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arXiv:2004.05657 · Physical Review A

Optimizing the spatial spread of a quantum walk

Gonzalo Martín-Vázquez, Javier Rodríguez-Laguna

Published Aug 24, 2020• 10 citations
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Abstract

We devise a protocol to build one-dimensional time-dependent quantum walks, maximizing the spatial spread throughout the procedure. We allow only one of the physical parameters of the coin-tossing operator to vary, i.e., the angle $\ensuremath{\theta}$, such that for $\ensuremath{\theta}=0$ we have the ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\sigma}}}_{z}$, while for $\ensuremath{\theta}=\ensuremath{\pi}/4$ we obtain the Hadamard gate. The optimal $\ensuremath{\theta}$ sequences present nontrivial patterns, with mostly $\ensuremath{\theta}\ensuremath{\approx}0$ alternating with $\ensuremath{\theta}\ensuremath{\approx}\ensuremath{\pi}/4$ values after increasingly long periods. We provide an analysis of the entanglement properties, quasienergy spectrum, and survival probability, providing a full physical picture.

DOI
10.1103/physreva.102.022223

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We devise a protocol to build one-dimensional time-dependent quantum walks, maximizing the spatial spread throughout the procedure. We allow only one of the physical parameters of the coin-tossing operator to vary, i.e., the angle \ensuremathθ\ensuremath{\theta}\ensuremathθ, such that for \ensuremathθ=0\ensuremath{\theta}=0\ensuremathθ=0 we have the \ensuremathσ\ifmmode^\elseˆ\fiz{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\sigma}}}_{z}\ensuremathσ\ifmmode^\elseˆ\fi​z​, while for \ensuremathθ=\ensuremathπ/4\ensuremath{\theta}=\ensuremath{\pi}/4\ensuremathθ=\ensuremathπ/4 we obtain the Hadamard gate. The optimal \ensuremathθ\ensuremath{\theta}\ensuremathθ sequences present nontrivial patterns, with mostly \ensuremathθ\ensuremath≈0\ensuremath{\theta}\ensuremath{\approx}0\ensuremathθ\ensuremath≈0 alternating with \ensuremathθ\ensuremath≈\ensuremathπ/4\ensuremath{\theta}\ensuremath{\approx}\ensuremath{\pi}/4\ensuremathθ\ensuremath≈\ensuremathπ/4 values after increasingly long periods. We provide an analysis of the entanglement properties, quasienergy spectrum, and survival probability, providing a full physical picture.
We analyse the quantum walk in higher spatial dimensions and compare classical and quantum spreading as a function of time. Tensor products of Hadamard transformations and the discrete Fourier transform arise as natural extensions of the 'quantum coin toss' in the one-dimensional walk simulation, and other illustrative transformations are also investigated. We find that entanglement between the dimensions serves to reduce the rate of spread of the quantum walk. The classical limit is obtained by introducing a random phase variable.
Quantum walks and random walks bear similarities and divergences. One of the most remarkable disparities affects the probability of finding the particle at a given location: typically, almost a flat function in the first case and a bell-shaped one in the second case. Here I show how one can impose any desired stochastic behavior (compatible with the continuity equation for the probability function) on both systems by the appropriate choice of time- and site-dependent coins. This implies, in particular, that one can devise quantum walks that show diffusive spreading without losing coherence as well as random walks that exhibit the characteristic fast propagation of a quantum particle driven by a Hadamard coin.
Quantum walks are a powerful tool for developing efficient algorithms in quantum computing. This research explores two discrete-time one-dimensional quantum walks where the coin operator varies along even and odd positions on the line. We find closed-form expressions for the coefficients of the wave function for both walks and also arrive at a formula for the probability distribution for one of the walks. A significant discovery is a way to model the well-known Hadamard walk using two alternating coins.
Discrete-time quantum walks are powerful platforms for simulating quantum transport and information processing. Here we introduce a walker on a one-dimensional lattice whose motion is controlled by three entangled coins, each initialized with the Hadamard gate, aiming to maximize information flow. The walker moves only when all three coins yield the same outcome (HHH or TTT), thus coupling the 8-dimensional coin Hilbert space to the position degree of freedom. By analyzing fully separable, fully entangled (GHZ-type) and intermediate initial states, and using the von Neumann entropy of reduced subsystems, we compute the mutual information I(C;P;t)I(C;P;t)I(C;P;t) between coin and position. The results show that initial three-partite entanglement accelerates the growth of mutual information by up to 18\% after ten steps (when compared to the lower of the two separable states), although it exhibits short-term non-monotonic dynamics due to quantum interference. For the first time, we introduce a tunable parameter ααα (amplitude of non-displacement states) and show that the GHZ state reaches a maximum of mutual information at α≈0.71α\approx 0.71α≈0.71 - a key finding for optimal control of information flow. Finally, an integrated photonic implementation using polarization, spatial modes and time bins is proposed, where ααα can be tuned with nonlinear or electro-optic elements. A scalable numerical framework (Python code) for simulations up to t=5t = 5t=5 steps is provided. Our findings establish three-partite entanglement as a dynamical resource for maximizing information flow and spatial spreading, with direct applications in quantum state transfer, entanglement-assisted sensing and programmable photonic quantum processors.
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We have recently proposed a two-dimensional quantum walk where the requirement of a higher dimensionality of the coin space is substituted with the alternance of the directions in which the walker can move [C. Di Franco, M. Mc Gettrick, and Th. Busch, Phys. Rev. Lett. 106, 080502 (2011)]. For a particular initial state of the coin, this walk is able to perfectly reproduce the spatial probability distribution of the nonlocalized case of the Grover walk. Here, we present a more detailed proof of this equivalence. We also extend the analysis to other initial states in order to provide a more complete picture of our walk. We show that this scheme outperforms the Grover walk in the generation of xxx-yyy spatial entanglement for any initial condition, with the maximum entanglement obtained in the case of the particular aforementioned state. Finally, the equivalence is generalized to wider classes of quantum walks and a limit theorem for the alternate walk in this context is presented.