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Universal and optimal coin sequences for high entanglement generation in 1D discrete time quantum walks

Aikaterini Gratsea, Friederike Metz, Thomas Busch

Published Oct 12, 2020• 21 citations
OpenAlex

Abstract

Entanglement is a key resource in many quantum information applications and achieving high values independently of the initial conditions is an important task. Here we address the problem of generating highly entangled states in a discrete time quantum walk irrespective of the initial state using two different approaches. First, we present and analyze a deterministic sequence of coin operators which produces high values of entanglement in a universal manner for a class of localized initial states. In a second approach, we optimize the discrete sequence of coin operators using a reinforcement learning algorithm. While the amount of entanglement produced by the deterministic sequence is fully independent of the initial states considered, the optimized sequences achieve in general higher average values of entanglement that do however depend on the initial state parameters. Our proposed sequence and optimization algorithm are especially useful in cases where the initial state is not fully known or entanglement has to be generated in a universal manner for a range of initial states.

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Universal and optimal coin sequences for high entanglement generation in 1D discrete time quantum walks

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Abstract Entanglement is a key resource in many quantum information applications and achieving high values independently of the initial conditions is an important task. Here we address the problem of generating highly entangled states in a discrete time quantum walk irrespective of the initial state using two different approaches. First, we present and analyze a deterministic sequence of coin operators which produces high values of entanglement in a universal manner for a class of localized initial states. In a second approach, we optimize the discrete sequence of coin operators using a reinforcement learning algorithm. While the amount of entanglement produced by the deterministic sequence is fully independent of the initial states considered, the optimized sequences achieve in general higher average values of entanglement that do however depend on the initial state parameters. Our proposed sequence and optimization algorithm are especially useful in cases where the initial state is not fully known or entanglement has to be generated in a universal manner for a range of initial states.

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Abstract. The coin-position entanglement generated by the evolution operator of a discrete–time quantum walk is quantified, using the von Neumann entropy of the reduced density operator (entropy of entanglement). In the case of a single walker, the entropy of entanglement converges, in the long time limit, to a well defined value which depends on the initial state. Exact expressions are obtained for local and non-local initial conditions. We also discuss the asymptotic bi-partite entanglement generated by non-separable coin operations for two coherent quantum walkers. In this case, the entropy of entanglement is observed to increase logarithmically with time. 1.

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ASYMPTOTIC ENTANGLEMENT IN 1D QUANTUM WALKS WITH A TIME-DEPENDENT COINED

S. Salimi, R. YOSEFJANI

Discrete-time quantum walk evolve by a unitary operator which involves two operators a conditional shift in position space and a coin operator. This operator entangles the coin and position degrees of freedom of the walker. In this paper, we investigate the asymptotic behavior of the coin position entanglement (CPE) for an inhomogeneous quantum walk which determined by two orthogonal matrices in one-dimensional lattice. Free parameters of coin operator together provide many conditions under which a measurement perform on the coin state yield the value of entanglement on the resulting position quantum state. We study the problem analytically for all values that two free parameters of coin operator can take and the conditions under which entanglement becomes maximal are sought.

arXiv:1012.4566•International Journal of Modern Physics B•Jul 18, 2012•24 citations
Entanglement is a key resource in many quantum information applications and achieving high values independently of the initial conditions is an important task. Here we address the problem of generating highly entangled states in a discrete time quantum walk irrespective of the initial state using two different approaches. First, we present and analyze a deterministic sequence of coin operators which produces high values of entanglement in a universal manner for a class of localized initial states. In a second approach, we optimize the discrete sequence of coin operators using a reinforcement learning algorithm. While the amount of entanglement produced by the deterministic sequence is fully independent of the initial states considered, the optimized sequences achieve in general higher average values of entanglement that do however depend on the initial state parameters. Our proposed sequence and optimization algorithm are especially useful in cases where the initial state is not fully known or entanglement has to be generated in a universal manner for a range of initial states.
Abstract Entanglement is a key resource in many quantum information applications and achieving high values independently of the initial conditions is an important task. Here we address the problem of generating highly entangled states in a discrete time quantum walk irrespective of the initial state using two different approaches. First, we present and analyze a deterministic sequence of coin operators which produces high values of entanglement in a universal manner for a class of localized initial states. In a second approach, we optimize the discrete sequence of coin operators using a reinforcement learning algorithm. While the amount of entanglement produced by the deterministic sequence is fully independent of the initial states considered, the optimized sequences achieve in general higher average values of entanglement that do however depend on the initial state parameters. Our proposed sequence and optimization algorithm are especially useful in cases where the initial state is not fully known or entanglement has to be generated in a universal manner for a range of initial states.
We study maximal coin-position entanglement generation via a discrete-time quantum walk, in which the coin operation is randomly selected from one of two coin operators set at each step. We solve maximal entanglement generation as an optimization problem with quantum process fidelity as the cost function. Then we determine the maximal entanglement that can be rigorously generated for any step beyond the second regardless of initial conditions with appropriate coin sequences. The simplest coin sequence comprising Hadamard and identity operations is equivalent to the generalized elephant quantum walk, which exhibits an increasingly faster spreading in terms of probability distribution. Experimentally, we demonstrate a ten-step quantum walk driven by such coin sequences with linear optics and thereby show the desired high-dimensional bipartite entanglement as well as the transport behavior of faster spreading.
Quantum walks can be used either as tools for quantum algorithm development or as entanglement generators, potentially useful to test quantum hardware. We present a novel algorithm based on a discrete Hadamard quantum walk on a line with one coin and two walkers whose purpose is to generate entanglement between walkers. We provide several classical computer simulations of our quantum algorithm in which we show that, although the asymptotical amount of entanglement generated between walkers does not reach the highest degree of entanglement possible at each step for either coin measurement outcome, the entanglement ratio (entanglement generated/highest value of entanglement possible, for each step) tends to converge, and the actual convergence value depends on the coin initial state and on the coin measurement outcome. Furthermore, our numerical simulations show that, for the quantum walks used in our algorithm, the value towards which entanglement ratio converges also depends on the position probability distribution symmetry of a quantum walk computed with one single walker and the same coin initial state employed in the corresponding quantum walk with two walkers.
Abstract. The coin-position entanglement generated by the evolution operator of a discrete–time quantum walk is quantified, using the von Neumann entropy of the reduced density operator (entropy of entanglement). In the case of a single walker, the entropy of entanglement converges, in the long time limit, to a well defined value which depends on the initial state. Exact expressions are obtained for local and non-local initial conditions. We also discuss the asymptotic bi-partite entanglement generated by non-separable coin operations for two coherent quantum walkers. In this case, the entropy of entanglement is observed to increase logarithmically with time. 1.
Discrete-time quantum walk evolve by a unitary operator which involves two operators a conditional shift in position space and a coin operator. This operator entangles the coin and position degrees of freedom of the walker. In this paper, we investigate the asymptotic behavior of the coin position entanglement (CPE) for an inhomogeneous quantum walk which determined by two orthogonal matrices in one-dimensional lattice. Free parameters of coin operator together provide many conditions under which a measurement perform on the coin state yield the value of entanglement on the resulting position quantum state. We study the problem analytically for all values that two free parameters of coin operator can take and the conditions under which entanglement becomes maximal are sought.