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Entropy

Quantum Walk on the Generalized Birkhoff Polytope Graph

Rafael Cação, Lucas R. C. T. Cortez, Ismael de Farias, Ernée Kozyreff Filho, Jalil Khatibi Moqadam, Renato Portugal

Published Sep 23, 2021• 1 citation
Publisher

Abstract

We study discrete-time quantum walks on generalized Birkhoff polytope graphs (GBPGs), which arise in the solution-set to certain transportation linear programming problems (TLPs). It is known that quantum walks mix at most quadratically faster than random walks on cycles, two-dimensional lattices, hypercubes, and bounded-degree graphs. In contrast, our numerical results show that it is possible to achieve a greater than quadratic quantum speedup for the mixing time on a subclass of GBPG (TLP with two consumers and m suppliers). We analyze two types of initial states. If the walker starts on a single node, the quantum mixing time does not depend on m, even though the graph diameter increases with it. To the best of our knowledge, this is the first example of its kind. If the walker is initially spread over a maximal clique, the quantum mixing time is O(m/ϵ), where ϵ is the threshold used in the mixing times. This result is better than the classical mixing time, which is O(m1.5/ϵ).

DOI
10.3390/e23101239

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We study discrete-time quantum walks on generalized Birkhoff polytope graphs (GBPGs), which arise in the solution-set to certain transportation linear programming problems (TLPs). It is known that quantum walks mix at most quadratically faster than random walks on cycles, two-dimensional lattices, hypercubes, and bounded-degree graphs. In contrast, our numerical results show that it is possible to achieve a greater than quadratic quantum speedup for the mixing time on a subclass of GBPG (TLP with two consumers and m suppliers). We analyze two types of initial states. If the walker starts on a single node, the quantum mixing time does not depend on m, even though the graph diameter increases with it. To the best of our knowledge, this is the first example of its kind. If the walker is initially spread over a maximal clique, the quantum mixing time is O(m/ϵ), where ϵ is the threshold used in the mixing times. This result is better than the classical mixing time, which is O(m1.5/ϵ).
Quantum mixing is a category of processes of reaching a stationary state when starting from a single initial vertex via quantum walks. In this thesis, we introduce the quantum mixing processes based on the framework of controlled quantum walks in [DH17a]. We first discuss why Szegedy's walks are not well-suited for the quantum mixing processes. We then show that the controlled quantum walk is a suitable framework for achieving a quadratic speed-up in the quantum mixing processes, because of its relationship with interpolated walks [KMOR16] and the phase estimation algorithm [CEMM98]. Accordingly, we give two definitions of quantum mixing times via controlled quantum walks: instant quantum mixing time and average quantum mixing time. We later prove the bounds of these two quantum mixing times on complete graphs. We also prove the upper bound of the average quantum mixing times on cycle graphs. To have clearer views on the bounds of instant and average quantum mixing times, we then discuss and present the simulations of quantum mixing processes, with four types of graphs (complete graphs, cycle graphs, 2-dimensional tori and dumbbell graphs) as examples. The simulations include the quantum mixing processes with the same ~θ (an important parameter in the controlled quantum walk operator) as in [DH17a], and the suitable range of ~θ that achieves the quadratic speed-up for different types of graphs. Inspired by the quantum search algorithm via controlled quantum walks in [DH17a], we adapt this algorithm to a new algorithm of quantum mixing, which generates samples from the stationary distribution of the Markov chain, with its runtime upper bounded by the quantum hitting time, which is quadratically faster than the classical hitting process.
We set the ground for a theory of quantum walks on graphs- the generalization of random walks on finite graphs to the quantum world. Such quantum walks do not converge to any stationary distribution, as they are unitary and reversible. However, by suitably relaxing the definition, we can obtain a measure of how fast the quantum walk spreads or how confined the quantum walk stays in a small neighborhood. We give definitions of mixing time, filling time, dispersion time. We show that in all these measures, the quantum walk on the cycle is almost quadratically faster then its classical correspondent. On the other hand, we give a lower bound on the possible speed up by quantum walks for general graphs, showing that quantum walks can be at most polynomially faster than their classical counterparts.
We set the ground for a theory of quantum walks on graphs-the generalization of random walks on finite graphs to the quantum world. Such quantum walks do not converge to any stationary distribution, as they are unitary and reversible. However, by suitably relaxing the definition, we can obtain a measure of how fast the quantum walk spreads or how confined the quantum walk stays in a small neighborhood. We give definitions of mixing time, filling time, dispersion time. We show that in all these measures, the quantum walk on the cycle is almost quadratically faster then its classical correspondent. On the other hand, we give a lower bound on the possible speed up by quantum walks for general graphs, showing that quantum walks can be at most polynomially faster than their classical counterparts.
Quantum walks on graphs have been shown in certain cases to mix quadratically faster than their classical counterparts. Lifted Markov chains, consisting of a Markov chain on an extended state space which is projected back down to the original state space, also show considerable speedups in mixing time. Here, we construct a lifted Markov chain on a graph with n2D(G)n^2 D(G)n2D(G) vertices that mixes exactly to the average mixing distribution of a quantum walk on the graph GGG with nnn vertices, where D(G)D(G)D(G) is the diameter of GGG. Moreover, the mixing time of this chain is D(G)D(G)D(G) timesteps, and we prove that computing the transition probabilities for the lifted chain takes time polynomial in nnn. As an immediate consequence, for every quantum walk there is a lifted Markov chain with a faster mixing time that is polynomial-time computable, as the quantum mixing time is trivially lower bounded by the graph diameter. The result is based on a lifting presented by Apers, Ticozzi and Sarlette (arXiv:1705.08253).
Several inequalities are proved for the mixing time of discrete-time quantum walks on finite graphs. The mixing time is defined differently than in Aharonov et al (2002 Proc. 33rd STOC ( 2001 ) (New York: ACM) pp 50–9 arXiv:quant-ph/0012090v2) and it is found that for particular examples of walks on a cycle, a hypercube and a complete graph, quantum walks provide no speedup in mixing over the classical counterparts. In addition, non-unitary quantum walks (i.e. walks with decoherence) are considered and a criterion for their convergence to the unique stationary distribution is derived.
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.