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arXiv:1910.01563 · Physical Review A

Quantum-classical dynamical distance and quantumness of quantum walks

Valentina Gualtieri, Claudia Benedetti, Matteo G. A. Paris

Published Jul 1, 2020• 19 citations
arXivHTMLPDF

Abstract

We introduce a fidelity-based measure, ${\text{D}}_{\mathrm{QC}}(t)$, to quantify the differences in the dynamics of classical versus quantum walks over a graph. We provide universal, graph-independent, analytic expressions of this quantum-classical dynamical distance, showing that at short times ${\text{D}}_{\mathrm{QC}}(t)$ is proportional to the coherence of the walker, i.e., a genuine quantum feature, whereas at long times it depends only on the size of the graph. At intermediate times, ${\text{D}}_{\mathrm{QC}}(t)$ does depend on the graph topology through its algebraic connectivity. Our results show that the difference in the dynamical behavior of classical and quantum walks is entirely due to the emergence of quantum features at short times. In the long-time limit, quantumness and the different nature of the generators of the dynamics, e.g., the open-system nature of classical walks and the unitary nature of quantum walks, are instead contributing equally.

DOI
10.1103/physreva.102.012201

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We introduce a fidelity-based measure, DQC(t){\text{D}}_{\mathrm{QC}}(t)DQC​(t), to quantify the differences in the dynamics of classical versus quantum walks over a graph. We provide universal, graph-independent, analytic expressions of this quantum-classical dynamical distance, showing that at short times DQC(t){\text{D}}_{\mathrm{QC}}(t)DQC​(t) is proportional to the coherence of the walker, i.e., a genuine quantum feature, whereas at long times it depends only on the size of the graph. At intermediate times, DQC(t){\text{D}}_{\mathrm{QC}}(t)DQC​(t) does depend on the graph topology through its algebraic connectivity. Our results show that the difference in the dynamical behavior of classical and quantum walks is entirely due to the emergence of quantum features at short times. In the long-time limit, quantumness and the different nature of the generators of the dynamics, e.g., the open-system nature of classical walks and the unitary nature of quantum walks, are instead contributing equally.
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.
Dynamical evolution of systems with sparse Hamiltonians can always be recognized as continuous-time quantum walks (CTQWs) on graphs. In this paper, we analyze the short-time asymptotics of CTQWs. In recent studies, it was shown that for the classical diffusion process the short-time asymptotics of the transition probabilities follows power laws whose exponents are given by the usual combinatorial distances of the nodes. Inspired by this result, we perform a similar analysis for CTQWs in both closed and open systems, including time-dependent couplings. For time-reversal symmetric coherent quantum evolutions, the short-time asymptotics of the transition probabilities is completely determined by the topology of the underlying graph analogously to the classical case, but with a doubled power-law exponent. Moreover, this result is robust against the introduction of on-site potential terms. However, we show that time-reversal symmetry-breaking terms and noncoherent effects can significantly alter the short-time asymptotics. The analytical formulas are checked against numerics, and excellent agreement is found. Furthermore, we discuss in detail the relevance of our results for quantum evolutions on particular network topologies.
Continuous-time quantum walks (CTQWs) provide a valuable model for quantum transport, universal quantum computation, and quantum spatial search, among others. Recently, the empowering role of new degrees of freedom in the Hamiltonian generator of CTQWs, which are the complex phases along the loops of the underlying graph, was acknowledged for its interest in optimizing or suppressing transport on specific topologies. We argue that the quantum-classical distance, a figure of merit which was introduced to capture the difference in dynamics between a CTQW and its classical, stochastic counterpart, guides the optimization of parameters of the Hamiltonian to achieve better quantum transport on cycle graphs and spatial search for the quantum speed limit without an oracle on complete graphs, the latter also implying fast uniform mixing. We compare the variations of this quantity with the 1-norm of coherence and the inverse participation ratio, showing that the quantum-classical distance is linked to both, but in a topology-dependent relation, which is key to spot the most interesting quantum evolution in each case.
Continuous-time open quantum walks (CTOQW) are introduced as the formulation of quantum dynamical semigroups of trace-preserving and completely positive linear maps (or quantum Markov semigroups) on graphs. We show that a CTOQW always converges to a steady state regardless of the initial state when a graph is connected. When the graph is both connected and regular, it is shown that the steady state is the maximally mixed state. The difference of long-time behaviors between CTOQW and other two continuous-time processes on graphs is exemplified. The examples demonstrate that the structure of a graph can affect a quantum coherence effect on CTOQW through a long time run. Precisely, a quantum coherence effect persists throughout the evolution of the CTOQW when the underlying topology is certain irregular graphs (such as a path or a star as shown in the examples). In contrast, a quantum coherence effect will eventually vanish from the open quantum system when the underlying topology is a regular graph (such as a cycle).
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.
Quantum walks act in obviously different ways from their classical counterparts, but decoherence will lessen and close this gap between them. To understand this process, it is necessary to investigate the evolution of quantum walks under different decoherence situations. In this article, we study a non-Markovian decoherent quantum walk on a line. In a short time regime, the behavior of the walk deviates from both ideal quantum walks and classical random walks. The position variance as a measure of the quantum walk collapses and revives for a short time, and tends to have a linear relation with time. That is, the walker's behavior shows a diffusive spread over a long time limit, which is caused by non-Markovian dephasing affecting the quantum correlations between the quantum walker and his coin. We also study both quantum discord and measurement-induced disturbance as measures of the quantum correlations, and observe both collapse and revival in the short time regime, and the tendency to be zero in the long time limit. Therefore, quantum walks with non-Markovian decoherence tend to have diffusive spreading behavior over long time limits, while in the short time regime they oscillate between ballistic and diffusive spreading behavior, and the quantum correlation collapses and revives due to the memory effect.