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

Rylan Malarchick

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.

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 ( for a single transmon with leakage, for two-qubit, and for three-qubit), the dominant cost per timestep is a complex matrix-vector multiplication: a , , or 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 ( FLOP/byte in the large- 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 -- 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.
Paper Structure (16 sections, 8 equations, 2 figures, 4 tables)

This paper contains 16 sections, 8 equations, 2 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Background
  3. The Lindblad propagation kernel
  4. Cache hierarchy and the Roofline model
  5. Compiler auto-vectorization of complex matvec
  6. Methods
  7. C library implementation
  8. Compiler flag experiments
  9. Benchmarking methodology
  10. Results
  11. Roofline characterization
  12. Compiler vectorization analysis
  13. Discussion
  14. Recommendations for quantum simulation library authors
  15. Conclusion
  16. ...and 1 more sections

Figures (2)

  • Figure 1: Achieved bandwidth (GB/s) for each layout variant at -O3 -march=native -ffast-math, grouped by system dimension. The $x$-axis labels indicate the cache level that holds the working set. SoA outperforms both AoS and hand-written AVX2 at every size. Bandwidth decreases from L1 to L3, consistent with the cache hierarchy.
  • Figure 2: Achieved bandwidth (GB/s) for the SoA variant across compiler flag configurations and system dimensions. The -ffast-math flag produces the dominant speedup. Note the anomalous regression at $d = 9$ when adding -march=native without -ffast-math (see text).