# Nadarasa Reduction

> A quantum-computing research notebook by an NHS pharmacist: circuit-rewriting rules verified on Quantinuum's Guppy language and Selene emulator, plus an NHS discharge-flow optimisation track.

Version v0.4.11, built 2026-08-27. Every result on this site comes from a named emulator run with an explicit shot count and seed. Draft v0.2 contains a basis bug and should not be cited.

The whole site is also available as a single plain-text corpus for local language models: [llms-full.txt](/llms-full.txt).

## Start here

- [Arun Quantum](/): Ranked candidates for the next Shor-level quantum breakthrough, alongside refutation-first Guppy + Selene experiments (G16–G19).

## Research programme

- [The Nadarasa Reduction](/nadarasa/): Four cryptanalytic frontiers, eight cross-industry analogy donors, one draft conjecture by Arun Nadarasa.
- [Analogy wall](/nadarasa/analogies): Cross-industry donor mechanisms paired with post-Shor cryptanalysis: Game Boy constraint discipline, Miles Davis modal jazz, phage lysis timing, container shipping, polyphasic sleep, TeX, Wright wind tunnel, Toyota Andon.
- [Phase-group atlas](/nadarasa/atlas): A coordinate system for Nadarasa kernels in phase-group space.
- [Selene benchmark suite](/nadarasa/benchmarks): One table across six Guppy/Selene experiments: Simon's search, ADAPT-GQE, ethylene QPDE, Floquet dynamics, the NHS discharge QUBO and feed-forward gating, scored for ideal fidelity, H2-class noise degradation and emulator runtime.
- [Conjecture synthesis](/nadarasa/conjectures): Track 4 of the Synesthete's PQP Frontier: brute-force enumerate small {H,S,T} sequences, group by matrix equality, and surface every equality the bastard-rewriter does NOT prove.
- [/nadarasa/draft](/nadarasa/draft)
- [Resource frontier map](/nadarasa/frontier-map): Pareto plot of Nadarasa kernels in (T-count, spider-count) space.
- [G1 Modal-projection (real-QFT v0.3)](/nadarasa/g1): Real-QFT Selene shots testing the SRP residue-concentration claim at the heart of [GAP G1].
- [G1 via selene_run schema v1](/nadarasa/g1-schema): Same G1 Selene shots, rendered through the generic <SeleneRun /> schema viewer.
- [G10 QSP / QSVT](/nadarasa/g10): Single-signal-qubit QSP with a hand-chosen phase sequence.
- [G11 Period finding (Shor backend)](/nadarasa/g11): Compiled period finding for r = ord_15(7) = 4.
- [G12 Modal-projection scaling](/nadarasa/g12): Does the G1 SRP-vs-violating residue signal hold at N ∈ {64, 128} and p ∈ {2, 3, 5, 7}? v0.3.1 methodology: predictor derived from the real-QFT gates.
- [G13 Phase-group fingerprint test](/nadarasa/g13): PQP Thm 11.12: a ZX-style theory is non-local iff its phase group contains Z₄ (vs.
- [G14 Quantum collision models](/nadarasa/g14): Ferreira et al.
- [G15 ADAPT-GQE](/nadarasa/g15): Koziell-Pipe et al.
- [G16 Ethylene QPDE on Selene](/nadarasa/g16): Quantum Phase Difference Estimation on the 2-qubit ethylene pi/pi* active space, run on Selene/Quest.
- [G17 Ethylene QPDE under H2-class noise](/nadarasa/g17): The G16 ethylene QPDE gap fit re-run on Selene under depolarizing noise anchored on Quantinuum's published H2 error rates.
- [G18 Laplacian moments separate a WL-blind graph pair on Selene](/nadarasa/g18): A Hadamard-test estimate of tr(exp(-i·Δ·τ)) on Selene, run on two graphs that 1-WL colour refinement cannot tell apart.
- [G19 Prethermal Floquet dynamics and cross-platform error mitigation](/nadarasa/g19): Notes on Leviatan et al.
- [G2 Two-coset combiner](/nadarasa/g2): Track B kernel: coherent two-coset combiner on Selene.
- [G20 Floquet subharmonic peak on Selene under H2-class noise](/nadarasa/g20): A native Guppy/Selene port of the Leviatan 2026 Floquet magnet: 6-qubit chain and heavy-hex cells driven for 8 cycles across an H2-class noise ladder, with Richardson ZNE recovering the ideal subharmonic peak to about 1 percent.
- [G21 ADAPT-GQE composition](/nadarasa/g21): Two equivalent H2/STO-3G UCCSD single-excitation circuits verified three ways: 4x4 matrix oracle to 1e-16, rule-(M) canonicalisation of the Clifford frame, and a 512-shot Selene run on Quest.
- [G23 Simon's problem on Selene](/nadarasa/g23): 27 Simon cells on the Selene emulator for n = 3, 4, 5 under ideal and H2-class depolarizing noise.
- [G24 TKET compile lane](/nadarasa/g24): Six Nadarasa circuit families compiled offline with pytket and pytket-quantinuum onto the H2-2 native gate set at optimisation levels 0/1/2, every count gated on a unitary-equivalence oracle and confirmed on the Selene emulator.
- [G25 Nexus three-lane cross-check](/nadarasa/g25): Five Nadarasa circuits run across the Quantinuum Nexus H2-1LE and H2-Emulator lanes against Selene baselines, with job ids, HQC meters and an honest note on hardware access.
