aiqbench analysis
This paper is a mathematical analysis of discrete-time quantum walks on hypercubes, using quantum Bernoulli noise and symmetric-group representation theory to reduce a high-dimensional walk to an eight-dimensional effective dynamics in odd dimensions. Quantum walks are a standard building block for quantum search and algorithm design, so the results are relevant to theorists, but this is foundational analysis with no hardware or algorithmic speedup claimed. Readers should treat the reduced spectra and long-time behavior as structural results, not as evidence of practical advantage.
Harmonic Reduction of Spin-Coupled Grover Walks on Hypercubes via Quantum Bernoulli Noises
arXiv:2610.14467v1 Announce Type: new Abstract: 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.
Abstract: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.
Quantum Physics (quant-ph); Mathematical Physics (math-ph); Combinatorics (math.CO); Probability (math.PR); Representation Theory (math.RT)
arXiv:2610.14467 [quant-ph]
(or arXiv:2610.14467v2 [quant-ph] for this version)
https://doi.org/10.48550/arXiv.2610.14467 Focus to learn more arXiv-issued DOI via DataCite
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