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arXiv:2005.06351 · Physical Review A

Continuous-time quantum walks in the presence of a quadratic perturbation

Alessandro Candeloro, Luca Razzoli, Simone Cavazzoni, Paolo Bordone, Matteo G. A. Paris

Published Oct 14, 2020• 10 citations
arXivHTMLPDF

Abstract

We address the properties of continuous-time quantum walks with Hamiltonians of the form $\mathcal{H}=L+\ensuremath{\lambda}{L}^{2}$, with $L$ the Laplacian matrix of the underlying graph and the perturbation $\ensuremath{\lambda}{L}^{2}$ motivated by its potential use to introduce next-nearest-neighbor hopping. We consider cycle, complete, and star graphs as paradigmatic models with low and high connectivity and/or symmetry. First, we investigate the dynamics of an initially localized walker. Then we devote attention to estimating the perturbation parameter $\ensuremath{\lambda}$ using only a snapshot of the walker dynamics. Our analysis shows that a walker on a cycle graph spreads ballistically independently of the perturbation, whereas on complete and star graphs one observes perturbation-dependent revivals and strong localization phenomena. Concerning the estimation of the perturbation, we determine the walker preparations and the simple graphs that maximize the quantum Fisher information. We also assess the performance of position measurement, which turns out to be optimal, or nearly optimal, in several situations of interest. Besides fundamental interest, our study may find applications in designing enhanced algorithms on graphs.

DOI
10.1103/physreva.102.042214

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We address the properties of continuous-time quantum walks with Hamiltonians of the form H=L+\ensuremathλL2\mathcal{H}=L+\ensuremath{\lambda}{L}^{2}H=L+\ensuremathλL2, with LLL the Laplacian matrix of the underlying graph and the perturbation \ensuremathλL2\ensuremath{\lambda}{L}^{2}\ensuremathλL2 motivated by its potential use to introduce next-nearest-neighbor hopping. We consider cycle, complete, and star graphs as paradigmatic models with low and high connectivity and/or symmetry. First, we investigate the dynamics of an initially localized walker. Then we devote attention to estimating the perturbation parameter \ensuremathλ\ensuremath{\lambda}\ensuremathλ using only a snapshot of the walker dynamics. Our analysis shows that a walker on a cycle graph spreads ballistically independently of the perturbation, whereas on complete and star graphs one observes perturbation-dependent revivals and strong localization phenomena. Concerning the estimation of the perturbation, we determine the walker preparations and the simple graphs that maximize the quantum Fisher information. We also assess the performance of position measurement, which turns out to be optimal, or nearly optimal, in several situations of interest. Besides fundamental interest, our study may find applications in designing enhanced algorithms on graphs.
Abstract A fully connected vertex w in a simple graph G of order N is a vertex connected to all the other N − 1 vertices. Upon denoting by L the Laplacian matrix of the graph, we prove that the continuous-time quantum walk (CTQW)—with Hamiltonian H = γL —of a walker initially localized at | w ⟩ does not depend on the graph G . We also prove that for any Grover-like CTQW—with Hamiltonian H = γL + ∑ w λ w | w ⟩⟨ w |—the probability amplitude at the fully connected marked vertices w does not depend on G . The result does not hold for CTQW with Hamiltonian H = γA (adjacency matrix). We apply our results to spatial search and quantum transport for single and multiple fully connected marked vertices, proving that CTQWs on any graph G inherit the properties already known for the complete graph of the same order, including the optimality of the spatial search. Our results provide a unified framework for several partial results already reported in literature for fully connected vertices, such as the equivalence of CTQW and of spatial search for the central vertex of the star and wheel graph, and any vertex of the complete graph.
Abstract We address continuous-time quantum walks on graphs, and discuss whether and how quantum-limited measurements on the walker may extract information on the tunnelling amplitude between the nodes of the graphs. For a few remarkable families of graphs, we evaluate the ultimate quantum bound to precision, i.e. we compute the quantum Fisher information (QFI), and assess the performances of incomplete measurements, i.e. measurements performed on a subset of the graph’s nodes. We also optimize the QFI over the initial preparation of the walker and find the optimal measurement achieving the ultimate precision in each case. As the topology of the graph is changed, a non-trivial interplay between the connectivity and the achievable precision is uncovered.
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.
Continuous-time quantum walks (QWs) on cycle graphs are investigated as a versatile metrological platform. The parameters of interest are a local on-site potential (electric-like perturbation) and the overall chiral phase (magnetic-like perturbation). For single-parameter estimation, we compare static strategies, in which information is encoded in the ground state, with dynamical protocols, in which the system evolves from the unperturbed ground state under the perturbed Hamiltonian. For the electric perturbation, the dynamical approach yields a quantum Fisher information (QFI) that scales quadratically in time, saturating the Heisenberg limit. In contrast, estimation of the magnetic perturbation alone is unfeasible in both static and dynamical settings as the eigenstates are independent of the chiral phase. Assisted estimation (in which a known perturbation is used to enhance sensitivity to the other) restores quadratic scaling for both fields, with optimal performance achieved by tuning the assisting parameter near critical values. Conversely, joint estimation of both fields performs rather poorly: the model exhibits substantial sloppiness over large parameter ranges, indicating sensitivity only to specific parameter combinations. Finally, the magnetically perturbed ring is re-analyzed as a quantum thermometer. The magnetic flux tunes the energy gap, enabling near-optimal low-temperature sensitivity that follows the Landau bound. Overall, quantum walks on cycle graphs represent a flexible class of sensors ideally suited for (assisted) single-parameter estimation.
We address continuous-time quantum walks on graphs in the presence of time- and space-dependent noise. Noise is modeled as generalized dynamical percolation, i.e. , classical time-dependent fluctuations affecting the tunneling amplitudes of the walker. In order to illustrate the general features of the model, we review recent results on two paradigmatic examples: the dynamics of quantum walks on the line and the effects of noise on the performances of quantum spatial search on the complete and the star graph. We also discuss future perspectives, including extension to many-particle quantum walk, to noise model for on-site energies and to the analysis of different noise spectra. Finally, we address the use of quantum walks as a quantum probe to characterize defects and perturbations occurring in complex, classical and quantum, networks.
Continuous-time quantum walks provide a natural framework to tackle the fundamental problem of finding a node among a set of marked nodes in a graph, known as spatial search. Whether spatial search by continuous-time quantum walk provides a quadratic advantage over classical random walks has been an outstanding problem. Thus far, this advantage is obtained only for specific graphs or when a single node of the underlying graph is marked. In this Letter, we provide a new continuous-time quantum walk search algorithm that completely resolves this: our algorithm can find a marked node in any graph with any number of marked nodes, in a time that is quadratically faster than classical random walks. The overall algorithm is quite simple, requiring time evolution of the quantum walk Hamiltonian followed by a projective measurement. A key component of our algorithm is a purely analogue procedure to perform operations on a state of the form e^{-tH^{2}}|ψ⟩, which, for a given Hamiltonian H, only requires evolving H for time scaling as sqrt[t]. This allows us to quadratically fast-forward the dynamics of a continuous-time classical random walk. The applications of our Letter thus go beyond the realm of quantum walks and can lead to new analog quantum algorithms for preparing ground states of Hamiltonians or solving optimization problems.