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Journal of Physics Communications

Quantum walk as generalized evolution and measurement device

Xiang Zhan

Published Aug 1, 2019• 2 citations
Publisher

Abstract

Abstract We show that arbitrary quantum evolution, both unitary and non-unitary ones, can be efficiently realized via one-dimensional quantum walk. This is done based on the equivalence between quantum evolution and positive-operated valued measurements, and an algorithm of properly controlling one-dimensional discrete-time quantum walk to realize arbitrary positive-operated valued measurements. Furthermore, our method based on one-dimensional quantum walk can be easily generalized to other platforms Our method enriches technics for quantum information processes, and the applications of quantum walk.

DOI
10.1088/2399-6528/ab3c00

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Abstract We show that arbitrary quantum evolution, both unitary and non-unitary ones, can be efficiently realized via one-dimensional quantum walk. This is done based on the equivalence between quantum evolution and positive-operated valued measurements, and an algorithm of properly controlling one-dimensional discrete-time quantum walk to realize arbitrary positive-operated valued measurements. Furthermore, our method based on one-dimensional quantum walk can be easily generalized to other platforms Our method enriches technics for quantum information processes, and the applications of quantum walk.
We show that a one-dimensional discrete time quantum walk can be used to implement a generalized measurement in terms of a positive operator value measure (POVM) on a single qubit. More precisely, we show that for a single qubit any set of rank 1 and rank 2 POVM elements can be generated by a properly engineered quantum walk. In such a scenario the measurement of a particle at a position x=i corresponds to a measurement of a POVM element E_{i} on a qubit. Since the idea of quantum walks originates from the von Neumann model of measurement, in which the change of the position of the pointer depends on the state of the system that is being measured, we argue that von Neumann measurements can be naturally extended to POVMs if one includes the internal evolution of the system in the model.
Quantum measurements play a fundamental role in quantum mechanics. Especially, generalized quantum measurements provide a powerful and versatile tool to extract information from quantum systems. However, how to realize them on an arbitrary higher-dimensional quantum system remains a challenging task. Here we propose a simple recipe for the implementation of a general positive-operator valued measurement (POVM) on a higher-dimensional quantum system via a one-dimensional discrete-time quantum walk with a two-dimensional coin. Furthermore, using single photons and linear optics, we realize experimentally a symmetric, informationally complete (SIC) POVM on a three-dimensional system with high fidelity. As an application, we realize a qutrit state tomography with SIC-POVM and confirm that the infidelity scaling achieved by the qutrit SIC-POVM is as good as that based on mutually unbiased bases. Our study paves the way to explore physics and information in higher-dimensional quantum systems and finds applications in various quantum information processing tasks that rely on generalized quantum measurements.
We demonstrate an implementation of unambiguous state discrimination of two equally probable single-qubit states via a one-dimensional photonic quantum walk experimentally. Furthermore we experimentally realize a quantum walk algorithm for implementing a generalized measurement in terms of positive operator value measurement on a single qubit. The measurement of the single-photons' positions corresponds to a measurement of an element of the positive operator value measurement on the polarizations of the single-photons.
Quantum and random walks were proven to be equivalent on finite graphs by demonstrating how to construct a time-dependent random walk sharing the exact same evolution of vertex probability of any given discrete-time coined quantum walk. Such equivalence stipulated a deep connection between the processes that is far stronger than simply considering quantum walks as quantum analogues of random walks. This article expands on the connection between quantum and random walks by demonstrating a procedure that constructs a time-dependent quantum walk matching the evolution of vertex probability of any given random walk in a unitary way. It is a trivial fact that a quantum walk measured at all time steps of its evolution degrades to a random walk. More interestingly, the method presented describes a quantum walk that matches a random walk without measurement operations, such that the unitary evolution of the quantum walk captures the probability evolution of the random walk. The construction procedure is general, covering both homogeneous and non-homogeneous random walks. For the homogeneous random walk case, the properties of unitary evolution imply that the quantum walk described is time-dependent since homogeneous quantum walks do not converge for arbitrary initial conditions
Quantum walks are a promising framework for developing quantum algorithms and quantum simulations. Quantum walks represent an important test case for the application of quantum computers. Here we present different forms of discrete-time quantum walks and show their equivalence for physical realizations. Using an appropriate digital mapping of the position space on which a walker evolves onto the multi-qubit states in a quantum processor, we present different configurations of quantum circuits for the implementation of discrete-time quantum walks in one-dimensional position space. With example circuits for a five qubit machine we address scalability to higher dimensions and larger quantum processors.
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.