Close exact Walsh-character and controlled-phase semantics.
Acceptance: All basis entries, bit order and j=0 agree with the frozen source; constant functions and constant-plus-one-parity are regression cases.
planned; no claimed closure
Structure before circuit tricks
Walsh-series state preparation: phase is not amplitude
Short Walsh expansions can avoid enumerating all amplitudes, but a shallow phase circuit is only the first part of the loader.
research target setting:real-walsh-infidelity-postselection
\[p=\frac{1}{N}\sum_x\sin^2(\tau f_S(x)/2),\quad N=2^n\]
Frozen target and input model
Formalize the source Walsh Series Loader, including its constant mode, normalized approximation and success-dependent complexity.
Access model: A real sampled function on the source dyadic grid, explicitly supplied Walsh coefficients, n data qubits and a signal ancilla; charge classical coefficient generation and primitive rotations.
- Pin Walsh ordering and its map to physical wires
- Declare epsilon as target state infidelity, not operator-norm or vector error
- Retain the zero-order phase under control; f_S and its accepted branch must be nonzero
- Smoothness, Walsh sparsity, Hamming weight and norm/filling assumptions are distinct parameters
Desired result, not an achieved bound
Reproduce the pinned source's per-attempt and repeat-until-success/amplified resource bounds with all function-dependent constants and gate-model assumptions visible; no new speedup claimed.
The state after postselection is proportional to -i(I-exp(-i tau F_S))|+^n>, not exactly to f_S at finite tau. Increasing fidelity can reduce success probability. The zero-order-term implementation issue must be tested explicitly. Local staging now checks the actual H-controlled-diagonal-H-phase composition, constant relative phase and exact sine-squared accepted mass; independent source/semantic admission and approximation/resources remain open.
Dependency-ready execution route
Derive the actual ancilla branches and normalized approximation theorem.
Acceptance: Success probability, zero branches, phases and infidelity/truncation errors are explicit; finite plots are not proofs.
planned; no claimed closure
Compose phase synthesis with RUS or amplitude amplification.
Acceptance: Report per-attempt and successful-preparation gates/depth separately, with coefficient preprocessing, inverses/reflections and finite-gate error charged.
planned; no claimed closure
Next bounded advance: Review the staged literal interference chain and constant-mode discriminator, then supply the Walsh parity/wire map and nonzero normalized/error/resource bridges; do not re-prove the already checked branch identity.
Reusable mathematics
No local transport theorem is bound to this record. Do not infer formal truth from its position in the atlas.
Lower-bound comparison contract
source-audit-pending
Compare sparse/smooth promised inputs with matched accuracy and success costs, never with a free arbitrary-vector table.
Same-model key: setting:real-walsh-infidelity-postselection
Diagnostic examples, not proofs
- Nonzero constant function (zero-mode regression)
- Constant plus a single Walsh character
- Localized smooth function with a small filling fraction
- Bit-reversed two-qubit ordering discriminator
Primary-source ledger
primary-text-checked arXiv v3, 9 November 2023; checked 29 September 2026: Theorem 1, Corollary 1, Table I and Appendix B (B14–B19, B28).
Walsh Series Loader: epsilon is state infidelity. Smooth fixed-function single-attempt depth O(epsilon^-1/2), size O(n+epsilon^-1/2), success Theta(epsilon); expected depth O(epsilon^-3/2), expected size O((n+epsilon^-1/2)/epsilon). Constants retain function/derivative/nonzero-mass assumptions. Sparse variants require supplied coefficients and Hamming-weight bounds. Amplification also charges reflections and inverses.
primary-text-checked arXiv v1 full HTML, Sections II.2–II.3, Eq. (11), Fig. 2: controlled zero-order term; checked 29 September 2026.
Companion adversarial implementation review, not a refutation of the WSL theory. Retain the constant Walsh coefficient in a controlled phase circuit; use constant and constant-plus-parity regression cases.
Download bounded agent / contributor packet
python3 website/scripts/research_atlas.py context --route spw-walsh