slug: laneve-2024-qsp-collapse-conjecture-refutation
bibkey: laneve2024multivariateqsp
doi: 10.22331/q-2025-02-20-1641
url: https://arxiv.org/abs/2407.20823v2
triage: theorem
motivation_gids:
  - D5/S3/Quantum/Algebra/MultivariateQSPCollapseRefutation.result

A three-level QSP segment that collapses without a common monomial factor

Problem

L. Laneve and S. Wolf, On multivariate polynomials achievable with quantum signal processing, arXiv:2407.20823v2 (Quantum 9, 1641 (2025)), Conjecture 8: if a Protocol C segment $A_m\stackrel{\cdots A1W}{W}$ ($\tilde{W}=\mathrm{diag}(1,a,b)$, $A_k\in SU(3)$) maps a polynomial state of effective dimension $\le 2$ to one of effective dimension $\le 2$ while every intermediate state has effective dimension $>2$, then the segment acts as $a^k{b}^h\cdot U$ on some pair of subspaces $H\to H^{\prime }$ of dimension $d\ge 2$. The verbatim statements are in the literature note.

Issue #14369 reads the conclusion permissively — subspaces $H,H^{\prime }$ of ${\mathbb{C}}^3$ with its Euclidean (Hermitian) norm, any linear isometry $U:H\to H^{\prime }$ and any $k,h\in \mathbb{N}$ — so that its refutation covers the printed form, where $U\in SU(d)$ is taken in orthonormal bases of $H$ and $H^{\prime }$.

Motivation

The paper states that Conjecture 8 is the final step it could not prove towards Conjecture 7: two-dimensional outputs of three-level multivariate QSP would then be reachable by two-level QSP, since a segment acting as $a^k{b}^h\cdot U$ can be replaced by one that never leaves two dimensions.

Gap

The authors’ own example $(1,a,b)\mapsto (1,ab,ab)$ acts as $ab\cdot \mathrm{id}$ on a two-dimensional subspace and fits the conjecture. Issue #14369 records the literature check before any Lean: no later work settles Conjecture 7 or 8; not-found-in-searched-scope.

Route

$m=2$, $A_1$ and $A_2$ the two cyclic permutations of the basis, ${\gamma }^{\prime }=(3,4,12a)/13$: ${\gamma }_1=(12ab,3,4a)/13$ has effective dimension 3, $\gamma =(3a,4ab,12ab)/13$ and ${\gamma }^{\prime }$ have effective dimension 2, and the segment is $T=\mathrm{diag}(a,ab,b)$. At $(a,b)=(1,1)$ the conclusion forces $U={\mathrm{id}}_H$; at $(i,-1)$ it forces $\mathrm{diag}(i,-i,-1)$ to act as a scalar on $H$, so $\dim H\le 1$.

Falsifier

The refutation concerns Conjecture 8 as printed. It does not address Conjecture 7: the output $(3a,4ab,12ab)/13$ may still be reachable by two-level QSP through another protocol.

Evidence

The canonical source is D5/S3/Quantum/Algebra/MultivariateQSPCollapseRefutation.lean, with public claim and result : ¬ claim. The axiom closure of result is exactly propext, Classical.choice and Quot.sound; there is no sorry, native_decide or new axiom. The module statement is sha256:30394a20a7a75b501d527feeca513db9e18f852500e33db4ae96536cd83dfc8d, the result statement sha256:3857e9bd93ab7e36a288e7abb485c451cee11d2c9849049ca152f850a3884479 and the claim statement sha256:d0840cb210e71fa32ee6b5f0eb5beaddfee40e196b37d6a608dd60764a2a5b41. The Freeze event is sha256:d6f5d294c0eb7f64c338a3ac5e4e77109f0eb17bd5a46c0fa309af32d4cf87bf; it has no project-level prerequisite (the module imports Mathlib only).

Triage

Tier 1 auxiliary conjecture of a 2024 paper (kept in the Quantum version), preregistered in issue #14369 before any Lean. theorem; resolution refuted.

The private theorem scalar_subspace_dimension carries the content: a diagonal matrix with pairwise distinct entries acts as a scalar only on subspaces of dimension at most one. It is not a Mathlib statement and lies on the live proof of result. The remaining private theorems (coefficient_span, initial_normalized, stages, transfer_diagonal, initial_dimension, middle_dimension, final_dimension, distinct_diagonal) are bind-only instantiations and normalizations, each used on the proof of result. The settlement is admitted as an external open-problem resolution.

Utility is kind=certified-instance; basis=refutes (the claim and its refutation). There is no digestion atom.

What the refutation shows

Proved by result: Conjecture 8 fails for a two-step segment whose processing operators are permutation matrices.

Argued, not formalized.

• Mechanism. The output returns to effective dimension 2 because two output coordinates acquire the same monomial $ab$ from different sources: one from the operator factor $ab$ acting on the constant coefficient, the other from the operator factor $b$ acting on the coefficient that already carries $a$. A collision of monomials in the state does not require a common monomial factor of the operator on any fixed subspace, which is the step the conjecture assumes.
• Conjecture 7. The output is $\gamma =a\cdot (3,4b,12b)/13$; the example shows only that this segment does not have the structure Conjecture 8 asserts, not that $\gamma$ is unreachable by two-level QSP. It removes the proposed route to Conjecture 7 without refuting Conjecture 7.

Open. Conjecture 7 is not determined here, nor is Conjecture 8 restricted to segments whose input state does not depend on $a,b$ (the setting of the authors’ own example).

ASSUMED-UNVERIFIED

The bounded literature check does not establish exhaustive worldwide novelty, priority, or the absence of an independent proof.