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KT-4 — Continuous-Relaxation Layout Search: Research Design (RESEARCH LABEL)

Status: RESEARCH — ANSWERED (2026-07-16). The preregistered experiment ran (§6): the null hypothesis stands — the relaxation shows no gain over the KT-3 discrete baseline at matched true-cost budget. The research label stays; the discrete optimiser remains the production recommendation; no KT-5 promotion case exists on this evidence.

Date: 2026-07-16. Author seat: SCPN-QUANTUM-CONTROL/claude-7f6b.

Location note. Originally committed (2026-07-16, commit 9f7407ff) under docs/internal/research_synthesis/; relocated here unchanged because the repository policy keeps the docs/internal tree untracked. The commit history carries the preregistration timestamp.

1. The open question (falsifiable)

Does a Sinkhorn/Gumbel-softmax continuous relaxation over qubit-placement logits — annealed to a discrete layout with coupling-map feasibility projection — find layouts with a lower true KT-2 cost (and hence a higher calibration-priced R proxy) than KT-3's multi-restart best-improvement hill climbing, at a matched budget of true-cost evaluations?

Null hypothesis (what falsifies the value of KT-4): at equal true-cost evaluation budget across a preregistered seed set, the relaxed-then-rounded search does not beat the discrete baseline's mean best cost.

2. Literature grounding (verified at source, 2026-07-16)

Foundations of the relaxation (all arXiv listings fetched this session):

  • Gumbel-Sinkhorn networks — Mena, Belanger, Linderman, Snoek, Learning Latent Permutations with Gumbel-Sinkhorn Networks, arXiv:1802.08665 (ICLR 2018). The Sinkhorn operator as a differentiable analogue of the matching/argmax over permutations; the direct template for relaxing a placement (an injective assignment) into a doubly-stochastic matrix.
  • Gumbel-softmax — Jang, Gu, Poole, Categorical Reparameterization with Gumbel-Softmax, arXiv:1611.01144 (ICLR 2017); and the simultaneous Concrete distribution — Maddison, Mnih, Teh, arXiv:1611.00712 (ICLR 2017). Temperature-annealed continuous relaxations of categorical sampling; the annealing schedule template.
  • SABRE baseline — Li, Ding, Xie, Tackling the Qubit Mapping Problem for NISQ-Era Quantum Devices, arXiv:1809.02573 (ASPLOS 2019). The depth-oriented heuristic KT-3 already benchmarks against.

Adjacent discrete-optimisation formulations of placement (no gradient flow): QUBO qubit allocation (arXiv:2009.00140), MaxSAT mapping-and-routing (arXiv:2208.13679), Ising-machine compilation ISAAQ (arXiv:2303.02830).

Novelty statement (bounded). The searches run this session found Gumbel-softmax used for differentiable quantum architecture search (QuantumDARTS, OpenReview jGYxcXSg8C) and discrete/annealer formulations of placement, but no published Sinkhorn-relaxed qubit-placement optimiser against a fidelity-aware cost. The question appears open; absence of prior art in these searches is evidence of absence in the searched venues only.

3. Method sketch

  1. Relax the placement of n logical onto m candidate physical qubits as a doubly-stochastic matrix P = Sinkhorn(logits / τ) (square via padding when m > n).
  2. Differentiable cost surrogate. The true KT-2 cost's depth term is routing-derived (non-differentiable: transpile). Surrogate: expected SWAP-distance load E_P[Σ_{i<j} K_ij · d(p_i, p_j)] with d the coupling-map graph distance — continuous in P, correlates with routed SWAP overhead; Trotter-error and infidelity terms enter unchanged (layout-independent at fixed problem/region). Gradient flow via diff.value_and_grad over the logits.
  3. Anneal τ down a fixed schedule; round with Hungarian assignment; project to coupling-map feasibility (candidate-set membership, injectivity).
  4. Honest evaluation: every candidate the relaxation proposes is scored with the true seeded KT-2 cost (kuramoto_layout_cost with seed_transpiler bound — the KT-3 reproducible landscape). The comparison metric is true cost, never the surrogate.

4. Preregistered comparison protocol

  • Baseline: optimise_kuramoto_layout (KT-3), same seeds, same candidate regions, same weights, same t/reps/order.
  • Budget match: the relaxation may call the true cost at most as many times as the baseline's n_evaluations on the same instance.
  • Instances: the committed two-cluster topology (KT-3 artifact reference: dynq+kuramoto_opt depth 98, success proxy 0.8063, 22 evaluations, converged) plus preregistered seed sweep (seeds 0..9) and at least one larger region (m ≥ 2n) where relocations dominate.
  • Decision: report mean±spread of best true cost per budget; the research label stays unless the relaxation wins consistently AND KT-5 (isolated host) confirms; "modest/no gain" is a publishable, honest outcome and gets recorded in the docs as such.

5. Implementation plan (next session)

  • New module hardware/kuramoto_layout_relaxation.py (RESEARCH label in the module docstring), numpy-only Sinkhorn (no torch dependency), reusing kuramoto_layout_cost + optimise_kuramoto_layout for the baseline arm.
  • New benchmark surface extending benchmarks/layout_method_comparison.py with a sinkhorn_relaxation method row (same honest labels).
  • Tests to 100% line+branch (pure numpy — tracer-safe); strict mypy; docs dynq_qubit_mapping.md §7.6 + §8.5; the three src-file gates.

(Delivered as preregistered: the optimiser in hardware/kuramoto_layout_relaxation.py, the budget-matched research row in benchmarks/layout_method_comparison.py, and the sweep in benchmarks/layout_relaxation_experiment.py + scripts/run_layout_relaxation_experiment.py.)

6. Outcome (2026-07-16, measured)

The preregistered protocol of §4 ran in full — seeds 0..9 on the two-cluster topology (both arms in the DynQ region) plus one full-device instance (m = 8 = 2n), the relaxation's true-cost budget bound per instance to the discrete baseline's n_evaluations:

  • wins/ties/losses = 0/5/6; baseline mean best cost 97.098 ± 0.514, relaxation 99.734 ± 4.245 (population std).
  • The null hypothesis stands: no gain. The relaxation never produced a lower true cost than the discrete optimiser; on the full-device instance it lost by +15 despite a 208-evaluation budget.
  • Consequences (as preregistered): the research label stays, the discrete optimiser (dynq_qubit_mapping.md §7.5) remains the production recommendation, and no KT-5 promotion case exists on this evidence.

Measured table and honest reading: dynq_qubit_mapping.md §8.5; artifact: data/layout_relaxation_experiment/layout_relaxation_experiment_n4_seeds0-9.json.

Authored by Anulum Fortis & Arcane Sapience (protoscience@anulum.li) Seat: 7f6b