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slug: connelly-2017-property-t-universal-state-transfer bibkey: connelly2017universality doi: 10.1016/j.laa.2017.06.015 url: https://arxiv.org/abs/1701.04145v2 triage: theorem motivation_gids:

  • D5/S3/Quantum/Dynamics/PropertyTCirculantUniversalTransfer.result

A unit-gain circulant on three vertices with universal perfect state transfer

Problem

E. Connelly, N. Grammel, M. Kraut, L. Serazo and C. Tamon (arXiv:1701.04145, quant-ph and math.CO; Linear Algebra Appl. 531 (2017) 516–532) study the continuous-time quantum walk U(t) = exp(−i t A) of a graph with Hermitian adjacency matrix A. The walk has universal perfect state transfer if for all vertices u, v some time gives |U(t)_{v,u}| = 1. A graph has property 𝕋 if every nonzero entry of A has modulus 1. The paper states:

\begin{conjecture} $\Circ(0,-\ii,\ii)$ is the only circulant with property
$\TT$ which has universal perfect state transfer. \end{conjecture}

and, in its introduction,

The only known examples of complex unit gain graphs
with the universal property are the circulants $K_{2}$ and $\Circ(0,-\ii,\ii)$.
We conjecture that this set is unique.

Issue #11528 fixes the reading. Circ(a) has entries C_{jk} = a_{k−j} over ℤ/nℤ, “the only” is read up to the paper’s switching equivalence (MA = BM with M a permutation matrix times an invertible diagonal matrix) and together with K₂, the order is n ≥ 2, and the circulant is Hermitian without loops. The formal claim is: every such circulant with property 𝕋 and universal perfect state transfer is switching equivalent to K₂ or to Circ(0, −i, i). The result refutes it.

Motivation

Universal perfect state transfer is rare: real symmetric matrices on more than two vertices never have it, and a later paper (arXiv:2301.01473) proved that among oriented graphs, with weights ±i, only K₂ and the oriented triangle have it. The property-𝕋 conjecture allows arbitrary unit phases. The frozen declaration D5/S3/Quantum/Dynamics/PropertyTCirculantUniversalTransfer.result shows that it fails already on three vertices.

Gap

Issue #11528 preregisters the reading, the counterexample and the literature check. The conjecture is in the latest arXiv version. Semantic Scholar lists 15 citing records. The arXiv sources of the 10 with arXiv identifiers were searched at every mention of the paper, of universality, of unit gains and of unimodular weights; none states or refutes the property-𝕋 conjecture. arXiv:2002.04666 restates it for unweighted oriented graphs, and arXiv:2301.01473 proves that oriented version. The witness follows from the Cameron et al. eigenvalue criterion that the paper quotes, and Zimborás et al. (arXiv:1208.4049) remark that the chiral triangle reaches perfect transfer for phases found by phase matching; no publication found states that the conjecture fails.

These readings are not-found-in-searched-scope; they do not establish an exhaustive worldwide literature search, priority, or the absence of an independent answer.

Route

  1. Take A = Circ(0, α, ᾱ) with α = (−4√3 + i)/7. Then |α| = 1, so A is Hermitian, has no loops and has property 𝕋.
  2. The Fourier matrix V_{km} = ω^{km}, ω = e^{2πi/3}, diagonalizes A with eigenvalues (√3/7)(−8, 3, 5).
  3. At t₁ = 14√3π/9 the phases e^{−i t₁ λ_m} are ω², 1, ω, so U(t₁) = ω² P with P the cyclic permutation matrix, and U(2t₁) = U(t₁)². With U(0) = I this gives universal perfect state transfer.
  4. det A = α³ + ᾱ³ = −360√3/343 ≠ 0 = det Circ(0, −i, i), and a monomial matrix is invertible, so A is not switching equivalent to Circ(0, −i, i); it is not switching equivalent to K₂ since 3 ≠ 2.

Falsifier

The answer would change if “circulant” were restricted to weights in {0, ±i}: that oriented statement is a theorem of arXiv:2301.01473. It does not change under rescaling of A: for c ≠ 0, det(cA) = c³ det A ≠ 0 while det Circ(0, −i, i) = 0. The literal sentence of the conjecture omits K₂; the reading includes K₂, as the paper’s introduction does.

Evidence

Exact SymPy arithmetic (issue #11528) gives the eigenvalues [−8√3/7, 3√3/7, 5√3/7], U(14√3π/9) = ω̄ P, U(28√3π/9) = ω P² and det A = −360√3/343. As a positive control, the same code on Circ(0, −i, i) reproduces the paper’s known example.

The canonical source is D5/S3/Quantum/Dynamics/PropertyTCirculantUniversalTransfer.lean. Its public declarations are PropertyT, UPST, SwitchingEquivalent, K2, orientedTriangle, claim and result. The frozen module state has statement identity sha256:0626c620229dbc592b50fc061705092ee9cf1f27255588fcd0766c2e39a139bc. The result declaration has statement identity sha256:b351231e29348c584acf93cdc8498913b76e7c7689ef35db3a0ce464419bfda6. The Freeze event is sha256:2f26f5e60d2e2790d6be55801b59731d2967ebb2b5805966b46a5507ec3f2e7c. Its project-level frozen prerequisite is the module providing hamiltonianPropagator. The proof uses only the standard axioms propext, Classical.choice and Quot.sound; no sorry, native_decide, or new axiom.

Triage

Tier 1 conjecture from a 2017 quant-ph and math.CO paper, preregistered in issue #11528 before any Lean. theorem; resolution refuted. The public theorem has proof_shape: bind-only: the Fourier diagonalization, the three phases and the matrix exponential lemmas of Mathlib give the closed form of the walk at t₁, and every step is a local have of result. Its admission basis is open-problem-resolution. Utility kind=certified-instance; basis=refutes with typed claim and result.

What the settlement shows

Refuted: D5/S3/Quantum/Dynamics/PropertyTCirculantUniversalTransfer.result shows that the property-𝕋 conjecture fails already for circulants of order 3. Inspection of the checked proof, not a separate theorem: the witness’s eigenvalues are √3/7 times the integers −8, 3, 5, which have distinct residues mod 3; this is the Cameron et al. criterion quoted by the paper, and it does not depend on the weights being ±i.

Surviving restriction: for weights ±i (oriented graphs) the uniqueness statement is the theorem of arXiv:2301.01473; the witness has unit weights other than ±i and does not bear on it.

Bounded source consequences: the witness is a Hermitian graph on 3 vertices with unit weights and universal perfect state transfer that is not switching equivalent to Circ(0, −i, i), so it also contradicts Fact 5 of arXiv:1701.04145v2 §8 (l. 1117–1119: $\Circ(0,-\ii,\ii)$ is the only graph on $3$ vertices with universal perfect state transfer, up to switching equivalence) when graphs carry unit gains. The Fact’s proof shows that such a graph is a circulant diagonalized by the Fourier matrix, but it does not fix the eigenvalues. This consequence follows by inspection of the checked witness and is not a separate formal theorem.

Open extent: the classification, up to switching equivalence, of the property-𝕋 circulants of order n ≥ 3 with universal perfect state transfer is not settled here.

ASSUMED-UNVERIFIED

The published Linear Algebra Appl. text was not read, so it is unverified whether the conjecture was edited there. The bounded literature check does not establish exhaustive worldwide novelty, priority, or the absence of an independent answer. The Lean kernel does not authenticate the external source or its version history.