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REVIEW 2 major objections 6 minor 20 references

A reproducible unanchored optimizer recovers exact three-basis MUBs in dimension six, a recurrent partial-exact four-basis hub-and-triangle, and no near-exact pairs for five or six bases under the reported campaigns.

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2026-07-14 10:26 UTC pith:54W2BON5

load-bearing objection Solid, carefully scoped mathematical-software paper that ships a usable AMUB lab and honest multi-seed evidence for d=6; finite sampling is owned, not hidden. the 2 major comments →

arxiv 2607.10615 v1 pith:54W2BON5 submitted 2026-07-12 cs.MS

A Reproducible Software Workflow for Unanchored Approximate MUB Optimization: A Case Study in Dimension Six

Abdul Fatah , Ian McLoughlin , Saim Ghafoor This is my paper
classification cs.MS
keywords mutually unbiased basesapproximate MUBdimension sixLie-algebra parameterizationunanchored optimizationpairwise defect geometryreproducible scientific softwarequantum hardware execution
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper offers a portable, artifact-driven software workflow for searching approximate mutually unbiased bases without fixing any basis in advance. Bases are built as unitaries via Lie-algebra exponentiation, optimized jointly with Adam, and diagnosed as weighted graphs of pairwise defects rather than by a single scalar loss. In dimension six, across one hundred random seeds in both double and single complex precision, the workflow recovers exact triples, repeatedly lands on a four-basis pattern with three near-exact edges and a defective triangle, and never finds a near-exact pair once five or six bases are requested under the primary numerical tolerance. A short quantum-hardware check shows that compiling the best four-basis transitions into three-qubit circuits yields a noise floor large enough to erase the classical distinction between good and bad pairs. The result is not a proof that complete seven-basis sets fail to exist; it is a shareable computational map of the basins this optimizer actually reaches, plus an honest first look at running those unitaries on present-day hardware.

Core claim

Under a fixed unanchored Lie-exponential parameterization, Adam schedule, seed set, and primary tolerance of 10^{-6}, the sampled optimization landscape in dimension six yields exact three-basis configurations for many seeds, a recurrent four-basis partial-exact hub-and-triangle with three near-exact pairs, and fully defective configurations with zero near-exact pairs for every seed when five or six bases are requested, in both complex128 and complex64 arithmetic.

What carries the argument

Unanchored Lie-algebra unitary parameterization: each candidate basis is Uk = exp(i Hk) with Hermitian generators obtained by symmetrizing unconstrained complex matrices, so unitarity is enforced by the forward model; the total AMUB loss is the sum of pairwise Frobenius defects of the entrywise-squared Gram matrices from the uniform 1/d target, retained as a weighted complete graph of pairwise defects.

Load-bearing premise

That one hundred random seeds, a fixed Adam learning rate and step budget, and this particular Lie-exponential parameterization are enough to treat the total absence of near-exact pairs for five and six bases as a stable feature of the landscape the method samples.

What would settle it

A single optimized configuration for n=5 or n=6, produced under the same parameterization and primary tolerance τ=10^{-6}, that contains at least one pairwise maximum entrywise deviation below 10^{-6} would falsify the reported fully defective transition; conversely, an independent optimizer or longer budget that recovers an exact four-MUB would show the hub-and-triangle is only a local basin of this workflow.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Exact three-MUB configurations in d=6 are routinely reachable by unanchored gradient search without anchoring to Fourier or Hadamard families.
  • Four-basis searches under this workflow should be expected to land in a three-edge hub plus defective triangle rather than an exact four-MUB.
  • Scalar aggregate loss alone is insufficient; pairwise defect graphs are required to separate exact triples, partial-exact hubs, and fully defective configurations.
  • Current compiled three-qubit embeddings of d=6 transitions (tens of native two-qubit gates) produce a noise floor that washes out classical near-exact versus defective distinctions.
  • The same parameterized stack can be rerun for other composite dimensions or larger seed budgets without rewriting the unitary model or artifact pipeline.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The recurrent hub-and-triangle for n=4 is consistent with known rigidity or non-extendability phenomena for MUBs in dimension six; the workflow may be rediscovering a structural obstruction rather than an optimizer artifact.
  • If longer multi-seed n=7 campaigns continue to show zero near-exact pairs, the software becomes a practical stress test for any future analytic construction claiming a complete set.
  • Precision sensitivity at n=4 (median near-exact count drops from three in complex128 to zero in complex64) suggests that reduced-precision accelerator runs need relaxed classification tolerances or higher Taylor order before they can be trusted for fine defect geometry.
  • Embedding d=6 into three qubits with post-selection is a concrete benchmark for whether future lower-depth unitary synthesis can bring the hardware noise floor below classical AMUB defect scales.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 6 minor

Summary. The paper presents a reproducible, parameter-driven software workflow for unanchored approximate mutually unbiased basis (AMUB) optimization in arbitrary dimension d, using Lie-algebra unitary parameterization (Uk = exp(iHk)) and a pairwise Frobenius defect loss. The implementation supports CPU, MPS, CUDA, and HPC backends, with a Taylor-series matrix-exponential layer for accelerator compatibility; structural results use the CPU/native torch.matrix_exp pathway. As a d=6 case study, 100-seed campaigns for n=3,4,5,6 in complex128 and complex64 recover exact three-basis configurations for many seeds, identify a recurrent four-basis partial-exact hub-and-triangle structure (three near-exact pairs, three defective), and report no near-exact pairs for n=5 or n=6 under the primary tolerance τ=10^{-6}. A hardware check embeds a representative n=4 configuration into three-qubit circuits on ibm_marrakesh; measured QPU losses sit in a 0.02–0.08 noise floor that obscures classical near-exact vs defective structure. Full run artifacts, code, and Zenodo archive are provided.

