Abstract
The preparation of initial superposition states of discrete-time quantum walks (DTQWs) is necessary for the study and applications of DTQWs. Based on an encoding method, here, we propose a DTQW protocol in linear optics, which enables the preparation of arbitrary initial superposition states of the walker and the coin and enables to obtain the states of the DTQW in addition to the probability distribution of the walker. With this protocol, we experimentally demonstrate the DTQW in the polarization space with both the walker and the coin initially in superposition states, by using only passive linear-optical elements. The effects of the walker’s different initial superposition states on the spread speed of the DTQW and on the entanglement between the coin and the walker are also experimentally investigated, which have not been reported before. When the walker starts with superposition states, we show that the properties of DTQWs are very different from those of DTQWs starting with a single position. Our findings reveal different properties of DTQWs and pave an avenue to study DTQWs with arbitrary initial states. Moreover, this encoding method enables one to encode an arbitrary high-dimensional quantum state, using a single physical qubit, and may be adopted to implement other quantum information tasks.
Highlights
IntroductionQuantum walks (QWs)[1] are extensions of the classical random walk and have wide applications in quantum algorithms,[2,3,4] quantum simulations,[5,6,7,8,9,10] quantum computation,[11,12,13] and so on.[14,15] In standard one-dimensional (1D) discrete-time quantum walks (DTQWs),[16,17] the walker’s position can be denoted as |x〉 (x is an integer) and the coin can be described with the basis states |0〉c and |1〉c
In each step of a discrete-time quantum walks (DTQWs) in polarization space, the state of the system can be expressed as |Ψ0〉p|0〉 + |Ψ1〉p|1〉, with jΨ0ip
Our results indicate that the protocol be initialized by using two half-wave plates (HWPs) on the two paths of the introduced here is realizable in experiments, and DTQW with small
Summary
Quantum walks (QWs)[1] are extensions of the classical random walk and have wide applications in quantum algorithms,[2,3,4] quantum simulations,[5,6,7,8,9,10] quantum computation,[11,12,13] and so on.[14,15] In standard one-dimensional (1D) discrete-time quantum walks (DTQWs),[16,17] the walker’s position can be denoted as |x〉 (x is an integer) and the coin can be described with the basis states |0〉c and |1〉c. The evolutions of the walker and the coin are usually characterized by a time-independent unitary operator U = TSc(ψ). The result of the DTQW with a finite number of steps is determined by the initial states of the coin and the walker, as well as the operator U.
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