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arXiv:quant-ph/0205083

Quantum Random Walks Hit Exponentially Faster

Julia Kempe

Published May 14, 2002• 95 citations
arXivHTMLPDF

Abstract

We show that the hitting time of the discrete time quantum random walk on the n-bit hypercube from one corner to its opposite is polynomial in n. This gives the first exponential quantum-classical gap in the hitting time of discrete quantum random walks. We provide the framework for quantum hitting time and give two alternative definitions to set the ground for its study on general graphs. We then give an application to random routing.

DOI
10.48550/arxiv.quant-ph/0205083

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We show that the hitting time of the discrete time quantum random walk on the n-bit hypercube from one corner to its opposite is polynomial in n. This gives the first exponential quantum-classical gap in the hitting time of discrete quantum random walks. We provide the framework for quantum hitting time and give two alternative definitions to set the ground for its study on general graphs. We then give an application to random routing.
It has been proved by Kempe that discrete quantum walks on the hypercube (HC) hit exponentially faster than the classical analog. The same was also observed numerically by Krovi and Brun for a slightly different property, namely, the expected hitting time. Yet, to what extent this striking result survives in more general graphs is to date an open question. Here, we tackle this question by studying the expected hitting time for quantum walks on HCs that are embedded into larger symmetric structures. By performing numerical simulations of the discrete quantum walk and deriving a general expression for the classical hitting time, we observe an exponentially increasing gap between the expected classical and quantum hitting times, not only for walks on the bare HC, but also for a large family of embedded HCs. This suggests that the quantum speedup is stable with respect to such embeddings.
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.
Recently, it has been shown that one-dimensional quantum walks can mix more quickly than classical random walks, suggesting that quantum Monte Carlo algorithms can outperform their classical counterparts. We study two quantum walks on the n-dimensional hypercube, one in discrete time and one in continuous time. In both cases we show that the quantum walk mixes in (π/4)n steps, faster than the O(n log n) steps required by the classical walk. In the continuous-time case, the probability distribution is {\em exactly} uniform at this time. More importantly, these walks expose several subtleties in the definition of mixing time for quantum walks. Even though the continuous-time walk has an O(n) instantaneous mixing time at which it is precisely uniform, it never approaches the uniform distribution when the stopping time is chosen randomly as in [AharonovAKV2001]. Our analysis treats interference between terms of different phase more carefully than is necessary for the walk on the cycle; previous general bounds predict an exponential, rather than linear, mixing time for the hypercube.
We demonstrate an implementation of the hitting time of a discrete time quantum random walk on cubelike graphs using IBM's Qiskit platform. Our implementation is based on efficient circuits for the Grover and Shift operators. We verify the results about the one-shot hitting time of quantum walks on a hypercube as proved in [https://link.springer.com/article/10.1007/s00440-004-0423-2]. We extend the study to another family of cubelike graphs called the augmented cubes [https://onlinelibrary.wiley.com/doi/abs/10.1002/net.10033]. Based on our numerical study, we conjecture that for all families of cubelike graphs there is a linear relationship between the degree of a cubelike graph and its hitting time which holds asymptotically. That is, for any cubelike graph of degree ΔΔΔ, the probability of finding the quantum random walk at the target node at time πΔ2\frac{πΔ}{2}2πΔ​ approaches 1 as the degree ΔΔΔ of the cubelike graph approaches infinity.
Hitting times are the average time it takes a walk to reach a given final vertex from a given starting vertex. The hitting time for a classical random walk on a connected graph will always be finite. We show that, by contrast, quantum walks can have infinite hitting times for some initial states. We seek criteria to determine if a given walk on a graph will have infinite hitting times, and find a sufficient condition, which for discrete time quantum walks is that the degeneracy of the evolution operator be greater than the degree of the graph. The set of initial states which give an infinite hitting time form a subspace. The phenomenon of infinite hitting times is in general a consequence of the symmetry of the graph and its automorphism group. Using the irreducible representations of the automorphism group, we derive conditions such that quantum walks defined on this graph must have infinite hitting times for some initial states. In the case of the discrete walk, if this condition is satisfied the walk will have infinite hitting times for any choice of a coin operator, and we give a class of graphs with infinite hitting times for any choice of coin. Hitting times are not very well defined for continuous time quantum walks, but we show that the idea of infinite hitting-time walks naturally extends to the continuous time case as well.
The expected hitting time of discrete quantum walks on a hypercube (HC) is numerically known to be exponentially shorter than that of their classical analogs in terms of the scaling with the HC dimension. Recent numerical analyses illustrated that this scaling exists not only on the bare HC, but also when the HC graph is symmetrically and locally embedded into larger graphs. The present work investigates the necessity of symmetry and locality for the speed-up by considering embeddings that are nonsymmetric or nonlocal. We provide numerical evidence that the exponential speed-up survives also in these cases. Furthermore, our numerical simulations demonstrate that removing a single edge from the HC also does not destroy the exponential speed-up. In the nonlocal embedding of the HC we encounter dark states, which we analyze. We provide a general and detailed presentation of the mapping that reduces the exponentially large Hilbert space of the quantum walk to an effective subspace of polynomial scaling. This mapping is our essential tool to numerically study quantum walks in such high-dimensional structures.