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Quantum Information and Computation

Periodicity and perfect state transfer in quantum walks on variants of cycles

Katharine E. Barr, Tim Proctor, Daniel Allen, Viv M. Kendon

Published May 1, 2014• 18 citations
Publisher

Abstract

We systematically investigated perfect state transfer between antipodal nodes of discrete time quantum walks on variants of the cycles $C_4$, $C_6$ and $C_8$ for three choices of coin operator. Perfect state transfer was found, in general, to be very rare, only being preserved for a very small number of ways of modifying the cycles. We observed that some of our useful modifications of $C_4$ could be generalised to an arbitrary number of nodes, and present three families of graphs which admit quantum walks with interesting dynamics either in the continuous time walk, or in the discrete time walk for appropriate selections of coin and initial conditions. These dynamics are either periodicity, perfect state transfer, or very high fidelity state transfer. These families are modifications of families known not to exhibit periodicity or perfect state transfer in general. The robustness of the dynamics is tested by varying the initial state, interpolating between structures and by adding decoherence.

DOI
10.26421/qic14.5-6-3

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We systematically investigated perfect state transfer between antipodal nodes of discrete time quantum walks on variants of the cycles C4C_4C4​, C6C_6C6​ and C8C_8C8​ for three choices of coin operator. Perfect state transfer was found, in general, to be very rare, only being preserved for a very small number of ways of modifying the cycles. We observed that some of our useful modifications of C4C_4C4​ could be generalised to an arbitrary number of nodes, and present three families of graphs which admit quantum walks with interesting dynamics either in the continuous time walk, or in the discrete time walk for appropriate selections of coin and initial conditions. These dynamics are either periodicity, perfect state transfer, or very high fidelity state transfer. These families are modifications of families known not to exhibit periodicity or perfect state transfer in general. The robustness of the dynamics is tested by varying the initial state, interpolating between structures and by adding decoherence.
We systematically investigated perfect state transfer between antipodal nodes of discrete time quantum walks on variants of the cycles C_4, C_6 and C_8 for three choices of coin operator. Perfect state transfer was found, in general, to be very rare, only being preserved for a very small number of ways of modifying the cycles. We observed that some of our useful modifications of C_4 could be generalised to an arbitrary number of nodes, and present three families of graphs which admit quantum walks with interesting dynamics either in the continuous time walk, or in the discrete time walk for appropriate selections of coin and initial conditions. These dynamics are either periodicity, perfect state transfer, or very high fidelity state transfer. These families are modifications of families known not to exhibit periodicity or perfect state transfer in general. The robustness of the dynamics is tested by varying the initial state, interpolating between structures and by adding decoherence.
Abstract. We systematically investigated perfect state transfer between antipodal nodes of discrete time quantum walks on variants of the cycles C4, C6 and C8 for three choices of coin operator. Perfect state transfer was found, in general, to be very rare, only being preserved for a very small number of ways of modifying the cycles. We observed that some of our useful modifications of C4 could be generalised to an arbitrary num-ber of nodes, and present three families of graphs which admit quantum walks with interesting dynamics either in the continuous time walk, or in the discrete time walk for appropriate selections of coin and initial conditions. These dynamics are either periodicity, perfect state trans-fer, or very high fidelity state transfer. These families are modifications of families known not to exhibit periodicity or perfect state transfer in general. The robustness of the dynamics is tested by varying the initial state, interpolating between structures and by adding decoherence. 1
We present three families of graphs which admit quantum walks with interesting dynamics either in the continuous time walk, or in the discrete time walk for appropriate selections of coin and initial conditions. These dynamics are either periodicity, perfect state transfer, or very high fidelity state transfer. These families are modifications of families known not to exhibit periodicity or perfect state transfer in general. The robustness of the dynamics is tested by varying the initial state and by adding decoherence. The perfect state transfer is found to be very robust to small disturbances. The families of graphs found to exhibit useful properties are generalisations of small graphs. By varying the strength of edges in each family, interpolation between the families can be carried out, enabling further understanding of the factors affecting perfect state transfer. The perfect state transfer in these families is found to be destroyed by this variation, indicating that the transport strongly depends on the graph structure. A brief summary of other small graphs investigated but found not to exhibit such properties is included for completeness.
We provide a brief survey of perfect state transfer in quantum walks on finite graphs. The ability to transfer a quantum state from one part of a quantum computer to another is a key ingredient of scalable architectures. Transfer through structures that do not require locally varying dynamic control simplifies the design and hence reduces the opportunities for errors to arise. Continuous time walks quantum walks on highly structured graphs exhibit perfect state transfer for the complete graph of size 2, the path of length 3, and the cycle of size 4. From these, larger graphs can be constructed, and the use of edge weights widens this set considerably. Discrete-time quantum walks have more flexibility through exploiting the coin degrees of freedom, but with the disadvantage that local control of the coin is required if the degree of the vertices varies. The closely related property of periodicity (exact return to the starting state) is also mentioned.
A continuous-time quantum random walk describes the motion of a quantum mechanical particle on an underlying graph. The graph itself is associated with a Hilbert space of dimension equal to the number of vertices. The dynamics of the walk is governed by the unitary operator U(t)=eiAt\mathcal{U}(t) = e^{iAt}U(t)=eiAt, where AAA is the adjacency matrix of the graph. An important notion in the quantum random walk is the transfer of a quantum state from one vertex to another. If the fidelity of the transfer is unity, we call it a perfect state transfer. Many graph families have been shown to admit PST or periodicity, including cubelike graphs. These graphs are unweighted. In this paper, we generalize the PST or periodicity of cubelike graphs to that of weighted cubelike graphs. We characterize the weights for which they admit PST or show periodicity, both at time t=π2t=\fracπ{2}t=2π​.
Let G be a graph and let t be a positive real number. Then the evolution of the continuous quantum walk defined on G is described by the transition matrix U(t)=exp(itH).The matrix H is called Hamiltonian. So far the most studied quantum walks are the ones whose Hamiltonians are the adjacency matrices of the underlying graphs and initial states are vertex states e_a, with e_a being the characteristic vector of vertex a. \n \nThis thesis focuses on Laplacian edge state transfer, that is, the quantum walks whose initial states are edge states e_a-e_b and Hamiltonians are the Laplacians of the underlying graphs. So far the research about perfect state transfer only involves vertex states and Laplacian edge state transfer has not been studied before. We extend the known results of perfect vertex state transfer and periodicity of vertex states to Laplacian edge state transfer. \n \nWe prove two useful closure properties for perfect Laplacian edge state transfer. One is that complementation preserves perfect edge state transfer. The other is that if G has perfect Laplacian edge state transfer at time τ and H has perfect Laplacian vertex state transfer also at time τ, then with some mild assumption on the pairs of vertex states and edge states that have perfect state transfer, the Cartesian product G □ H also admits perfect edge state transfer. Those two properties provide us new ways to construct graphs with perfect Laplacian edge state transfer. We also observe one phenomenon that happens in Laplacian edge state transfer which never happens in vertex state transfer: if there is perfect state transfer from e_a-e_b to e_α-e_β and also from e_b-e_c to e_β-e_γ at the same time t in G, then G admits perfect state transfer from e_a-e_c to e_α-e_γ at time t. \n \nWe give characterizations of perfect Laplacian edge state transfer in cycles, paths and complete bipartite graphs K_{2,4n}. We study perfect state transfer and periodicity on edge states with special spectral features. We also consider the case when the unsigned Laplacian is Hamiltonian and initial states are plus states of the form e_a+e_b. In this case, we characterize perfect state transfer in paths, cycles and bipartite graphs. \nWe close this thesis by a list of open questions.