical fidelity of Fc > 0.97. These results demonstrate that attention-based neural networks can robustly track complex, reversible quantum arithmetic without suffering exponential error accumulation, thereby establishing a stricter classical baseline for proving practical quantum advantage.