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arXiv:1805.09784 · npj Quantum Information

Experimental demonstration of quantum walks with initial superposition states

Qi-Ping Su, Yu Zhang, Li Yu, Jia-Qi Zhou, Jinshuang Jin, Xiao-Qiang Xu, Shao-Jie Xiong, QingJun Xu, Zhe Sun, Kefei Chen, Franco Nori, Chui-Ping Yang

Published May 17, 2019• 34 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1038/s41534-019-0155-x

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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.
Quantum walks have wide applications in quantum information, such as universal quantum computation, so it is important to explore properties of quantum walks thoroughly. We propose a novel method to implement discrete-time quantum walks (DTQWs) using only a single qubit, in which both coin and walker are encoded in the two-dimensional state space of a single qubit, operations are realized using single-qubit gates only, and high-dimensional final states of DTQWs can be obtained naturally. With this "one-qubit" approach, DTQW experiments can be realized much more easily, compared with previous methods, in most quantum systems and all properties based on quantum states of DTQWs (such as quantum correlation and coherence) can be investigated. By this approach, we experimentally implement one-particle and two-particle DTQWs with seven steps using single photons. Furthermore, we systematically investigate quantum correlations and coherence (based on the full state of the coin and walker) of the DTQW systems with different initial states of the coin, which have not been obtained and studied in DTQW experiments. As an application, we also study the assisted distillation of quantum coherence using the full state of the two-particle DTQW from the experiment. The maximal increase in distillable coherence for high-dimensional mixed states is investigated for the first time by obtaining its upper and lower bounds. Our work opens a new door to implement DTQW experiments and to better explore properties of quantum walks.
Quantum state preparation in high-dimensional systems is an essential requirement for many quantum-technology applications. The engineering of an arbitrary quantum state is, however, typically strongly dependent on the experimental platform chosen for implementation, and a general framework is still missing. Here we show that coined quantum walks on a line, which represent a framework general enough to encompass a variety of different platforms, can be used for quantum state engineering of arbitrary superpositions of the walker's sites. We achieve this goal by identifying a set of conditions that fully characterize the reachable states in the space comprising walker and coin and providing a method to efficiently compute the corresponding set of coin parameters. We assess the feasibility of our proposal by identifying a linear optics experiment based on photonic orbital angular momentum technology.
We analyze the process of a discrete-time quantum walk over 4 steps and 5 positions with linear optics elements. The quantum walk is characterized by a ballistic spread of wavepackets along 4 steps. By employing different initial coin states, we observe non-Gaussian distribution of the walkers' finial position, which characterizes a quadratic enhancement of the spread of photon wavepackets compared to a classical random walk. By introducing controllable decoherence, we observe the quantum-to-classical transmission in a quantum walk architecture.
In this paper, we investigate the enhancement of delocalization and coin-position entanglement in asymmetric discrete-time quantum walks (DTQWs). The asymmetry results from asymmetric coin operations, asymmetric initial states, and asymmetric polarization-dependent losses. By varying these asymmetry factors, the inverse participation ratio and entanglement entropy of the walker are numerically calculated for different coin and loss parameters, both for symmetric and asymmetric initial states. We then experimentally implement a 16-step asymmetric DTQW using a time-multiplexing fiber loop structure. By choosing an asymmetric initial state, both coin-position entanglement and delocalization are simultaneously enhanced under specific coin parameters. Moreover, we observe that with finite asymmetric polarization-dependent loss, the photon probability on the left side decreases significantly, while that on the right side increases and becomes more localized. Interestingly, under specific coin parameters, the entanglement and delocalization exhibit improved robustness against polarization-dependent loss. These results demonstrate that the DTQWs constitute an ideal platform for investigating photonic delocalization and hybrid entanglement.
In this paper, we investigate the enhancement of delocalization and coin-position entanglement in asymmetric discrete-time quantum walks (DTQWs). The asymmetry results from asymmetric coin operations, asymmetric initial states, and asymmetric polarization-dependent losses. By varying these asymmetry factors, the inverse participation ratio and entanglement entropy of the walker are numerically calculated for different coin and loss parameters, both for symmetric and asymmetric initial states. We then experimentally implement a 16-step asymmetric DTQW using a time-multiplexing fiber loop structure. By choosing an asymmetric initial state, both coin-position entanglement and delocalization are simultaneously enhanced under specific coin parameters. Moreover, we observe that with finite asymmetric polarization-dependent loss, the photon probability on the left side decreases significantly, while that on the right side increases and becomes more localized. Interestingly, under specific coin parameters, the entanglement and delocalization exhibit improved robustness against polarization-dependent loss. These results demonstrate that the DTQWs constitute an ideal platform for investigating photonic delocalization and hybrid entanglement.
We explore a discrete-time, coined quantum walk on a quantum network where the coherent superposition of walker moves originates from the unitary interaction of the walker coin with the qubit degrees of freedom in the quantum network. The walk dynamics leads to a growth of entanglement between the walker and the network on one hand and, on the other, between the network qubits among themselves. The initial entanglement among the network qubits plays a crucial role in determining the asymptotic values of these entanglement measures and the quantum walk statistics. Specifically, the entanglement entropy of the walker-network state and the negativity of the quantum network-qubit state saturate to values increasing with the initial network entanglement. The asymptotic time-averaged walker-position probability distribution shows increasing localization around the initial walker position with higher initial network entanglement. A potential application of these results as a characterization tool for quantum network properties is suggested.