Significance. If the reported numerical landscape is accepted as scoped, the work supplies a useful, portable mathematical-software artifact for AMUB exploration: unanchored Lie-exponential optimization, pairwise defect geometry diagnostics, precision-aware multi-seed campaigns, and backend-portable execution with explicit separation of reference (CPU/native) vs accelerator (Taylor) pathways. The careful non-claim of existence/nonexistence for complete MUBs in d=6, the public artifact pipeline, and the honest QPU noise-floor assessment are strengths appropriate to cs.MS and computational quantum information. The recurrent n=4 hub-and-triangle observation and the clean n≥5 disappearance of near-exact pairs under a fixed protocol are concrete, checkable empirical findings that others can extend with different optimizers or budgets. Significance is primarily as reproducible infrastructure and landscape sampling rather than as a resolution of the d=6 MUB problem.

major comments (2)
  1. [§5.8, §6.1] The software is advertised for arbitrary d, yet the only systematic multi-seed structural campaigns are in d=6. Section 5.8 mentions positive-control configurations in dimensions where complete MUB sets are known, but no multi-seed recovery of complete (d+1)-MUB sets in prime or prime-power dimensions (e.g. d=2,3,5,7) is reported. A short validation campaign recovering known complete sets would substantially strengthen the claim that the unanchored Lie-exponential + Adam protocol is a reliable general workflow, not only a d=6 sampler.
  2. [§6.1 Table 3; §6.4; Fig. 1] The 'recurrent hub-and-triangle' claim for n=4 rests on median near-exact count = 3 (Table 3) and representative spectra/heatmaps. The paper does not report how often the three near-exact edges form a star (one hub) versus other three-edge graphs across the 100 seeds, nor whether the defective triangle is always the complement of a single hub. A brief combinatorial summary of edge patterns over seeds would make the structural claim load-bearing rather than representative-run based.
minor comments (6)
  1. [§6] Section 6 opens with a formatting glitch: 'subsectionSingle-Seed Sweep overn= 2,...,7' (missing space and subsection markup). Fix before production.
  2. [Abstract; §6.5] Backend name is written both as ibm-marrakesh and ibm_marrakesh; standardize to the IBM platform identifier used in the citation.
  3. [Table 1; Tables 5–6] Table 1 lists Taylor order N=20 for accelerator timing only; a one-line reminder in the Table 5/6 captions that structural AMUB conclusions do not use the Taylor pathway would reduce misreading by skimmers.
  4. [Fig. 1; Fig. 2] Figures 1–2 are labeled as pairwise-loss heatmaps but the color scale is written '|Ui Uj|^2'; clarify whether the plotted quantity is the full overlap matrix or the scalar ℓij per pair, and ensure axis labels match the caption.
  5. [Table 2; §6.0–6.1] The single-seed n=3 run with s=1234 is defective (Table 2) while the multi-seed median is exact; a short cross-reference in §6.0 to the multi-seed basin diversity in §6.1 would help readers who stop at the validation sweep.
  6. [§1; §3.3] Related-work citations on computational MUB searches in d=6 are present but brief; a sentence locating the unanchored approach relative to anchored Hadamard-family searches (beyond the gauge discussion in §3.3) would improve orientation for the quantum-information audience.

Circularity Check

0 steps flagged

No significant circularity: reported AMUB structures are optimizer outputs under a fixed loss and protocol, not quantities forced by definition or fit.

full rationale

The paper’s load-bearing chain is: (i) define the standard pairwise Frobenius AMUB defect ℓ_ij and total loss L_n from the classical mutual-unbiasedness condition |U†_i U_j|² = (1/d) 1; (ii) parameterize unitaries by Lie-algebra exponentiation U_k = exp(i H_k) with H_k Hermitian by construction; (iii) minimize L_n with Adam under a fixed hyperparameter and seed protocol; (iv) post-process pairwise δ_ij against a stated tolerance τ and report observed basin structure (exact triples, n=4 hub-and-triangle, fully defective n=5,6). None of these steps defines the reported structures in terms of themselves, fits a free parameter that is later renamed a prediction, or imports a uniqueness/ansatz result from the same authors. Near-exact classification is an explicit post-hoc threshold on recorded δ_ij, not a fitted target. The hub-and-triangle pattern is an empirical description of saved pairwise losses after optimization. The authors repeatedly scope the claim as reproducible numerical evidence under a fixed workflow, not existence/nonexistence of complete MUBs. Citations are external MUB and numerical literature. Score 0 with empty steps is therefore the correct outcome.

