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arXiv:1004.0206

k-Boson Quantum Walks Do Not Distinguish Arbitrary Graphs

Jamie Smith

Published Apr 1, 2010• 9 citations
arXivHTMLPDF

Abstract

In this paper, we define k-equivalence, a relation on graphs that relies on their associated cellular algebras. We show that a k-Boson quantum walk cannot distinguish pairs of graphs that are k- equivalent. The existence of pairs of k-equivalent graphs has been shown by Ponomarenko et al. [2, 6]. This gives a negative answer to a question posed by Gamble et al. [7].

DOI
10.48550/arxiv.1004.0206

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In this paper, we define k-equivalence, a relation on graphs that relies on their associated cellular algebras. We show that a k-Boson quantum walk cannot distinguish pairs of graphs that are k- equivalent. The existence of pairs of k-equivalent graphs has been shown by Ponomarenko et al. [2, 6]. This gives a negative answer to a question posed by Gamble et al. [7].
Quantum random walks have been shown to be powerful quantum algorithms for certain tasks on graphs like database searching, quantum simulations etc. In this work we focus on its applications for the graph isomorphism problem. In particular we look at how we can compare multi-particle quantum walks and well known classical WL tests and how quantum walks can be used to distinguish hard graphs like CFI graphs which k-WL tests cannot distinguish. We provide theoretical proofs and empirical results to show that a k-QW with input superposition states distinguishes k-CFI graphs. In addition we also prove that a k-1 QW with localized input states distinguishes k-CFI graphs. We also prove some additional results about strongly regular graphs (SRGs).
We show that the quantum dynamics of interacting and noninteracting quantum particles are fundamentally different in the context of solving a particular computational problem. Specifically, we consider the graph isomorphism problem, in which one wishes to determine whether two graphs are isomorphic (related to each other by a relabeling of the graph vertices), and focus on a class of graphs with particularly high symmetry called strongly regular graphs (SRGs). We study the Green's functions that characterize the dynamical evolution single-particle and two-particle quantum walks on pairs of nonisomorphic SRGs and show that interacting particles can distinguish nonisomorphic graphs that noninteracting particles cannot. We obtain the following specific results. (1) We prove that quantum walks of two noninteracting particles, fermions or bosons, cannot distinguish certain pairs of nonisomorphic SRGs. (2) We demonstrate numerically that two interacting bosons are more powerful than single particles and two noninteracting particles, in that quantum walks of interacting bosons distinguish all nonisomorphic pairs of SRGs that we examined. By utilizing high-throughput computing to perform over 500 million direct comparisons between evolution operators, we checked all tabulated pairs of nonisomorphic SRGs, including graphs with up to 64 vertices. (3) By performing a short-time expansion of the evolution operator, we derive distinguishing operators that provide analytic insight into the power of the interacting two-particle quantum walk.
Berry and Wang [Phys. Rev. A {\bf 83}, 042317 (2011)] show numerically that a discrete-time quantum random walk of two noninteracting particles is able to distinguish some non-isomorphic strongly regular graphs from the same family. Here we analytically demonstrate how it is possible for these walks to distinguish such graphs, while continuous-time quantum walks of two noninteracting particles cannot. We show analytically and numerically that even single-particle discrete-time quantum random walks can distinguish some strongly regular graphs, though not as many as two-particle noninteracting discrete-time walks. Additionally, we demonstrate how, given the same quantum random walk, subtle differences in the graph certificate construction algorithm can nontrivially impact the walk's distinguishing power. We also show that no continuous-time walk of a fixed number of particles can distinguish all strongly regular graphs when used in conjunction with any of the graph certificates we consider. We extend this constraint to discrete-time walks of fixed numbers of noninteracting particles for one kind of graph certificate; it remains an open question as to whether or not this constraint applies to the other graph certificates we consider.
A distinguishability operator is defined for the continuous-time quantum walk (CTQW) of a bipartite quantum walker on two simply connected graphs, WGi,Gj=UGi(t)⊗UGj(t′)−UGj(t′)⊗UGi(t)W_{G_i,G_j} = U_{G_i}\left(t\right) \otimes U_{G_j}\left(t'\right) - U_{G_j}\left(t'\right) \otimes U_{G_i}\left(t\right)WGi​,Gj​​=UGi​​(t)⊗UGj​​(t′)−UGj​​(t′)⊗UGi​​(t), where UGi(t)U_{G_i}\left(t\right)UGi​​(t) is the unitary CTQW operator for a labeled graph GiG_iGi​ over a time interval ttt. The null space of WGi,GjW_{G_i,G_j}WGi​,Gj​​ defines the vector space of initial bipartite states whose time development is either constant or only dependent on t+t′t + t't+t′ and is invariant to which quantum walker subsystem goes with each graph. The set of null spaces corresponding with a set of WGi,GjW_{G_i,G_j}WGi​,Gj​​ have interesting relations as subspaces, intersections between subspaces, and subspaces of intersections. These relations are depicted as Euler diagrams for labeled graphs of three and four vertices.
We consider the definition of quantum walks on directed graphs. Call a directed graph reversible if, for each pair of vertices (vi,vj)(v_i, v_j)(vi​,vj​), if viv_ivi​ is connected to vjv_jvj​ then there is a path from vjv_jvj​ to viv_ivi​. We show that reversibility is a necessary and sufficient condition for a directed graph to allow the notion of a discrete-time quantum walk, and discuss some implications of this condition. We present a method for defining a "partially quantum'' walk on directed graphs that are not reversible.