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arXiv:1509.05816

Discrete-time quantum walks as generators of multipartite entanglement

Joshua Lockhart, Mauro Paternostro

Published Sep 18, 2015• 3 citations
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Abstract

We extend the idea of a discrete-time quantum walk on a graph by placing a qubit on each vertex, and allowing the walker to interact with the qubit at its current position. We show that allowing for a controlled-Z interaction at each time step between the walker and the vertex qubits generates multipartite entanglement between the qubits. We demonstrate that for particular coin operators the system generates a highly entangled cluster state, of use in measurement based quantum computation.

DOI
10.48550/arxiv.1509.05816

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We extend the idea of a discrete-time quantum walk on a graph by placing a qubit on each vertex, and allowing the walker to interact with the qubit at its current position. We show that allowing for a controlled-Z interaction at each time step between the walker and the vertex qubits generates multipartite entanglement between the qubits. We demonstrate that for particular coin operators the system generates a highly entangled cluster state, of use in measurement based quantum computation.
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.
Discrete-time quantum walks provide a versatile framework for investigating the generation, redistribution, and transport of quantum correlations in composite quantum systems. Here, we study the dynamics of bipartite and genuine multipartite entanglement in a two-walker discrete-time quantum walk on a one-dimensional lattice. By employing logarithmic negativity and the generalized geometric measure (GGM), we systematically characterize the redistribution of bipartite entanglement among different subsystem partitions and the emergence of genuine multipartite entanglement involving the two coin and two position degrees of freedom. We show that the entanglement dynamics are strongly influenced by the lattice topology. The open-boundary regime exhibits a monotonic redistribution of quantum correlations, whereas the closed-boundary regime gives rise to pronounced oscillatory behavior due to boundary-induced interference and recurrent wave-packet overlap. In the open-boundary regime, the GGM rapidly approaches its theoretical maximum value of 1/21/21/2 and remains largely insensitive to the choice of the initial Bell state as well as to continuous variations of the local coin operator over a broad parameter range, except near the Pauli-XXX coin. These results demonstrate that maximal genuine multipartite entanglement generation is a robust and generic feature of open-boundary two-walker discrete-time quantum walks, establishing them as promising platforms for engineering multipartite quantum correlations in quantum information processing and quantum simulation.
Quantum magic, which accounts for the nonstabilizer content of a state, is essential for universal quantum computation beyond classically simulable resources. However, the way magic builds up during structured unitary dynamics remains largely open. Here, we investigate the generation and evolution of quantum magic in discrete-time quantum walks (DTQWs)—a simple, tunable unitary model realizable on a wide range of architectures. We use stabilizer Rényi entropy as a measure of quantum magic and investigate single- and two-walker quantum walks on a one-dimensional lattice, considering a wide range of initial coin states. Our results reveal that DTQWs can dynamically generate significant magic, with the amount and structure dependent on the initial state of the coin. In the case of a single walker, we found a nontrivial and complementary relationship between magic and entanglement at long times. In the two-walker setting, even states close to stabilizer form can evolve into highly magical states when subjected to quantum walk protocols. The generation of magic can occur independently of entanglement growth, emphasizing its distinct role as a quantum resource. We further show that the dynamical magic generation is robust under realistic noise—specifically, decoherence in the coin degree of freedom—demonstrating that this process persists across noisy settings. Finally, we discuss an experimentally feasible scheme to measure magic using current technology. Our findings position DTQWs as accessible and controllable platforms for producing quantum magic, offering a fresh perspective on their role in quantum information processing and reliable quantum computation.
We explore a discrete-time, coined quantum walk on a quantum network where the coherent superposition of walker moves originates from the unitary interaction of the walker coin with the qubit degrees of freedom in the quantum network. The walk dynamics leads to a growth of entanglement between the walker and the network on one hand and, on the other, between the network qubits among themselves. The initial entanglement among the network qubits plays a crucial role in determining the asymptotic values of these entanglement measures and the quantum walk statistics. Specifically, the entanglement entropy of the walker-network state and the negativity of the quantum network-qubit state saturate to values increasing with the initial network entanglement. The asymptotic time-averaged walker-position probability distribution shows increasing localization around the initial walker position with higher initial network entanglement. A potential application of these results as a characterization tool for quantum network properties is suggested.
Quantum walks have wide applications in quantum information, such as universal quantum computation, so it is important to explore properties of quantum walks thoroughly. We propose a novel method to implement discrete-time quantum walks (DTQWs) using only a single qubit, in which both coin and walker are encoded in the two-dimensional state space of a single qubit, operations are realized using single-qubit gates only, and high-dimensional final states of DTQWs can be obtained naturally. With this "one-qubit" approach, DTQW experiments can be realized much more easily, compared with previous methods, in most quantum systems and all properties based on quantum states of DTQWs (such as quantum correlation and coherence) can be investigated. By this approach, we experimentally implement one-particle and two-particle DTQWs with seven steps using single photons. Furthermore, we systematically investigate quantum correlations and coherence (based on the full state of the coin and walker) of the DTQW systems with different initial states of the coin, which have not been obtained and studied in DTQW experiments. As an application, we also study the assisted distillation of quantum coherence using the full state of the two-particle DTQW from the experiment. The maximal increase in distillable coherence for high-dimensional mixed states is investigated for the first time by obtaining its upper and lower bounds. Our work opens a new door to implement DTQW experiments and to better explore properties of quantum walks.