Benchmarks

All benchmarks use the same Lindblad master equation for $M$ coupled cavities with Fock truncation $d=3$:

\[\dot{\rho} = -i[H, \rho] + \gamma \sum_m \left( a_m \rho a_m^\dagger - \frac{1}{2}\{a_m^\dagger a_m, \rho\} \right)\]

with $H = \sum_m \chi\, n_m(n_m - 1) + \sum_{n \ne m} \kappa\, a_n^\dagger a_m$, $\chi = 2\pi \times 0.2$, $\kappa = 2\pi \times 0.1$, $\gamma = 0.01$. Initial state: $|1,0,\ldots,0\rangle\langle 1,0,\ldots,0|$.

When to Use cuDensityMat

cuDensityMat uses tensor network contraction — it never materializes the full Liouvillian superoperator. For small systems ($M \le 6$), the per-action overhead makes it slower than sparse matrix approaches. For large systems ($M \ge 8$), it is the only viable option because the sparse Liouvillian exceeds GPU memory.

A100: CuQuantum.jl vs QuantumToolbox.jl

Same hardware (NVIDIA A100-SXM4-40GB, 12 vCPUs), same RK4 integrator, same time steps.

Full 100-step RK4 Simulation

At $M=6$, cuSPARSE is 8× faster than cuDensityMat. At $M=8$, the Liouvillian is $43\text{M} \times 43\text{M}$ sparse (~77 GB) — only cuDensityMat can run.

Single $L[\rho]$ Action

Batched Evolution (A100)

Many density matrices evolved with different coupling strengths in one kernel launch — the parameter sweep use case for quantum optimal control.

At small system sizes ($M=2\text{--}4$), native batching amortizes kernel launch overhead by up to 186×. At $M=6$, the per-action cost dominates and batching gives ~1.7× speedup.

Cross-Framework Comparison (Tesla T4)

Five frameworks, same problem. cuDensityMat benchmarks use RK4; QuantumToolbox.jl and QuTiP use adaptive solvers.

Full 100-step Simulation

On T4, CPU sparse wins at all sizes. The T4 has only 0.25 TFLOPS FP64 — cuDensityMat's tensor network contraction needs A100-class throughput to be competitive.

Single $L[\rho]$ Action (T4)

Wrapper Overhead

All three cuDensityMat wrappers call the same C library. The overhead at $M=6$ on T4:

Julia is faster due to direct ccall without Python GIL overhead. The gap narrows at large sizes where the cuDensityMat kernel dominates.

Validation

CPU and GPU produce bitwise-identical results for the static Liouvillian at $t=0$. All frameworks agree on final populations to within solver tolerances across 100 time steps.

Environment

Reproduction

Benchmark scripts are in benchmark/comparison/ and benchmark/batched/. See the comparison README for setup and run instructions.