Cache Hierarchy and Vectorization Analysis of Lindblad Master Equation Simulation for Near-Term Quantum Control

Abstract: Simulation of open quantum systems via the Lindblad master equation is a computational bottleneck in near-term quantum control workflows, including optimal pulse engineering (GRAPE), trajectory-based robustness analysis, and feedback controller design. For the system sizes relevant to near-term quantum control ($d = 3$ for a single transmon with leakage, $d = 9$ for two-qubit, and $d = 27$ for three-qubit), the dominant cost per timestep is a $(d^2 \times d^2)$ complex matrix-vector multiplication: a $9\times9$, $81\times81$, or $729\times729$ dense matvec, respectively. The working set sizes (1.5 KB, 105 KB, and 8.1 MB) straddle the L1, L2, and L3 cache boundaries of modern CPUs, making this an ideal system for cache-hierarchy performance analysis. We characterize the arithmetic intensity ($\approx 1/2$ FLOP/byte in the large-$d$ limit), construct a Roofline model for the propagation kernel, and systematically vary compiler flags and data layout to isolate the contributions of auto-vectorization, fused multiply-add, and structure-of-arrays (SoA) memory layout. We show that SoA layout combined with -O3 -march=native -ffast-math yields $2$--$4\times$ speedup over scalar array-of-structures baselines, and that -ffast-math is essential for enabling GCC auto-vectorization of complex arithmetic. These results motivate a set of concrete recommendations for authors of quantum simulation libraries targeting near-term system sizes.