Axiom & Free-Parameter Ledger

5 free parameters · 3 axioms · 1 invented entities

The paper rests on standard unitary Lie-algebra facts, the classical definition of mutual unbiasedness, and a small set of optimizer and classification hyperparameters that are fully disclosed. No new physical entities are postulated; the hub-and-triangle is a descriptive label for observed pairwise-defect graphs.

free parameters (5)
  • Adam learning rate η = 0.02
    Fixed at 0.02 for all structural campaigns (Table 1); controls basin selection.
  • Initialization scale = 0.05
    Fixed at 0.05; determines starting distribution of generators.
  • Step counts = 1500/2000
    1500 steps for n<5, 2000 for n≥5; finite budget that may miss deeper basins.
  • Primary near-exact tolerance τ = 1e-6
    10^{-6} used for all headline near-exact counts; relaxed tolerances examined only for sensitivity.
  • Taylor expansion order N = 20
    Order 20 used only for accelerator timing benchmarks, not structural claims.
axioms (3)
  • standard math Two orthonormal bases are mutually unbiased iff every entry of |U_i^† U_j|^2 equals 1/d.
    Classical definition (Eqs. 1–3); used throughout.
  • standard math exp(iH) is unitary whenever H is Hermitian.
    Lie-group fact used to enforce unitarity by construction (Eq. 7).
  • domain assumption The unanchored multi-seed Adam landscape with the stated hyperparameters is a meaningful sample of accessible AMUB configurations.
    Explicitly acknowledged as a limitation in §5.10; underpins the ‘no near-exact pairs’ claim for n=5,6.
invented entities (1)
  • hub-and-triangle partial-exact configuration no independent evidence
    purpose: Descriptive label for the recurrent n=4 pairwise-defect graph (three near-exact edges from one hub, three defective edges forming a triangle).
    Purely observational; no new mathematical object is postulated beyond the recorded pairwise losses.

pith-pipeline@v1.1.0-grok45 · 28893 in / 2658 out tokens · 39085 ms · 2026-07-14T10:26:54.785481+00:00 · methodology

0 comments
read the original abstract

We present a reproducible, parameter-driven software workflow for optimizing approximate mutually unbiased basis (AMUB) configurations in arbitrary dimensions d using a Lie-algebra unitary parameterization. The workflow is designed for portable execution across CPU, Apple MPS, CUDA-capable GPU, and HPC backends, using a Taylor-series matrix exponential layer as an accelerator compatibility pathway. As a dimension-six case study, we optimize unanchored configurations across 100 random seeds for basis counts n = 3, 4, 5, 6 in complex128 and complex64 arithmetic. The workflow recovers exact three-basis configurations, identifies a recurrent four-basis partial-exact hub-and-triangle structure, and finds no near-exact pairs for n = 5 or n = 6 in the reported campaigns under the primary tolerance. As a hardware-execution check, we embed the representative d = 6, n = 4 transition unitaries into three-qubit 8x8 unitaries and execute the resulting circuits on the 156-qubit Heron processor ibm-marrakesh using subspace post-selection. The measured QPU pairwise losses are dominated by a hardware and compilation noise floor of approximately 0.02-0.08, associated with compiled circuits averaging 37 native CZ gates, which obscures the distinction between classically near-exact and defective pairs. The results provide a reproducible computational framework for exploring AMUB landscapes, together with an initial assessment of the challenges involved in executing optimized dimension-six unitaries on current quantum hardware.

Figures

Figures reproduced from arXiv: 2607.10615 by Abdul Fatah, Ian McLoughlin, Saim Ghafoor.

Figure 2
Figure 2. Figure 2: Representative pairwise-loss heatmap for [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗
Figure 1
Figure 1. Figure 1: Representative pairwise-loss heatmap for [PITH_FULL_IMAGE:figures/full_fig_p019_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: Classical and QPU pairwise-loss comparison [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Best AMUB loss summary in d = 6. The plot shows median best loss with min–max range across 100 seeds for complex128 and complex64. 7 Conclusion This paper presented a reproducible mathematical-software workflow for unan￾chored approximate mutually unbiased basis optimization. The workflow is parameterized by dimension, candidate basis count, random seed, numerical precision, and backend policy. Candidate b… view at source ↗
Figure 5
Figure 5. Figure 5: Near-exact MUB pair counts in d = 6. The plot shows the median number of near-exact pairs with min–max ranges across 100 seeds. Under the primary tolerance, no near-exact pairs are observed for n = 5 or n = 6 in either precision. configurations are recovered in the reference cam￾paign, while four-basis runs repeatedly identify a structured partial-exact hub-and-triangle config￾uration with three near-exact… view at source ↗
Figure 6
Figure 6. Figure 6: Sorted pairwise AMUB loss spectra for representative complex128 configurations. Each bar corresponds [PITH_FULL_IMAGE:figures/full_fig_p025_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Sorted pairwise AMUB loss spectra for representative complex64 configurations. Machine-zero losses below [PITH_FULL_IMAGE:figures/full_fig_p026_7.png] view at source ↗

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