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arXiv:2503.08595

Limiting distributions of ergodic continuous-time quantum walks on periodic graphs

Anne Boutet de Monvel, Kiran Kumar A. S., Mostafa Sabri

Published Mar 11, 2025• 0 citations
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Abstract

In this expository note, we study several families of periodic graphs which satisfy a sufficient condition for the ergodicity of the associated continuous-time quantum walk. For these graphs, we compute the limiting distribution of the walk explicitly. We uncover interesting behavior where in some families, the walk is ergodic in both horizontal and sectional directions, while in others, ergodicity only holds in the horizontal (large N) direction. We compare this to the limiting distribution of classical random walks on the same graphs.

DOI
10.48550/arxiv.2503.08595

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In this expository note, we study several families of periodic graphs which satisfy a sufficient condition for the ergodicity of the associated continuous-time quantum walk. For these graphs, we compute the limiting distribution of the walk explicitly. We uncover interesting behavior where in some families, the walk is ergodic in both horizontal and sectional directions, while in others, ergodicity only holds in the horizontal (large N) direction. We compare this to the limiting distribution of classical random walks on the same graphs.
Quantum walks can be defined in two quite distinct ways: discrete-time and continuous-time quantum walks (DTQWs and CTQWs). For classical random walks, there is a natural sense in which continuous-time walks are a limit of discrete-time walks. Quantum mechanically, in the discrete-time case, an additional ``coin space'' must be appended for the walk to have nontrivial time evolution. Continuous-time quantum walks, however, have no such constraints. This means that there is no completely straightforward way to treat a CTQW as a limit of a DTQW, as can be done in the classical case. Various approaches to this problem have been taken in the past. We give a construction for walks on ddd-regular, ddd-colorable graphs when the coin flip operator is Hermitian: from a standard DTQW we construct a family of discrete-time walks with a well-defined continuous-time limit on a related graph. One can think of this limit as a ``coined'' continuous-time walk. We show that these CTQWs share some properties with coined DTQWs. In particular, we look at a spatial search by a DTQW over the two-dimensional (2D) torus (a grid with periodic boundary conditions) of size $\sqrt{N}\ifmmode\times\else\texttimes\fi{}\sqrt{N}$, where it was shown that a coined DTQW can search in time O(NlogN)O(\sqrt{N}logN)O(N​logN), but a standard CTQW takes $\mathrm{\ensuremath{\Omega}}(N)$ time to search for a marked element. The continuous limit of the DTQW search over the 2D torus exhibits the O(NlogN)O(\sqrt{N}logN)O(N​logN) scaling, like the coined walk it is derived from. We also look at the effects of graph symmetry on the limiting walk, and show that the properties are similar to those of the DTQW as shown in Krovi and Brun, Phys. Rev. A 75, 062332 (2007).
We undertake a detailed analysis of ergodicity for homogeneous discrete-time quantum walks on the integer lattice. The most significant result of our paper holds in dimension one, and gives a complete equivalence between the absolutely continuous spectrum of the unitary operator encoding the walk, and the equidistribution of its dynamics in position space, which appears for the first time in the context of large-volume quantum ergodicity. In higher dimensions, we give a criterion for full and partial ergodicity in terms of a finer property of the spectrum which we dub ``No Repeating Graphs'', and we distinguish how strongly the equidistribution is taking place (weak convergence vs total variation). Many examples are included to illustrate the criterion and to distinguish between the types of ergodicity.
We present three families of graphs which admit quantum walks with interesting dynamics either in the continuous time walk, or in the discrete time walk for appropriate selections of coin and initial conditions. These dynamics are either periodicity, perfect state transfer, or very high fidelity state transfer. These families are modifications of families known not to exhibit periodicity or perfect state transfer in general. The robustness of the dynamics is tested by varying the initial state and by adding decoherence. The perfect state transfer is found to be very robust to small disturbances. The families of graphs found to exhibit useful properties are generalisations of small graphs. By varying the strength of edges in each family, interpolation between the families can be carried out, enabling further understanding of the factors affecting perfect state transfer. The perfect state transfer in these families is found to be destroyed by this variation, indicating that the transport strongly depends on the graph structure. A brief summary of other small graphs investigated but found not to exhibit such properties is included for completeness.
The position density of a "particle" performing a continuous-time quantum walk on the integer lattice, viewed on length scales inversely proportional to the time , converges (as tends to infinity) to a probability distribution that depends on the initial state of the particle. This convergence behavior has recently been demonstrated for the simplest continuous-time random walk [N. Konno, Phys. Rev. E 72, 026113 (2005)]. In this Brief Report, we use a different technique to establish the same convergence for a very large class of continuous-time quantum walks, and we identify the limit distribution in the general case.