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arXiv:1608.00498 · Physical Review A

Perfect state transfer by means of discrete-time quantum walk search algorithms on highly symmetric graphs

M. Štefaňák, S. Skoupý

Published Aug 1, 2016• 53 citations
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Abstract

Perfect state transfer between two marked vertices of a graph by means of a discrete-time quantum walk is analyzed. We consider the quantum walk search algorithm with two marked vertices, sender and receiver. It is shown by explicit calculation that, for the coined quantum walks on a star graph and a complete graph with self-loops, perfect state transfer between the sender and receiver vertex is achieved for an arbitrary number of vertices $N$ in $O(\sqrt{N})$ steps of the walk. Finally, we show that Szegedy's walk with queries on a complete graph allows for state transfer with unit fidelity in the limit of large $N$.

DOI
10.1103/physreva.94.022301

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Perfect state transfer between two marked vertices of a graph by means of a discrete-time quantum walk is analyzed. We consider the quantum walk search algorithm with two marked vertices, sender and receiver. It is shown by explicit calculation that, for the coined quantum walks on a star graph and a complete graph with self-loops, perfect state transfer between the sender and receiver vertex is achieved for an arbitrary number of vertices NNN in O(N)O(\sqrt{N})O(N​) steps of the walk. Finally, we show that Szegedy's walk with queries on a complete graph allows for state transfer with unit fidelity in the limit of large NNN.
Perfect state transfer can be achieved between two marked vertices of graphs like a star graph, a complete graph with self-loops and a complete bipartite graph, and two-dimensional Lattice by means of discrete-time quantum walk. In this paper, we investigate the quality of quantum state transfer between two marked vertices of an unsymmetrical graph like the butterfly network. Our numerical results support the conjecture that the fidelity of state transfer depends on the quantum state to be transferred dynamically. The butterfly network is a typical example studied in networking coding. Therefore, these results can provide a clue to the construction of quantum network coding schemes.
Abstract Previously it was shown that (almost) perfect state transfer can be achieved on the complete bipartite graph by a discrete-time coined quantum walk based algorithm when both the sender and receiver vertices are in the same partition of the graph and when the sender and receiver are in opposite partitions of the same size. By changing the coin operator, we analyze the state transfer problem and we show that it is still possible to achieve state transfer with high fidelity even when the sender and receiver are in different partitions with different sizes. Moreover, it is also possible to use an active switch approach using lackadaisical quantum walks where the marked vertex is switched between the sender and receiver during the algorithm.
Search and state transfer between hubs, i.e., fully connected vertices, on otherwise arbitrary connected graph is investigated. Motivated by a recent result of Razzoli et al. [J. Phys. A: Math. Theor. 55, 265303 (2022)] on the universality of hubs in continuous-time quantum walks and spatial search, we extend the investigation to state transfer and also to the discrete-time case. We show that the continuous-time quantum walk allows for perfect state transfer between multiple hubs if the numbers of senders and receivers are close. Turning to the discrete-time case, we show that the search for hubs is successful provided that the initial state is locally modified to account for a degree of each individual vertex. Concerning state transfer using discrete-time quantum walk, it is shown that between a single sender and a single receiver one can transfer two orthogonal states in the same run time. Hence, it is possible to transfer an arbitrary quantum state of a qubit between two hubs. In addition, if the sender and the receiver know each other location, another linearly independent state can be transferred, allowing for exchange of a qutrit state. Finally, we consider the case of transfer between multiple senders and receivers. In this case we cannot transfer specific quantum states. Nevertheless, quantum walker can be transferred with high probability in two regimes---either when there is a similar number of senders and receivers, which is the same as for the continuous-time quantum walk, or when the number of receivers is considerably larger than the number of senders. Our investigation is based on dimensional reduction utilizing the invariant subspaces of the respective evolutions and the fact that for the appropriate choice of the loop weights the problem can be reduced to the complete graph with loops.
We investigate coined quantum-walk search and state-transfer algorithms, focusing on the complete MMM-partite graph with NNN vertices in each partition. First, it is shown that by adding a loop to each vertex, the search algorithm finds the marked vertex with unit probability in the limit of a large graph. Next, we employ the evolution operator of the search with two marked vertices to perform a state transfer between the sender and the receiver. We show that when the sender and the receiver are in different partitions, the algorithm succeeds with fidelity approaching unity for a large graph. However, when the sender and the receiver are in the same partition, the fidelity does not reach exactly 1. To solve this problem, we propose a state-transfer algorithm with an active switch, whose fidelity can be estimated based on the single vertex search alone.
Abstract Perfect state transfer has attracted a great deal of attention recently due to its crucial role in quantum communication and scalable quantum computation. In this paper, we propose the perfect state transfer algorithms with a pair of sender-receiver and two pairs of sender-receiver on the complete bipartite graph respectively. The algorithm with a pair of sender-receiver is implemented through discrete-time quantum walk, flexibly setting the coin operators based on the positions of the sender and receiver. The algorithm with two pairs of sender-receiver ensures that the two quantum states are distributed on both sides of the complete bipartite graph during the process, thereby achieving perfect state transfer. In addition, the quantum circuits corresponding to the algorithms are provided. The algorithms can transfer an arbitrary quantum state and can simultaneously transfer two arbitrary quantum states from the senders to the receivers in any case. Moreover, the algorithms are not only applicable to complete bipartite graphs but also to more graph structures with complete bipartite subgraphs, which will provide potential applications for quantum information processing.
Abstract We investigate state transfer on a hypercube by means of a quantum walk where the sender and the receiver vertices are marked by a weighted loops. First, we analyze search for a single marked vertex, which can be used for state transfer between arbitrary vertices by switching the weighted loop from the sender to the receiver after one run-time. Next, state transfer between antipodal vertices is considered. We show that one can tune the weight of the loop to achieve state transfer with high fidelity in shorter run-time in comparison to the state transfer with a switch. Finally, we investigate state transfer between vertices of arbitrary distance. It is shown that when the distance between the sender and the receiver is at least 2, the results derived for the antipodes are well applicable. If the sender and the receiver are direct neighbours the evolution follows a slightly different course. Nevertheless, state transfer with high fidelity is achieved in the same run-time.