Germanium Baseband iSWAP: Validating a 4-Day-Old Experimental Result

arXiv:2608.16716 (Massai et al., IBM Research Europe -- Zurich, 17-18 Aug 2026) demonstrates a real single-pulse baseband iSWAP gate (56 ns) in strained-germanium hole spin qubits, by orienting the magnetic field so the exchange interaction's longitudinal component J∥ and Zeeman detuning E_Δg both vanish, leaving a pure transverse J⊥ coupling. This experiment reproduces their result with dense_evolution.circuits.trotter, applied for the first time to a genuinely time-dependent pulse (previously only exercised against static Hamiltonians), and extends the analysis with four follow-up checks.

Scope note: this validates dense_evolution.circuits.trotter and dense_evolution.uhlmann_fidelity -- not dense_evolution.solvers.vhd_tb/harrison_tb ("tight-binding"). Those compute bulk-crystal band structure (sp3s* basis, no spin, no confinement) -- a different physics regime from this paper's confined two-spin exchange qubits, even though vhd_tb already has real germanium parameters.

Two real physics corrections to the naive Colab draft

1. The paper's transverse Hamiltonian is H⊥ = (1/2)J⊥(σ+₁σ-₂ + h.c.) = (1/4)J⊥(XX+YY) -- a coefficient of 1/4, not the naively-assumed 1/2.
2. The real pulse shape (Supp. Fig. 13) is 16 ns raised-cosine rise + 24 ns flat top + 16 ns raised-cosine fall (56 ns total) -- not a Tukey window spanning the whole duration.

A third bug (a factor-of-2 error in the peak-amplitude calibration, giving a 50% swap instead of 100%) was caught only by actually running the code, not by inspection.

Reference circuit and Trotterized pulse

The native iswap gate defines ground truth directly through dense_evolution's own gate set (drawn as a Quirk-style box diagram by this experiment's draw_circuit utility), sidestepping any basis-convention mismatch with the paper's matrix notation.

DenseSVSimulator(2).run_circuit([('x', 1), ('iswap', 0, 1)])

Each Trotterized pulse slice's Rxx/Ryy rotation is built from pauli_rotation_ops directly (not trotter_evolve_ops, since the coefficient varies per slice with the pulse envelope).

N slices	Gate count	Infidelity vs. iSWAP	−log₁₀(infidelity)
4	73	8.2 × 10⁻⁴
8	145	1.9 × 10⁻⁵
16	289	1.0 × 10⁻⁷
32	577	< 5 × 10⁻⁹ (print floor)
64	1153	< 5 × 10⁻⁹ (print floor)

Notable derived fact: X⊗X, Y⊗Y, and Z⊗Z always pairwise commute (simultaneously diagonal in the Bell basis). Since this operating point has J∥=0 by construction, the Trotter decomposition here is exact at any slice count -- fidelity is already >0.999 at just 4 slices, saturating to 1.0 by 32. The small residual is pure quadrature error from approximating the smooth envelope with piecewise-constant steps, not Trotter truncation error.

SPAM noise: the paper's own exact channel

D_p(ρ) = (1−p)ρ + (p/4)·I₄  —  paper's own global 2-qubit channel, not dense_evolution's per-qubit NoiseModel

p (source)	uhlmann_fidelity	Exact composition (hand-derived)	Paper's own approximation
0.2258 (from ⟨diag(PSPAM)⟩=0.69)	0.6996	0.6996	0.6900
0.2100 (as stated in text)	0.7181	0.7181	0.7098

dense_evolution.circuits.registry.NoiseModel's built-in 'depolarizing' model is a per-qubit local channel -- physically different from the paper's own global 2-qubit depolarizing model, D_p(ρ) = (1-p)ρ + (p/4)I₄ (Section XVIII.C). Reusing NoiseModel here would silently model different physics, so this experiment implements the paper's exact channel by hand and scores it with dense_evolution.uhlmann_fidelity -- verified to match a hand-derived exact sequential-composition formula to machine precision, and to differ, as expected, from the paper's own explicitly-labeled "back-of-envelope" approximation (which composes two independent survival probabilities rather than the true channel composition).

Combining the paper's own real measurements -- F_QPT = 60% (full process tomography) and ⟨diag(P_SPAM)⟩ = 0.69 (SPAM-only measurement) -- reproduces their reported F_iSWAP ≈ 87% exactly: 0.60 / 0.69 = 0.87.

Four follow-ups

1. Randomized benchmarking (Pauli frame). Directly answers the paper's own Section VII: "A more rigorous fidelity estimation via randomised benchmarking is left to follow-up work." Since the injected noise parameter is known here (p=0.21), this validates that the RB protocol via dense_evolution correctly recovers it: a fitted exponential decay gives r=0.7900, matching the theoretical r = 1-p = 0.7900 exactly. Not full 2-qubit Clifford RB (would need the full Clifford table) -- Pauli-frame RB (Pauli twirling), a real, valid protocol.

2. Per-state SPAM profile, not a uniform average. Fig. 5f shows four real, precisely-labeled diagonal SPAM values (0.17, 0.40, 0.39, 0.37 -- the states needing concatenated two-qubit rotations) alongside a mean of 0.69 over all 16 states. Reconstructing an honest two-tier profile (these 4 real values, the other 12 solved only to match the known mean -- never invented) gives a round-trip fidelity spread from 0.29 to 0.81, hidden entirely by the single uniform p used above.

3. Coherent vs. stochastic error. The paper's own caveat (i), Section XVIII.C: "assumes depolarising SPAM, whereas actual errors contain coherent components... not addressed here." A small 23.6% coherent pulse-amplitude miscalibration reproduces the same F_iSWAP≈87% just as well as the depolarizing-SPAM story -- the aggregate F_QPT number alone cannot distinguish the two mechanisms. (23.6% is also implausibly large for a calibrated pulse compared to typical experimental precision, which if anything favors the paper's own SPAM-dominant explanation.)

4. The general off-resonance regime -- where Trotter finally has real error. Neither the iSWAP point above nor a pure CPhase point (J⊥=0, purely diagonal Hamiltonian) ever show Trotter error, because XX/YY/ZZ always commute. Adding a nonzero Zeeman detuning term (Z⊗I - I⊗Z, which genuinely does not commute with XX+YY) finally produces real, non-trivial Trotter error that shrinks with slice count -- and order-2 Suzuki-Trotter is consistently ~40-1000x more accurate than order-1 at matched slice count, confirming dense_evolution.circuits.trotter's documented convergence claim for the first time on a genuinely time-dependent pulse.

Status

Validated in scripts/germanium_iswap_validation.py and tests/test_germanium_iswap_validation.py (8/8 passing). The box-diagram circuit-drawing utility (draw_circuit) and the global 2-qubit depolarizing channel (depolarize_2q) are candidates for promotion into Dense-Evolution proper if they prove broadly useful.

Reproduce

python scripts/germanium_iswap_validation.py
pytest tests/test_germanium_iswap_validation.py -v