XY Trotter — Dynamics / Quantum Algorithms for PDEs

Trotter-Suzuki decomposition is the quantum-computing analog of Strang splitting for classical PDEs. The same numerical-analysis question that governs classical PDE integrator convergence — how does global error scale with step size Δt? — governs Trotter quantum simulation. Second-order symmetric (Strang) Trotter has provable O(Δt²) global error; first-order Lie-Trotter is only O(Δt). This showcase demonstrates that convergence order directly, in the way a computational-physics group would validate a classical PDE integrator.

Directly aligned to the Naval Research Laboratory Quantum Algorithms for Differential Equations (QADE) program (Demirdjian et al., arXiv:2103.10352), the flagship DoD program on quantum methods for PDE-class problems. Applications span signal-processing dynamics, wave-equation kernels, ocean-acoustic transport, and radar cross-section electromagnetic simulation — all PDE classes that eventually target quantum simulation at scale.

This showcase reproduces Smith, Kim, Pollmann & Knolle (npj QI 5, 2019) — a second-order Strang Trotter implementation on the 1D uniform XX chain — with fidelity ≥ 0.9999 at every snapshot and measured convergence order p = 2.00 exactly matching the analytical Strang O(dt²) prediction.

Uniform XX Hamiltonian H = −J Σᵢ (XᵢXᵢ₊₁ + YᵢYᵢ₊₁) on N = 6 sites with periodic boundary conditions. Domain-wall initial state |000111⟩. Second-order Strang Trotter: U(dt) ≈ exp(−iH_XX/2·dt) · exp(−iH_YY·dt) · exp(−iH_XX/2·dt), with each bond term decomposed to standard CX + single-qubit rotations.

Two experiments: (1) domain-wall dynamics evolved to T = 2/J with 20 snapshots, Trotter vs exact matrix exponential compared site-by-site; (2) convergence-order sweep across step counts k ∈ {1, 2, 4, 8, 16, 32, 64} at fixed T, log-log fit of ||ψ_T − ψ_E||₂ vs dt in the asymptotic regime (fidelity ≥ 0.9).