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arXiv:1809.09211 · Journal of Physics A Mathematical and Theoretical

The walker speaks its graph: global and nearly-local probing of the tunnelling amplitude in continuous-time quantum walks

Luigi Seveso, Claudia Benedetti, Matteo G. A. Paris

Published Jan 24, 2019• 7 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1751-8121/ab0195

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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 address the properties of continuous-time quantum walks with Hamiltonians of the form $\mathcal{H}=L+\ensuremath{\lambda}{L}^{2}$, with LLL 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.
In discrete-time quantum walk (DTQW) the walker's coin space entangles with the position space after the very first step of the evolution. This phenomenon may be exploited to obtain the value of the coin parameter $\ensuremath{\theta}$ by performing measurements on the sole position space of the walker. In this paper, we evaluate the ultimate quantum limits to precision for this class of estimation protocols and use this result to assess measurement schemes having limited access to the position space of the walker in one dimension. We find that the quantum Fisher information (QFI) of the walker's position space ${H}_{w}(\ensuremath{\theta})$ increases with $\ensuremath{\theta}$ and with time, which, in turn, may be seen as a metrological resource. We also find a difference in the QFI of bounded and unbounded DTQWs and provide an interpretation of the different behaviors in terms of interference in the position space. Finally, we compare ${H}_{w}(\ensuremath{\theta})$ to the full QFI ${H}_{f}(\ensuremath{\theta})$, i.e., the QFI of the walkers position plus coin state, and find that their ratio is dependent on $\ensuremath{\theta}$, but saturates to a constant value, meaning that the walker may probe its coin parameter quite faithfully.
Discrete-time quantum walk (DTQW) represents a convenient mathematical framework for describing the motion of a particle on a discrete set of positions when this motion is conditioned by the values of certain internal degrees of freedom, which are usually referred to as the coin of the particle. As such, and owing to the inherent dependence of the position distribution on the coin degrees of freedom, DTQWs naturally emerge as promising candidates for quantum metrology. In this paper, we explore the use of DTQWs as quantum probes in scenarios where the parameter of interest is encoded in the internal degree of freedom of the walker and investigate the role of the topology of the walker's space on the attainable precision. In particular, we start considering the encoding of the parameter by rotations for a walker on the line and evaluate the quantum Fisher information (QFI) and the position Fisher information (FI), explicitly determining the optimal initial state in position space that maximizes the QFI across all encoding schemes. This allows us to understand the role of interference in the position space and to introduce an optimal topology, which maximizes the QFI of the coin parameter and makes the position FI equal to the QFI.
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.
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.
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.