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arXiv:2412.20753 · Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering Sciences

Simple quantum coins enable pretty good state transfer on every hypercube

Hanmeng Zhan

Published Oct 9, 2025• 0 citations
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Abstract

We consider pretty good state transfer in coined quantum walks between antipodal vertices on the hypercube [Formula: see text]. When [Formula: see text] is a prime, this was proven to occur in the arc-reversal walk with Grover coins. We extend this result by constructing weighted Grover coins that enable pretty good state transfer on every [Formula: see text]. Our coins are real and require modification of the weight on only one arc per vertex. We also generalize our approach and establish a sufficient condition for pretty good state transfer to occur on other graphs.This article is part of the theme issue 'Numerical analysis, spectral graph theory, orthogonal polynomials and quantum algorithms'.

DOI
10.1098/rsta.2024.0418

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We consider pretty good state transfer in coined quantum walks between antipodal vertices on the hypercube [Formula: see text]. When [Formula: see text] is a prime, this was proven to occur in the arc-reversal walk with Grover coins. We extend this result by constructing weighted Grover coins that enable pretty good state transfer on every [Formula: see text]. Our coins are real and require modification of the weight on only one arc per vertex. We also generalize our approach and establish a sufficient condition for pretty good state transfer to occur on other graphs.This article is part of the theme issue 'Numerical analysis, spectral graph theory, orthogonal polynomials and quantum algorithms'.
We study a transport phenomenon in certain coined quantum walks where a subspace of states localized at a vertex gets transferred to another vertex. We first develop characterizations for perfect and pretty good subspace state transfer using the spectral properties of a Hermitian weighted digraph obtained from the underlying graph. We then provide a polynomial-time algorithm that tests whether pointwise perfect subspace state transfer occurs at an integer step, given that the subspace and coins are rational. Finally, we construct several infinite families of examples that admit pointwise perfect ddd-dimensional subspace state transfer where d≥2d\ge 2d≥2.
Quantum computing is believed to provide many advantages over traditional computing, particularly considering the speed at which computations can be performed. One of the challenges that needs to be resolved in order to construct a quantum computer is the transmission of information from one part of the computer to another. Quantum walks, the quantum analogues of classical random walks, provide one potential method for resolving this challenge. \n \nIn this thesis, we use techniques from algebraic graph theory and number theory to analyze the mathematical model for continuous time quantum walks on graphs. For the continuous time quantum walk model, we define a transition operator, which is a function of a Hamiltonian. We focus on the cases where the adjacency matrix or the Laplacian of a graph act as the Hamiltionian. We mainly consider quantum walks on paths as a model for spin chains, which are the underlying basis of a quantum communication protocol. \n \nFor communication to be efficient, we desire states to be transferred with high fidelity, a measure of the amount of similarity between the transmitted state and the received state. At the maximum fidelity of 1, we say we have achieved perfect state transfer. Examples of perfect state transfer are relatively rare, so the concept of pretty good state transfer was introduced as a natural relaxation, which exists if fidelities arbitrarily close to 1 are obtained. \n \nOur first main result is to characterize pretty good state transfer on paths. Previously, pretty good state transfer on paths was considered mainly for the end vertices, though results for both models indicated that if there was pretty good state transfer between the end vertices, then there was pretty good state transfer between internal vertices equidistant from each end. We complete the characterization by demonstrating, for the adjacency matrix model, a family of paths where pretty good state transfer exists between internal vertices but not between end vertices, and verifying that no other example exists. For the Laplacian model, we show that there are no paths with pretty good state transfer between internal vertices but not between the end vertices. \n \nOur second main result considers initial states involving multiple vertices. Under the adjacency matrix model, we provide necessary and sufficient conditions for pretty good state transfer in a particular family of paths in terms of the eigenvalue support of the initial state. We also discuss recent results on fractional revival, which is another form of multiple qubit state transfer.
Abstract We establish the theory for pretty good state transfer in discrete-time quantum walks. For a class of walks, we show that pretty good state transfer is characterized by the spectrum of certain Hermitian adjacency matrix of the graph; more specifically, the vertices involved in pretty good state transfer must be strongly cospectral relative to this matrix, and the arccosines of its eigenvalues must satisfy some number theoretic conditions. Using normalized adjacency matrices, cyclic covers, and the theory on linear relations between geodetic angles, we construct several infinite families of walks that exhibits this phenomenon.
Abstract We investigate state transfer on a hypercube by means of a quantum walk where the sender and the receiver vertices are marked by a weighted loops. First, we analyze search for a single marked vertex, which can be used for state transfer between arbitrary vertices by switching the weighted loop from the sender to the receiver after one run-time. Next, state transfer between antipodal vertices is considered. We show that one can tune the weight of the loop to achieve state transfer with high fidelity in shorter run-time in comparison to the state transfer with a switch. Finally, we investigate state transfer between vertices of arbitrary distance. It is shown that when the distance between the sender and the receiver is at least 2, the results derived for the antipodes are well applicable. If the sender and the receiver are direct neighbours the evolution follows a slightly different course. Nevertheless, state transfer with high fidelity is achieved in the same run-time.