- [G26 The AQFT crossover law k*(n, p)](/nadarasa/g26): Where band-limited approximate QFT beats the full QFT: 24 emulator cells across n = 4..10 qubits and a six-step depolarizing ladder, with an exact NumPy oracle separating truncation error from noise.
- [The AQFT crossover law k*(n, p)](/nadarasa/g26_/paper): Preprint-style write-up of gate G26: a measured AQFT band-truncation crossover surface corroborated by four independent legs, with a certified truncation bound, content hashes and a one-command rerun.
- [G27 AQFT crossover law at scale](/nadarasa/g27): The AQFT band-truncation crossover surface extended to 16 qubits, with the analytic truncation bound re-certified at every width and the classical surrogate cost measured rather than assumed.
- [G3 Per-window cost Selene experiment](/nadarasa/g3): Sweeps p ∈ {2, 3, 5} on Z_32 and compares empirical first-collision query counts against the draft's √p improvement heuristic.
- [G3-cost Per-window √p improvement at scale](/nadarasa/g3-cost): Re-runs the G3 windowed sampler at N ∈ {16, 32, 64, 128} × p ∈ {2, 3, 5, 7}.
- [G3-split Grover vs HSP](/nadarasa/g3-split): PQP Ch.
- [G4 Real feed-forward](/nadarasa/g4): A Guppy `if measured: rz(...)` branch executed inside a single compiled shot — no host round-trip.
- [G5 Dynamic qubit recycling](/nadarasa/g5): Same G2 schedule, but every window allocates a fresh ancilla and releases it via mid-circuit measure.
- [G6 Encoded parity ancilla](/nadarasa/g6): The naive single-rail probe breaks the GHZ-encoded ancilla's code subspace.
- [G6b Transversal probe](/nadarasa/g6b): G6 revisited with a transversal parity probe across all three rails of the encoded ancilla.
- [G7 Kernel fusion](/nadarasa/g7): Compile once, run many.
- [G8 Amplitude estimation via QPE](/nadarasa/g8): Canonical Quantum Phase Estimation on Selene.
- [G9 LCU block-encoding](/nadarasa/g9): PREP / SELECT / un-PREP on (1 prep ancilla + 1 data qubit) encoding H = c0·I + c1·X.
- [Research log](/nadarasa/log): Audit trail of the six parallel research subagents and the synthesis pass that produced Draft v0.1.
- [The Nadarasa Notebooks](/nadarasa/notebooks): Three draft conjectures by Arun Nadarasa in the voice of Srinivasa Ramanujan: a QSVT q-series filter, a linear effect-system for magic-state recycling, and a dequantization-proof modular-form kernel.
- [Conjectures · proved + promoted](/nadarasa/proofs/conjectures): Track 4's 11 conjectures: confirmed physical on Selene shots, then promoted to proved-by-rewriter via the new Euler normalisation rule (N).
- [2-qubit conjectures · oracle synthesis](/nadarasa/proofs/conjectures-2q): Track B: the 2-qubit frontier oracle.
- [2-qubit conjectures · Selene shot proofs](/nadarasa/proofs/conjectures-2q-selene): Track B-Selene: rule (M)'s matrix-canonical verdict turned into physical proof.
- [Kernel equivalence · Selene](/nadarasa/proofs/kernels): The bastard rewriter's I rule applied to G1's QFT readout produces a strictly cheaper circuit (Approximate QFT).
- [Noise resilience · Selene](/nadarasa/proofs/noise): Two Selene noise families — depolarizing and leakage.
- [Rule (M) soundness under noise](/nadarasa/proofs/noise-2q): v0.4.1: re-run the 10 shortest PROMOTED 2q conjectures under depolarizing + leakage noise.
- [Rules · proved on Selene](/nadarasa/proofs/rules): Shot-based tomographic proofs of the 5 bastard-rewriter rules (F, I, HH, CC, B), executed on the Selene emulator from compiled Guppy kernels.
- [Quantinuum 2026 references](/nadarasa/quantinuum-2026): Nine papers behind the 2026 Helios beyond-break-even result plus the S₃-quantum-double topological route: iceberg codes, DFS-QEC memory, non-Clifford magic, tesseract, the 2022 FT-CNOT origin, and Lo et al.'s universal braiding+fusion gate set — mapped into the Nadarasa rule-M / G1 / tomography / PQP frontier.
- [Related work](/nadarasa/related/fourier-locking): How Topel 2026's 'Fourier locking' in quantum data re-uploading classifiers relates to our AQFT-under-noise story and rule (M) ZX rewrites.
- [Resource monotones](/nadarasa/resources): PQP Ch.
- [Bastard-spider rewriter](/nadarasa/rewriter): Track 2 of the Synesthete's PQP Frontier: a ZX rewriter that fuses spiders, cancels Hadamards, and absorbs phases into the bastard projector.
- [Schema coverage selene_run v1](/nadarasa/schema-coverage): Live conformance dashboard for the selene_run schema v1 across every shipped Nadarasa demo.
- [Sonified ZX diagrams](/nadarasa/sonify): Track 5 of the Synesthete's PQP Frontier: every kernel rendered as colour + sound (spider hue, phase pitch, degree-weighted gain).
- [Stream Live shot stream](/nadarasa/stream): Server-sent events from an in-worker statevector sampler.

