bibkey: wang2026hypercubelocalization authors: Ce Wang year: 2026 title: “Spectral Criterion for Disorder-Free Localization of Quantum Walks on Hypercube: QBN Approach” doi: null url: https://arxiv.org/abs/2609.07267v1 claim: “Conjecture 4.1 asserts a nonuniform fixed point for the all-plus spin-coupled Grover walk on every even hypercube of dimension at least four.” strata_touched:

• D5/S3/Quantum/Dynamics/SpinCoupledEvenHypercubeGroverFixedPoint license: citation-only triage: anchor

Spectral Criterion for Disorder-Free Localization of Quantum Walks on Hypercube: QBN Approach

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Source: https://arxiv.org/abs/2609.07267v1

Section 3, page 6, Proposition 3.4 (prop:reduced_evolution) gives the reduced evolution operator (3.14). Section 4, page 11, Proposition 4.3 gives the fixed-point equation (4.4), and Corollary 4.4 (cor:fixed_point_DFL) gives the implication from a nonzero fixed point with nonconstant position weights to disorder-free localization. Page 12 defines the Grover coin and the distance-dependent coupling. Propositions 4.5–4.6, pages 12–13, treat dimensions four and six. Conjecture 4.1 (conj:DFL_general) is on page 13.

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Source statement

For every even (d\ge 4), the spin-coupled Grover walk on the (d)-dimensional hypercube with coupling (\phi_\sigma=|\sigma|\pi/d) and all (+1) spin configuration exhibits disorder-free localization. Equivalently, the reduced evolution operator (\tilde W_{\mathbf s}) possesses a fixed point whose position distribution is non-uniform.

With all spins equal to +1, the coin-amplitude form of the reduced operator is

$$(Wu)(\tau ,k)=\exp \left(-i\pi |\tau △\{k\}|/d\right)\left(\frac{2}{d}\sum _{j}u(\tau △\{k\},j)-u(\tau △\{k\},k)\right).$$

A position vertex is a subset of the d coordinate indices. Its unnormalized position weight is the sum over coin coordinates of the squared complex norm. Corollary 4.4 normalizes a nonzero fixed point; division by its total squared norm preserves nonconstancy.

The construction in SpinCoupledEvenHypercubeGroverFixedPoint.result proves the equivalent fixed-point clause using paired-coordinate signs and three adjacent Hamming layers. The implication to disorder-free localization uses Corollary 4.4 of the paper. The Lean claim does not formalize the quantum Bernoulli noise Hilbert-space presentation or the full spectral criterion.