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arXiv:2609.07267

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

Ce Wang

quant-phmath-phmath.FAmath.PRSubmitted Sep 7, 2026• Announced Sep 8, 2026• 0 citations
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Abstract

We study disorder-free localization in a spin-coupled quantum walk on the hypercube within the quantum Bernoulli noises framework. We derive a general criterion for disorder-free localization: it occurs if and only if some spectral subspace of a reduced unitary evolution contains a vector whose position marginal probability is non-uniform. For the Grover walk with distance-dependent coupling \(φ_σ=|σ|π/(n+1)\) in dimension \(d=4\), we explicitly construct a non-uniform fixed point in the all \(+1\) spin configuration. This implies that disorder-free localization occurs in our model.

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We study disorder-free localization in a spin-coupled quantum walk on the hypercube within the quantum Bernoulli noises framework. We derive a general criterion for disorder-free localization: it occurs if and only if some spectral subspace of a reduced unitary evolution contains a vector whose position marginal probability is non-uniform. For the Grover walk with distance-dependent coupling (φ_σ=|σ|π/(n+1)) in dimension (d=4), we explicitly construct a non-uniform fixed point in the all (+1) spin configuration. This implies that disorder-free localization occurs in our model.
In this paper, we present a quantum Bernoulli noises approach to quantum walks on hypercubes. We first obtain an alternative description of a general hypercube, and then, based on the alternative description, we find that the operators ∂k*+∂k behave actually as the shift operators, where ∂k and ∂k* are the annihilation and creation operators acting on Bernoulli functionals, respectively. With the above-mentioned operators as the shift operators on the position space, we introduce a discrete-time quantum walk model on a general hypercube and obtain an explicit formula for calculating its probability distribution at any time. We also establish two limit theorems showing that the averaged probability distribution of the walk even converges to the uniform probability distribution. Finally, we show that the walk produces the uniform measure as its stationary measure on the hypercube provided its initial state satisfies some mild conditions. Some other results are also proven.
We consider spin-coupled Grover walks on hypercubes within the framework of quantum Bernoulli noises (QBN). We identify parity-dependent harmonic subspaces generated by the central Hamming layers: in even dimension they give a fixed-point subspace, whereas in odd dimension the walk reduces on the harmonic subspace to an explicit eight-dimensional reduced dynamics. Using symmetric-group representation theory, we characterize the corresponding isotypic structure and determine the reduced spectrum. The harmonic reduction also yields explicit long-time probability behavior and shows that the resulting states may occupy an asymptotically vanishing fraction of the hypercube while still spanning its full Hamming diameter.
We study quantum-assisted balanced rendezvous on the ddd-dimensional hypercube in a one-step two-agent quantum-walk framework, where the two agents move the same Hamming distance and meet at a common midpoint. The key structure is a projective ttt-sphere resolution, which governs exact finite-dimensional quantum attainment. Its scalar analogue recovers the corresponding classical characterization. We characterize finite-dimensional quantum attainment of the no-signalling benchmark, construct quantum strategies from projective sphere resolutions, and establish a rigidity theorem for optimal finite-dimensional strategies. A marked difference appears between nearest-neighbor and higher-cardinality updates: quantum advantage can occur in the former, whereas in the latter the geometry of the relevant Hamming spheres rules out exact finite-dimensional attainment of the no-signalling value. Together, these results connect quantum-walk dynamics, Hamming-sphere geometry, and quantum rendezvous within the quantum Bernoulli noises (QBN) framework.
In this paper, we introduce and investigate a model of magnetic quantum walk on a general hypercube. We first construct a set of unitary involutions associated with a magnetic potential ν by using quantum Bernoulli noises. And then, with these unitary involutions as the magnetic shift operators, we define the evolution operator W(ν) for the model, where ν is the magnetic potential. We examine the point-spectrum and approximate-spectrum of the evolution operator W(ν) and obtain their representations in terms of the coin operator system of the model. We show that the point-spectrum and approximate-spectrum of W(ν) are completely independent of the magnetic potential ν although W(ν) itself is dependent of the magnetic potential ν. Our work might suggest that a quantum walk perturbed by a magnetic field can have spectral stability with respect to the magnetic potential.
Open quantum walks (OQWs) (also known as open quantum random walks) are quantum analogs of classical Markov chains in probability theory, and have potential application in quantum information and quantum computation. Quantum Bernoulli noises (QBNs) are annihilation and creation operators acting on Bernoulli functionals, and can be used as the environment of an open quantum system. In this paper, by using QBNs as the environment, we introduce an OQW on a general higher-dimensional integer lattice. We obtain a quantum channel representation of the walk, which shows that the walk is indeed an OQW. We prove that all the states of the walk are separable provided its initial state is separable. We also prove that, for some initial states, the walk has a limit probability distribution of higher-dimensional Gauss type. Finally, we show links between the walk and a unitary quantum walk recently introduced in terms of QBNs.
We investigate the phenomenon of disorder-free localization in quantum systems with global permutation symmetry. We use permutation group theory to systematically construct permutation-symmetric many-fermion Hamiltonians and interpret them as generators of continuous-time quantum walks. When the number of fermions is very large we find that all the canonical basis states localize at all times, without the introduction of any disorder coefficients. This time-independent localization is not the result of any emergent disorder, distinguishing it from existing mechanisms for disorder-free localization. Next we establish the conditions under which the localization is preserved. We find that interactions that preserve and break the global permutation symmetry sustain localization. Furthermore, the basis states of systems with reduced permutation symmetry localize even for a small number of fermions when the symmetry-reducing parameters are tuned accordingly. We show that similar localization also occurs for a permutation-symmetric Heisenberg spin chain and permutation-symmetric bosonic systems, implying that the localization is independent of the superselected symmetry. Finally, we make connections of the Hamiltonians studied here to the adjacency matrices of graphs and use this to propose a prescription for disorder-free localization in continuous-time quantum walk systems. Many of the models proposed here feature all-to-all connectivity and can be potentially realized on superconducting quantum circuits, trapped ion systems, and ultracold atoms.