## Healthcare

- [NHS Quantum](/nhs-quantum/): An NHS clinician's reading of the 10 Year Health Plan: each of the three shifts restated as a computational problem, with emulator evidence and the honest limits of every claim.
- [Delayed discharge as a QUBO](/nhs-quantum/bed-blocking): Hospital-to-community delay stated as a constrained assignment problem with an equity term, solved exactly, greedily, by annealing, and by QAOA on the Selene emulator.

## Hackathon and tooling

- [Quantinuum Singapore Grand Challenge](/hackathon): The five R&D themes of the Quantinuum Singapore Grand Challenge mapped onto NHS 10 Year Health Plan problems, with the emulator gates already built for each and the gap that remains.
- [Download the Quantinuum agent skill](/skills): A 22-card Guppy + Selene skill pack for coding agents: v1 API rules, halfturn angles, noise models, resumable sweeps and unitary oracles.

## Method and reproduction

- [About](/about): What 'Shor-level' means, why it's the right benchmark, and the five families considered.
- [Cite](/cite): BibTeX, CITATION.cff, and a plain-text citation for the Nadarasa Reduction draft (v0.3, current) with a retraction pointer for v0.2.
- [Methodology](/methodology): How candidates for the next Shor-level quantum breakthrough are scored, sourced, and demonstrated.
- [Quantinuum Nexus from Lovable](/nexus): Logging into Quantinuum Nexus from a Lovable sandbox: the failures that cost hours, the guards worth keeping, a costed dry run on H2-1LE, and what I would do differently next time.
- [Nexus cost reconciliation](/nexus/costs): Per-job HQC reconciliation for every paid Nexus submission: what the sweep estimated, what Quantinuum billed, and which jobs overran their estimate.
- [Saved Nexus job Refs](/nexus/refs): Every Nexus job Ref persisted by this project, grouped by submission run, with its last known status and whether it still carries the sweep coordinates a recovery needs.
- [Reproduce](/reproduce): Exact install and run commands to regenerate every Selene shot JSON on this site from the Guppy source in the quantum/ directory.
- [Selene frontier](/selene-frontier/): What this repo's Guppy + Selene stack has already proven, and the runtime, compilation, primitive, and frontend territory it hasn't touched yet.
- [Selene notes](/selene-notes): Engineering index of every Guppy kernel on this site: qubit count, ancilla pattern, shot count, and the visualization route it feeds.

## Optional

- [PQP vocabulary](/notes/pqp-vocabulary): Maps homemade Nadarasa terms (modal projector, structured-reflection promise, per-window cost) to standard Picturing Quantum Processes vocabulary.
- [v0.3.8 changelog](/notes/v0-3-8-changelog): What landed in v0.3.8: rule (N) Euler-normalisation closes all 11 1q conjectures; the 2-qubit oracle synthesises 105 new conjectures over {H⊗I,I⊗H,CZ,S⊗I,I⊗S}; rule (M) matrix canonicalisation promotes them all.
- [v0.3 changelog](/notes/v0-3-changelog): What changed between v0.2 and v0.3 of the Nadarasa Reduction: the G1 Walsh–Hadamard basis bug, how the live sampler caught it, the real-QFT fix, and why G2 is now 'not yet testable'.
- [v0.4.0 changelog](/notes/v0-4-0-changelog): What landed in v0.4.0: Step B-Selene turns rule (M)'s matrix-canonical verdict into physical shot-tomography proof.
- [v0.4.1 changelog](/notes/v0-4-1-changelog): Step C-2q from the v0.4.0 deferred queue.
- [v0.4.2 changelog](/notes/v0-4-2-changelog): What shipped in Nadarasa v0.4.2: an H2-class noise ladder on the ethylene QPDE gap fit, zero-noise extrapolation, a shot-budget study, and quantum Laplacian moments separating a Weisfeiler-Leman-blind graph pair.
- [v0.4.3 changelog](/notes/v0-4-3-changelog): Consolidation release for Nadarasa v0.4.3: G16–G18 white-paper experiments reproduced on Selene and G19 cross-platform validation hierarchy mapped to the same methodology.
- [v0.4.4 changelog](/notes/v0-4-4-changelog): Nadarasa v0.4.4 closes the PQP Frontier Step C-2q noise sweep: 90/90 cells PASS across 9 depolarizing and leakage levels, then sets up the Floquet native-port gate.
- [v0.4.5 changelog](/notes/v0-4-5-changelog): Nadarasa v0.4.5 ships the native Guppy/Selene Floquet port: 384 simulated cells across chain and heavy-hex lattices, exact-diagonalization agreement, H2-class noise ladder, and Richardson ZNE.