Quantum walks with memory provided by recycled coins and a memory of the coin-flip history

Abstract

Quantum walks have emerged as an interesting approach to quantum information processing, exhibiting many unique properties compared to the analogous classical random walk. Here we introduce a model for a discrete-time quantum walk with memory by endowing the walker with multiple recycled coins and using a physical memory function via a history dependent coin flip. By numerical simulation we observe several phenomena. First in one dimension, walkers with memory have persistent quantum ballistic speed up over classical walks just as found in previous studies of multicoined walks with trivial memory function. However, measurement of the multicoin state can dramatically shift the mean of the spatial distribution. Second, we consider spatial entanglement in a two-dimensional quantum walk with memory and find that memory destroys entanglement between the spatial dimensions, even when entangling coins are employed. Finally, we explore behavior in the presence of spatial randomness and find that in the time regime where single-coined walks localize, multicoined walks do not and in fact a memory function can speed up the walk relative to a multicoin walker with no memory. We explicitly show how to construct linear optics circuits implementing the walks, and discuss prospects for classical simulation.

One-dimensional quantum walks with two-step memory

“…Other models of quantum walks with history dependence have been proposed by using multiple coins or mixing coin [5][6][7][8][9]. The distribution for 3-coin history-dependent quantum walk (HDQW for short) [5] seems close in shape to our result.…”

“…For concreteness, we focus on the QW2M with Hadamard coin, whose direction for each step is determined by (6). The first few steps of this walk are |0, 1, 0, 0 → 1/ √ 2 (|1, 0, −1, 0 + |1, 0, 1, 1 ) (7) →1/2(|0, −1, 0, 0 + |0, −1, −2, 1 + |0, 1, 2, 0 − |0, 1, 0, 1 ) (8) →1/(2 √ 2)(|−1, 0, 1, 0 + |−1, 0, −1, 1 + |−1, −2, −1, 0 − |−1, −2, −3, 1 + |1, 2, 1, 0 + |1, 2, 3, 1 − |1, 0, −1, 0 + |1, 0, 1, 1 ) (9) →1/4(|0, 1, 0, 0 + |0, 1, 2, 1 + |0, −1, −2, 0 − |0, −1, 0, 1 + |−2, −1, 0, 0 + |−2, −1, −2, 1 − |−2, −3, −2, 0 + |−2, −3, −4, 1 + |2, 1, 0, 0 + |2, 1, 2, 1 + |2, 3, 2, 0 − |2, 3, 4, 1 − |0, −1, 0, 0 − |0, −1, −2, 1 + |0, 1, 2, 0 − |0, 1, 0, 1 ) (10) →1/(4 √ 2)(|1, 0, −1, 0 + |1, 0, 1, 1 + |1, 2, 1, 0 − |1, 2, 3, 1 + |−1, −2, −1, 0 + |−1, −2, −3, 1 − |−1, 0, 1, 0 + |−1, 0, −1, 1 + |−1, 0, −1, 0 + |−1, 0, 1, 1 + |−1, −2, −3, 0 − |−1, −2, −1, 1 − |−3, −2, −1, 0 − |−3, −2, −3, 1 + |−3, −4, −3, 0 − |−3, −4, −5, 1 + |1, 0, 1, 0 + |1, 0, −1, 1 + |1, 2, 3, 0 − |1, 2, 1, 1 + |3, 2, 1, 0 + |3, 2, 3, 1 − |3, 4, 3, 0 + |3, 4, 5, 1 − |−1, 0, 1, 0 − |−1, 0, −1, 1 − |−1, −2, −1, 0 + |−1, −2, −3, 1 + |1, 2, 1, 0 + |1, 2, 3, 1 − |1, 0, −1, 0 + |1, 0, 1, 1 )…”

History dependent quantum random walks as quantum lattice gas automata

“…The memory of the velocities is updated each time a velocity is used to determine the next site. The QRW transition in Brun, Carteret and Ambainis' model [29] and in Rohde, Brennen and Gilchrist's model [32] (both described by case (a) above) scatters the velocity that happened N steps ago. Thus it is convenient to split the scattering into two distinct stages as in [32], the first of which is the selection of the velocity to scatter or the memory operation and the other is the ricochet operation that refers to the interaction between the particle and the site.…”

“…We therefore only consider velocity memory. In the case (a) models [29,30,32], in order to store the history, the single velocity qubit of a QRW is replaced with multiple velocity qubits, N C 2 . The Hilbert space for a QRW with particle history is then…”

“…is the symmetric scattering matrix, parameterized by θ k for the k-th velocity selection (all R k are identical in [32]),…”