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Infinite Dimensional Analysis Quantum Probability and Related Topics

Uniform measure for continuous-time quantum walk on hypercube

Ce Wang

Published Jul 28, 2026• 0 citations
Publisher

Abstract

In this paper, we introduce a generalized continuous-time quantum walk (CTQW) on a hypercube, which can be interpreted as a CTQW with a “coin”. We examine its properties from a perspective of probability distribution and spectral theory, and obtain formulae for calculating its probability distributions and spectra, respectively. In particular, we prove that, for a wide range of initial states, the walk produces the uniform measure on the hypercube as its a stationary measure. Finally, we show that the time-reversal symmetry of the walk can be described directly by a unitary operator. Some other results are also proven.

DOI
10.1142/s0219025726500153

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In this paper, we introduce a generalized continuous-time quantum walk (CTQW) on a hypercube, which can be interpreted as a CTQW with a “coin”. We examine its properties from a perspective of probability distribution and spectral theory, and obtain formulae for calculating its probability distributions and spectra, respectively. In particular, we prove that, for a wide range of initial states, the walk produces the uniform measure on the hypercube as its a stationary measure. Finally, we show that the time-reversal symmetry of the walk can be described directly by a unitary operator. Some other results are also proven.
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.
The mixing time of a discrete-time quantum walk on the hypercube is considered. The mean probability distribution of a Markov chain on a hypercube is known to mix to a uniform distribution in time O(n log n)O(n\text{ }\text{log}\text{ }n)O(n log n). We show that the mean probability distribution of a discrete-time quantum walk on a hypercube mixes to a (generally nonuniform) distribution $\ensuremath{\pi}(x)$ in time O(n)O(n)O(n), and the stationary distribution is determined by the initial state of the walk. An explicit expression for $\ensuremath{\pi}(x)$ is derived for the particular case of a symmetric walk. These results are consistent with those obtained previously for a continuous-time quantum walk. The effect of decoherence due to randomly breaking links between connected sites in the hypercube is also considered. We find that the probability distribution mixes to the uniform distribution as expected. However, the mixing time has a minimum at a critical decoherence rate $p\ensuremath{\approx}0.1$. A similar effect was previously reported for a quantum walk on an NNN-cycle with decoherence from repeated measurements of position. A controlled amount of decoherence helps in obtaining---and preserving---a uniform distribution over the 2n{2}^{n}2n sites of the hypercube in the shortest possible time.
Time-reversal symmetry is of fundamental importance to physics. In the classical theory of time-reversal symmetry, the time-reversal symmetry of a quantum system is described by an anti-unitary operator, which is known as the time-reversal operator of the system. In this paper, we introduce and study a model of continuous-time quantum walk on a special locally infinite graph. After examining its spectral property, we investigate the time-reversal symmetry of the model. To our surprise, we find that its time-reversal symmetry can be described directly by a unitary operator, which contrasts sharply with that in the classical theory of time-reversal symmetry. Some other related results are also proven.
In this paper, continuous-time quantum walk on hypercube is discussed in view of Cartesian product structure. We find that the [Formula: see text]-fold Cartesian power of the complete graph [Formula: see text] is the [Formula: see text]-dimensional hypercube, which give us new ideas for the study of quantum walk on hypercube. Combining the product structure, the spectral distribution of the graph and the quantum decomposition of the adjacency matrix, the probability amplitudes of the continuous-time quantum walker’s position at time [Formula: see text] are given, and it is discussed that the probability distribution for the continuous-time case is uniform when [Formula: see text]. The application of this product structure greatly improves the study of quantum walk on complex graphs, which has far-reaching influence and great significance.
In this paper, continuous-time quantum walk is discussed based on the view of quantum probability, i.e. the quantum decomposition of the adjacency matrix A of graph. Regard adjacency matrix A as Hamiltonian which is a real symmetric matrix with elements 0 or 1, so we regard [Formula: see text] as an unbiased evolution operator, which is related to the calculation of probability amplitude. Combining the quantum decomposition and spectral distribution [Formula: see text] of adjacency matrix A, we calculate the probability amplitude reaching each stratum in continuous-time quantum walk on complete bipartite graphs, finite two-dimensional lattices, binary tree, [Formula: see text]-ary tree and [Formula: see text]-fold star power [Formula: see text]. Of course, this method is also suitable for studying some other graphs, such as growing graphs, hypercube graphs and so on, in addition, the applicability of this method is also explained.
Hitting times for discrete quantum walks on graphs give an average time before the walk reaches an ending condition. To be analogous to the hitting time for a classical walk, the quantum hitting time must involve repeated measurements as well as unitary evolution. We derive an expression for hitting time using superoperators, and numerically evaluate it for the discrete walk on the hypercube. The values found are compared to other analogs of hitting time suggested in earlier work. The dependence of hitting times on the type of unitary ``coin'' is examined, and we give an example of an initial state and coin which gives an infinite hitting time for a quantum walk. Such infinite hitting times require destructive interference, and are not observed classically. Finally, we look at distortions of the hypercube, and observe that a loss of symmetry in the hypercube increases the hitting time. Symmetry seems to play an important role in both dramatic speed-ups and slow-downs of quantum walks.