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arXiv:2106.11685 · Physical Review A

Quantum-classical distance as a tool to design optimal chiral quantum walks

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

Published Mar 11, 2022• 15 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.105.032425

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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.
Abstract Quantum transport across discrete structures is a relevant topic that can be suitably studied in the context of continuous-time quantum walks. The addition of phase degrees of freedom, leading to chiral quantum walks, can also account for directional transport on graphs with loops. We discuss criteria for quantum transport and study the enhancement that can be achieved with chiral quantum walks on chain-like graphs, exploring different topologies for the chain units and optimizing over the phases. We select three candidate structures with optimal performances and we investigate their transport behaviour with Krylov reduction. While one of them can be reduced to a weighted line with minor couplings modulation, the other two are truly chiral quantum walks, with enhanced transport probability over long chain structures.
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.
Abstract We investigate novel transport properties of chiral continuous-time quantum walks (CTQWs) on graphs. By employing a gauge transformation, we demonstrate that CTQWs on chiral chains are equivalent to those on non-chiral chains, but with additional momenta from initial wave packets. This explains the novel transport phenomenon numerically studied in (Khalique et al 2021 New J. Phys. 23 083005). Building on this, we delve deeper into the analysis of chiral CTQWs on the Y-junction graph, introducing phases to account for the chirality. The phase plays a key role in controlling both asymmetric transport and directed complete transport among the chains in the Y-junction graph. We systematically analyze these features through a comprehensive examination of the chiral CTQW on a Y-junction graph. Our analysis shows that the CTQW on Y-junction graph can be modeled as a combination of three wave functions, each of which evolves independently on three effective open chains. By constructing a lattice scattering theory, we calculate the phase shift of a wave packet after it interacts with the potential-shifted boundary. Our results demonstrate that the interplay of these phase shifts leads to the observed enhancement and suppression of quantum transport. The explicit condition for directed complete transport or 100 % efficiency is analytically derived. Our theory has applications in building quantum versions of binary tree search algorithms.
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.
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.
Quantum walks can be defined in two quite distinct ways: discrete-time and continuous-time quantum walks (DTQWs and CTQWs). For classical random walks, there is a natural sense in which continuous-time walks are a limit of discrete-time walks. Quantum mechanically, in the discrete-time case, an additional ``coin space'' must be appended for the walk to have nontrivial time evolution. Continuous-time quantum walks, however, have no such constraints. This means that there is no completely straightforward way to treat a CTQW as a limit of a DTQW, as can be done in the classical case. Various approaches to this problem have been taken in the past. We give a construction for walks on ddd-regular, ddd-colorable graphs when the coin flip operator is Hermitian: from a standard DTQW we construct a family of discrete-time walks with a well-defined continuous-time limit on a related graph. One can think of this limit as a ``coined'' continuous-time walk. We show that these CTQWs share some properties with coined DTQWs. In particular, we look at a spatial search by a DTQW over the two-dimensional (2D) torus (a grid with periodic boundary conditions) of size $\sqrt{N}\ifmmode\times\else\texttimes\fi{}\sqrt{N}$, where it was shown that a coined DTQW can search in time O(NlogN)O(\sqrt{N}logN)O(N​logN), but a standard CTQW takes $\mathrm{\ensuremath{\Omega}}(N)$ time to search for a marked element. The continuous limit of the DTQW search over the 2D torus exhibits the O(NlogN)O(\sqrt{N}logN)O(N​logN) scaling, like the coined walk it is derived from. We also look at the effects of graph symmetry on the limiting walk, and show that the properties are similar to those of the DTQW as shown in Krovi and Brun, Phys. Rev. A 75, 062332 (2007).