Quantum walks are known to have nontrivial interactions with absorbing boundaries. In particular it has been shown that an absorbing boundary in the one-dimensional quantum walk partially reflects information, as observed by absorption probability computations. In this paper, we shift our sights from the local phenomena of absorption probabilities to the global behavior of finite absorbing quantum walks in one dimension. We conduct our analysis by approximating the eigenbasis of the associated absorbing quantum walk operator matrix ${Q}_{n}$ , where $n$ is the lattice size. The conditional probability distributions of these finite absorbing quantum walks exhibit distinct behavior at various timescales, namely wavelike reflections for times $t=O(n)$ , fractional quantum revivals for $t=O({n}^{2})$ , and stability for $t=O({n}^{3})$ . At the end of this paper, we demonstrate the existence of fractional quantum revivals in other sufficiently regular quantum walk systems.Quantum walks are known to have nontrivial interactions with absorbing boundaries. In particular it has been shown that an absorbing boundary in the one-dimensional quantum walk partially reflects information, as observed by absorption probability computations. In this paper, we shift our sights from the local phenomena of absorption probabilities to the global behavior of finite absorbing quantum walks in one dimension. We conduct our analysis by approximating the eigenbasis of the associated absorbing quantum walk operator matrix ${Q}_{n}$, where $n$ is the lattice size. The conditional probability distributions of these finite absorbing quantum walks exhibit distinct behavior at various timescales, namely wavelike reflections for times $t=O(n)$, fractional quantum revivals for $t=O({n}^{2})$, and stability for $t=O({n}^{3})$. At the end of this paper, we demonstrate the existence of fractional quantum revivals in other sufficiently regular quantum walk systems.

arXiv:1909.12680Physical Review AMar 6, 202011 citations

One-dimensional quantum walks

Andris Ambainis, Eric Bach, Ashwin Nayak +2 more

We define and analyze quantum computational variants of \nrandom walks on one-dimensional lattices. In particular, we analyze a quantum analog of the symmetric random \nwalk, which we call the Hadamard walk. Several striking \ndifferences between the quantum and classical cases are ob- served. For example, when unrestricted in either direction, \nthe Hadamard walk has position that is nearly uniformly \ndistributed in the range [-t/√2, t/√2] after t steps, which \nis in sharp contrast to the classical random walk, which has \ndistance O(√t) from the origin with high probability. With \nan absorbing boundary immediately to the left of the starting position, the probability that the walk exits to the left is 2/π, and with an additional absorbing boundary at location n, the probability that the walk exits to the left actually increases, approaching 1/√2 in the limit. In the classical case both values are 1.We define and analyze quantum computational variants of \nrandom walks on one-dimensional lattices. In particular, we analyze a quantum analog of the symmetric random \nwalk, which we call the Hadamard walk. Several striking \ndifferences between the quantum and classical cases are ob- served. For example, when unrestricted in either direction, \nthe Hadamard walk has position that is nearly uniformly \ndistributed in the range [-t/√2, t/√2] after t steps, which \nis in sharp contrast to the classical random walk, which has \ndistance O(√t) from the origin with high probability. With \nan absorbing boundary immediately to the left of the starting position, the probability that the walk exits to the left is 2/π, and with an additional absorbing boundary at location n, the probability that the walk exits to the left actually increases, approaching 1/√2 in the limit. In the classical case both values are 1.

ACM Symposium on Theory of Computing (STOC)Jul 6, 2001563 citations

Properties of long quantum walks in one and two dimensions

Hao Luo, Peng Xue

arXiv:1503.06293Quantum Information ProcessingSep 22, 20156 citations

One-dimensional quantum walks with two-step memory

Qing Zhou, Songfeng Lu

arXiv:1904.06528Quantum Information ProcessingOct 22, 20198 citations

{Q}_{n}

, where

n

is the lattice size. The conditional probability distributions of these finite absorbing quantum walks exhibit distinct behavior at various timescales, namely wavelike reflections for times

t=O(n)

, fractional quantum revivals for

t=O({n}^{2})

, and stability for

t=O({n}^{3})

. At the end of this paper, we demonstrate the existence of fractional quantum revivals in other sufficiently regular quantum walk systems.