lindblad-bench: Cache Hierarchy Analysis of Lindblad Simulation

Status: Preprint on arXiv

GitHub: github.com/rylanmalarchick/lindblad-bench

Paper: arXiv:2603.18052 -- "Cache Hierarchy and Vectorization Analysis of Lindblad Master Equation Simulation for Near-Term Quantum Control"

Overview

For near-term quantum control -- optimal pulse engineering, feedback controllers, robustness sweeps -- the simulation bottleneck is dense matrix-vector multiplication on small systems (d=3 to d=27). At these scales, Python dispatch overhead dominates actual arithmetic, and the working set fits entirely in CPU cache. This paper analyzes where the time actually goes and what compiler/memory layout choices matter.

The core observation: a 9x9 complex matrix-vector product (d=3 transmon) is 648 FLOPs and 1.3 KB. That fits in L1 cache. The bottleneck isn't the math -- it's everything around it.

Key Results

Cache Hierarchy Scaling

Approach

1. Bare-metal C library -- no dependencies, C11, cache-aligned allocations
2. Pade [13/13] matrix exponential -- Higham 2005 scaling-and-squaring for propagator precomputation
3. Roofline modeling -- measured arithmetic intensity vs peak throughput to identify bottleneck regime
4. Compiler flag sweep -- scalar, O2, O3, O3+native, O3+native+ffast-math across GCC builds
5. Assembly analysis -- Godbolt verification of AVX2 FMA generation (vfmadd231pd)

The key finding: -ffast-math is essential for GCC to auto-vectorize complex arithmetic. Without it, strict IEEE 754 compliance prevents the compiler from reordering floating-point operations needed for SIMD.

Structure-of-Arrays Layout

The standard approach stores complex numbers as interleaved (re, im) pairs (Array-of-Structures). Switching to separate real and imaginary arrays (Structure-of-Arrays) enables wider SIMD loads and eliminates cross-lane shuffles:

AoS: [re0, im0, re1, im1, re2, im2, ...]  -- 128-bit loads mix re/im
SoA: [re0, re1, re2, re3, ...]             -- 256-bit loads are pure re
     [im0, im1, im2, im3, ...]             -- 256-bit loads are pure im

Connection to QubitOS

This library is the standalone performance study informing the QubitOS Tier 2 solver (C fast path for d <= 27). The FFI interface in QubitOS is designed to call into this library. The cache hierarchy analysis explains why small-system simulation should use precomputed propagators rather than adaptive ODE integrators.

Technology Stack

Links

• GitHub Repository
• arXiv Preprint (arXiv:2603.18052)

Citation

Malarchick, R. (2026). "Cache Hierarchy and Vectorization Analysis of
Lindblad Master Equation Simulation for Near-Term Quantum Control."
arXiv:2603.18052.