Notes · PQP digest
PQP vocabulary mapping
Several Nadarasa terms are homemade. This page maps them to the standard vocabulary of Picturing Quantum Processes (Coecke & Kissinger, CUP 2017) so reviewers can locate each operation in the book. Built from the digest at quantum/PQP_DIGEST.md.
Term-by-term mapping
| Our term | PQP term | Chapter |
|---|---|---|
| modal projector | classical post-selection on a bastard-spider output | Ch. 8 §8.3.3 + §8.4 |
| predictor derived from the gates | spider fusion + doubling-Born | Cor. 8.35 + Eq. 6.4 |
| structured-reflection promise (SRP) | structure on the phase-group of the classical subgroup | Ch. 9 §9.3.6 + Ch. 11 Thm 11.12 |
| per-window √p improvement | amplitude-amplification cost (Grover) — NOT HSP | Ch. 12 §12.2.3 vs §12.2.4 |
| GAP ledger entry | A ⋡ B in (P, F) — resource gap under implemented rewrites | Ch. 13 |
| kernel cannot signal | causality (discard out = discard in) | Def 6.52 + Eq. 10.31 |
| live sampler / shot stream | ONB measurement as doubled effect | Def 7.1, Def 7.6 |
modal projector: Not a named primitive in PQP — it is a perfectly well-defined operation that the book never calls out by name. Bastard-spider fusion (Thm 8.72) is what gives the closed form.
predictor derived from the gates: v0.3 methodology rule operationalised. Not heuristic discipline — the dodo theorem (Thm 8.41) says the diagram uniquely determines the basis.
structured-reflection promise (SRP): Reads as a constraint on which phase-group elements survive spider fusion. The Z₄ phase-group fingerprint test is the same machinery applied as a litmus.
per-window √p improvement: The original G3 framing confused Grover-style √-cost with HSP-style log-cost. Ex. 12.22 explicitly disambiguates: Deutsch–Jozsa for n > 1 is NOT HSP.
GAP ledger entry: A gap becomes 'certified' when an additive monotone (T-count, spider-count) provably non-increasing under F separates the two sides. See /nadarasa/resources.
kernel cannot signal: Non-signalling is a derived consequence of causality (Thm 6.84), not an extra postulate.
live sampler / shot stream: State collapse as a one-line spider equation; tomography is the Ch. 7 §7.4 picture.
Citations the v0.3 rule depends on
- Eq. 6.4Born rule from the doubling constructionCh. 6 · Quantum processes
Every pure quantum map is f ⊗ f̄; composing an effect on a state then yields |⟨ψ|φ⟩|² automatically. This is why deriving the predictor from the gates is not heuristic — it is the definition.
Anchors: v0.3 methodology rule (predictor from gates) · G1 / G2-real / G12 closed-form derivations
- Thm 8.34 · Cor 8.35Spider fusion theoremCh. 8 · Classical-quantum processes
Any connected same-colour spider diagram fuses to a single spider determined only by leg count. Operationally: closed forms collapse to one expression per branch — exactly what the gate-derived predictors compute.
Anchors: v0.3 methodology rule · modal projector / G1 / G12 simplifications
- Thm 8.41Dodo theorem (no spider impostors)Ch. 8 · Classical-quantum processes
Any family of maps satisfying spider fusion IS a real spider family for some ONB. So 'derive the predictor from the gates' is not just heuristic discipline — the diagram uniquely determines the basis up to relabelling.
Anchors: v0.3 methodology rule
- Def 6.52 · Eq. 10.31Causality criterion (discard out = discard in)Ch. 6 / Ch. 10
A quantum process is causal iff discarding its output equals discarding its input. Non-signalling between subsystems is a derived consequence (Thm 6.84), not an extra postulate.
Anchors: G2-real signalling check · any 'kernel cannot signal' claim
- Thm 9.128Clifford-ZX completenessCh. 9 · Phases and complementarity
The 4-rule Clifford ZX-calculus is complete: any equality of Clifford circuits is provable by the 4 rules alone. Justifies treating rewrites as a sound monotone-tracking system on the Clifford fragment.
Anchors: resource-monotone certification · G13 phase-group test
- JPV 2017 + NW 2017Full-ZX completeness (post-publication)Successor results, not in PQP
Jeandel–Perdrix–Vilmart 2017 completed Clifford+T ZX; Ng–Wang 2017 completed full ZX. Both post-date PQP and are what makes modern rewriters (PyZX, quizx) theoretically grounded.
Post-pub: JPV: arXiv:1705.11151 · Ng–Wang: arXiv:1706.09877
Anchors: full-ZX rewriting
- Thm 11.12Phase-group fingerprint for non-localityCh. 11 · Quantum non-locality
A ZX-style theory exhibits GHZ-style non-locality iff its phase group contains Z₄ (vs. Z₂×Z₂ for Spekkens' toy theory). A single litmus test on the phase angles a kernel actually uses.
Anchors: G13 phase-group test
- §12.2.3 vs §12.2.4 · Ex. 12.22HSP exponential ≠ Grover square-rootCh. 12 · Quantum computation
HSP uses O(log|G/H|) queries (exponential). The √-factor only appears in Grover/amplitude amplification. Ex. 12.22 explicitly notes Deutsch–Jozsa for n>1 is NOT an HSP instance.
Anchors: G3 split: grover vs hsp
- Ch. 13Gaps as A ⋡ B in (P, F)Ch. 13 · Quantum resources
A gap is precisely A ⋡ B in the partial order of diagrams under the implemented rewrites F. Additive monotones (T-count, spider count) provably non-increasing under F certify a gap as real, not conjectured.
Anchors: resource-monotone certification
- §14.1–14.3Quantomatic, !-boxes, conjecture synthesisCh. 14 · Automation
!-boxes finitely encode the infinite spider-fusion family; conjecture synthesis discovers ~170 rules from generators alone. Practical successors (post-PQP): PyZX (Python), quizx (Rust).
Post-pub: PyZX: arXiv:1904.04735 · quizx: github.com/zxcalc/quizx
Anchors: scoping note: PyZX / quizx adoption
- Topel 2026Fourier locking in re-uploading classifiers (spectral homotopy)Related work · post-PQP preprint
Data re-uploading QML classifiers exhibit 'Fourier locking' — trainable spectrum collapses to a small integer lattice. Remedy is a continuous 'spectral homotopy' of the encoding schedule. Same family of question as our AQFT-under-noise cutoff and rule (M)'s discrete diagram deformations. See /nadarasa/related/fourier-locking.
Post-pub: Topel, S. (2026), CC-BY-4.0 · github.com/moth-quantum/fourier_locking_study
Anchors: G1 AQFT under noise · rule (M) as discrete spectral rewrite