Tracking Affine Subspace with Gaussian Elimination for Adaptive Quantum Circuit Simulation

Simulating quantum circuits on classical computers is challenging because conventional state-vector simulators are required to track 2 N amplitudes, a resource-intensive process. While sparse simulators that exploit state-support sparsity—where only a small subset of computational basis states carry nonzero amplitudes—offer highly efficient alternatives, they lose their advantage for circuits that generate dense quantum states. To address this, we propose an adaptive simulation technique that dynamically predicts state sparsity through a rapid pre-simulation assessment. Employing a novel application, Gaussian elimination on linear constraints, the proposed approach efficiently tracks an affine subspace of the state space to estimate the number of non-zero amplitudes without complex calculations. We emphasize that our technique specifically targets state-support sparsity rather than gate-level or unitary-matrix sparsity. Overall, this approach enables the system to select between full-state and sparse-state simulations, significantly improving speed and memory efficiency for sparse circuits as well as preserving dense-circuit performance.