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arXiv:2108.04923

Discrete-time quantum walk approach to high-dimensional quantum state transfer and quantum routing

Hengji Li, Jian Li, Xiu‐Bo Chen

Published Aug 10, 2021• 2 citations
arXivHTMLPDF

Abstract

High-dimensional quantum systems can offer extended possibilities and multiple advantages while developing advanced quantum technologies. In this paper, we propose a class of quantum-walk architecture networks that admit the efficient routing of high-dimensional quantum states. Perfect state transfer of an arbitrary unknown qudit state can be achieved between two arbitrary nodes via a one-dimensional lackadaisical discrete-time quantum walk. In addition, this method can be generalized to the high-dimensional lattices, where it allows distillable entanglement to be shared between arbitrary input and output ports. Implementation of our scheme is more feasible through exploiting the coin degrees of freedom and the settings of the coin flipping operators are simple. These results provide a direct application in a high-dimensional computational architecture to process much more information.

DOI
10.48550/arxiv.2108.04923

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High-dimensional quantum systems can offer extended possibilities and multiple advantages while developing advanced quantum technologies. In this paper, we propose a class of quantum-walk architecture networks that admit the efficient routing of high-dimensional quantum states. Perfect state transfer of an arbitrary unknown qudit state can be achieved between two arbitrary nodes via a one-dimensional lackadaisical discrete-time quantum walk. In addition, this method can be generalized to the high-dimensional lattices, where it allows distillable entanglement to be shared between arbitrary input and output ports. Implementation of our scheme is more feasible through exploiting the coin degrees of freedom and the settings of the coin flipping operators are simple. These results provide a direct application in a high-dimensional computational architecture to process much more information.
We show that a perfect state transfer of an arbitrary unknown two-qubit state can be achieved via a discrete-time quantum walk with various settings of coin flips and extend this method to the distribution of an arbitrary unknown multiqubit entangled state between every pair of sites in the multidimensional network. Furthermore, we study the routing of quantum information on this network in a quantum-walk architecture, which can be used as quantum information processors to communicate between separated qubits.
We propose a scheme for perfect transfer of an unknown qubit state via the discrete-time quantum walk on a line or a circle. For this purpose, we introduce an additional coin operator which is applied at the end of the walk. This operator does not depend on the state to be transferred. We show that perfect state transfer over an arbitrary distance can be achieved only if the walk is driven by an identity or a flip coin operator. Other biased coin operators and the Hadamard coin allow perfect state transfer over finite distances only. Furthermore, we show that quantum walks ending with a perfect state transfer are periodic.
We propose a scheme for perfect transfer of an unknown qubit state via the discrete-time quantum walk on a line or a circle. For this purpose, we introduce an additional coin operator which is applied at the end of the walk. This operator does not depend on the state to be transferred. We show that perfect state transfer over an arbitrary distance can be achieved only if the walk is driven by an identity or a flip coin operator. Other biased coin operators and the Hadamard coin allow perfect state transfer over finite distances only. Furthermore, we show that quantum walks ending with a perfect state transfer are periodic.
Efficient generation of arbitrary superposed quantum states in high-dimensional systems is crucial for various quantum technologies but remains fundamentally challenging because of the intricate and extensive parameter control. In this work, we propose a generalized discrete one-dimensional quantum walk model with time- and position-dependent coin operations, and varied shift lengths. This approach enables deterministic and direct preparation of any qudit state | ϕ 〉 = ∑ k = 0 d − 1 a k | k 〉 using binary or ternary tree structures, without solving quadratic equations. Our method requires only O ( log d ) steps and O ( d ) coin operations, significantly reducing the resource requirements compared to the previous approach of O ( d ) steps and O ( d 2 ) coin operations. Additionally, we propose an experimental scheme based on successive interferometers, demonstrating that our method is compatible with current state-of-the-art experimental technologies. This efficient method for preparing quantum superposition states lays a foundation for advancing quantum technologies, with broad applications in quantum information processing.
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.
We introduce some new perfect state transfer and teleportation schemes by quantum walks with two coins. Encoding the transferred information in coin 1 state and alternatively using two coin operators, we can perfectly recover the information on coin 1 state at target position only by at most two times of flipping operation. Based on quantum walks with two coins either on the line or on the NNN-circle, we can perfectly transfer any qubit state. In addition, using quantum walks with two coins either on complete graphs or regular graphs, we can first implement perfect qudit state transfer by quantum walks. Compared with existing schemes driven by one coin, more general graph structures can be used to perfectly transfer more general state. Moreover, the external control of coin operator during the transmitting process can be decreased greatly. Selecting coin 1 as the sender and coin 2 as the receiver, we also study how to realize generalized teleportation over long steps of walks by the above quantum walk models. Because quantum walks is an universal quantum computation model, our schemes may provide an universal platform for the design of quantum network and quantum computer.