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arXiv:2104.10091 · Physical Review A

Generalized quantum-classical correspondence for random walks on graphs

Massimo Frigerio, Claudia Benedetti, Stefano Olivares, Matteo G. A. Paris

Published Sep 16, 2021• 24 citations
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Abstract

We introduce a minimal set of physically motivated postulates that the Hamiltonian $H$ of a continuous-time quantum walk should satisfy in order to properly represent the quantum counterpart of the classical random walk on a given graph. We found that these conditions are satisfied by infinitely many quantum Hamiltonians, which provide novel degrees of freedom for quantum enhanced protocols, In particular, the on-site energies, i.e., the diagonal elements of $H$, and the phases of the off-diagonal elements are unconstrained on the quantum side. The diagonal elements represent a potential-energy landscape for the quantum walk and may be controlled by the interaction with a classical scalar field, whereas, for regular lattices in generic dimension, the off-diagonal phases of $H$ may be tuned by the interaction with a classical gauge field residing on the edges, e.g., the electromagnetic vector potential for a charged walker.

DOI
10.1103/physreva.104.l030201

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We introduce a minimal set of physically motivated postulates that the Hamiltonian HHH of a continuous-time quantum walk should satisfy in order to properly represent the quantum counterpart of the classical random walk on a given graph. We found that these conditions are satisfied by infinitely many quantum Hamiltonians, which provide novel degrees of freedom for quantum enhanced protocols, In particular, the on-site energies, i.e., the diagonal elements of HHH, and the phases of the off-diagonal elements are unconstrained on the quantum side. The diagonal elements represent a potential-energy landscape for the quantum walk and may be controlled by the interaction with a classical scalar field, whereas, for regular lattices in generic dimension, the off-diagonal phases of HHH may be tuned by the interaction with a classical gauge field residing on the edges, e.g., the electromagnetic vector potential for a charged walker.
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Abstract We address decoherence and classicalization of continuous-time quantum walks (CTQWs) on graphs. In particular, we investigate three different models of decoherence and employ the quantum-classical (QC) dynamical distance as a figure of merit to assess whether, and to which extent, decoherence classicalizes the CTQW, i.e. turns it into the analogue classical process. We show that the dynamics arising from intrinsic decoherence, i.e. dephasing in the energy basis, do not fully classicalize the walker and partially preserves quantum features. On the other hand, dephasing in the position basis, as described by the Haken–Strobl master equation or by the quantum stochastic walk (QSW) model, asymptotically destroys the quantumness of the walker, making it equivalent to a classical random walk. We also investigate how fast is the classicalization process and observe a larger rate of convergence of the QC-distance to its asymptotic value for intrinsic decoherence and the QSW models, whereas in the Haken–Strobl scenario, larger values of the decoherence rate induce localization of the walker.
I obtain the dynamics of the continuous time quantum walk on a d-dimensional lattice, with periodic boundary conditions, as an appropriate limit of the dynamics of the discrete time quantum walk on the same lattice. This extends the main result of [8] which proved this limit for the case of the quantum walk on the infinite line and the quantum cellular automaton proposed in [1]. By highlighting the main features of the limiting procedure, I then extend it to general graphs. For a given discrete time quantum walk on a general graph, I single out the type of continuous dynamics (Hamiltonians) that can be obtained as a limit of the discrete time dynamics. 1 Generalities on quantum walks; continuous and discrete time Consider a graph G: = {V, E} with a set of vertices V of cardinality N and a set of edges E. We assume G to be undirected and without self-loops. In the interval of time ∆t a certain fraction γ∆t of a quantity pj leaves the location of the j-th vertex to move to a neighboring vertex k. The quantity pj, j = 1,..., N, at time t + ∆t is given by 1 pj(t + ∆t) = pj(t) − deg (j)γ∆tpj(t) + ∑ (k,j)∈E γ∆tpk(t). (1) Define ⃗p the N−vector whose components are the pj’s quantities, and L the Laplacian matrix defined by Ljj = − deg (j), Ljk = 1 if (j, k) ∈ E and Ljk = 0 otherwise. L is
We investigate the genuinely quantum features of continuous-time quantum walks by combining a single-time and a multi-time quantifier of nonclassicality. On the one hand, we consider the quantum-classical dynamical distance DQC(t)D_{\mathrm{QC}}(t)DQC​(t), which measures the departure of the time-evolved quantum state of a continuous-time quantum walk from the classical state of a random walk on the same graph. On the other, we analyse the joint probability distributions associated with sequential measurements of the walker's position, assessing their violation of the classical Kolmogorov consistency conditions via a dedicated quantifier Kˉ(t)\bar{K}(t)Kˉ(t). We demonstrate a quadratic short-time scaling of Kˉ(t)\bar{K}(t)Kˉ(t), which differs from the known linear scaling of DQC(t)D_{\mathrm{QC}}(t)DQC​(t), but, as the latter, is fully determined by the degree of the initially occupied node and is independent of the global graph topology. At longer times, instead, Kˉ(t)\bar{K}(t)Kˉ(t) exhibits a pronounced topology-driven behavior: it is strongly suppressed on complete graphs while remaining finite and oscillatory on cycles, in contrast with the almost topology-independent asymptotics of DQC(t)D_{\mathrm{QC}}(t)DQC​(t). We then extend the analysis to Markovian open-system dynamics, focusing on dephasing in the position basis (Haken-Strobl model) and in the energy basis (intrinsic decoherence). Site dephasing drives both quantifiers to zero, with the decay of Kˉ(t)\bar{K}(t)Kˉ(t) controlled by the spectral gap of the corresponding Lindblad generator. By contrast, energy-basis dephasing preserves a finite asymptotic value of Kˉ(t)\bar{K}(t)Kˉ(t), depending on the overlap structure of the Laplacian eigenspaces with the site basis.
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.