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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

The role of measurement in the recurrence property of discrete timed quantum walks

L. Kecskés, T. Kiss, M. Štefaňák, Igor Jex

Published Apr 29, 2011• 0 citations
Publisher

Abstract

The quantum mechanical generalisation of random walks (called Quantum Walks) present us with a broader spectrum of possibilities compared to their classical counterparts. The aim of the presented study is to explore a new portion of this area by incorporating a new step in the process of the Quantum Walk unique to quantum mechanics: the measurement. Our focus lies in the characterising number of the recurrence behaviour of the walk (Polya-number). We observe the effect of the standard projective measurement, a yes-no measurement on the origin and the effect of different measurement schemes (periodic and random) on the definition and the numeric value of the Polya-number.

DOI
10.1117/12.886813

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The quantum mechanical generalisation of random walks (called Quantum Walks) present us with a broader spectrum of possibilities compared to their classical counterparts. The aim of the presented study is to explore a new portion of this area by incorporating a new step in the process of the Quantum Walk unique to quantum mechanics: the measurement. Our focus lies in the characterising number of the recurrence behaviour of the walk (Polya-number). We observe the effect of the standard projective measurement, a yes-no measurement on the origin and the effect of different measurement schemes (periodic and random) on the definition and the numeric value of the Polya-number.
Recurrence of quantum walks on lattices can be characterized by the generalized Pólya number. Its value reflects the difference between a classical and a quantum system. The dimension of the lattice is not a unique parameter in the quantum case; both the coin operator and the initial quantum state of the coin influence the recurrence in a nontrivial way. In addition, the definition of the Pólya number involves measurement of the system. Depending on how measurement is included in the definition, the recurrence properties vary. We show that in the limiting case of frequent, strong measurements, one can approach the classical dynamics. Comparing various cases, we have found numerical indication that our previous definition of the Pólya number provides an upper limit.
The measurement process plays a crucial role in quantum mechanics as it interrupts the unitary evolution of a state. The consequences are apparent when investigating the return probability (Polya-number) of a particle in a Hadamard walk on the line [1, 2]: Depending on whether the evolution is restarted or continued after the measurement, either a transient or a recurrent evolution can be observed. We investigate both cases experimentally for the first time and monitor the evolution of the walker over 39 steps, revealing the fundamental differences of the two cases as predicted by theory: In the restart-regime, the Polya-number will gradually approach 1. In contrast, the continue-regime yields an asymptotic value of 2/π.
We propose a definition for the P\'olya number of continuous-time quantum walks to characterize their recurrence properties. The definition involves a series of measurements on the system, each carried out on a different member from an ensemble in order to minimize the disturbance caused by it. We examine various graphs, including the ring, the line, the higher-dimensional integer lattices, and a number of other graphs, and we calculate their P\'olya number. For the timing of the measurements, a Poisson process as well as regular timing are discussed. We find that the speed of decay for the probability at the origin is the key for recurrence.
The measurement process plays a crucial role in quantum mechanics as it interrupts the unitary evolution of a state. The consequences are apparent when investigating the return probability (Polya-number) of a particle in a Hadamard walk on the line [1,2]: Depending on whether the evolution is restarted or continued after the measurement, either a transient or a recurrent evolution can be observed. We investigate both cases experimentally for the first time by introducing sinks in a time-multiplexed quantum walk setup using a fast-switching electro-optic modulator (EOM). Monitoring the evolution of the walker over 39 steps reveals the fundamental differences of the two cases as predicted by theory: In the restart-regime, the Polya-number will gradually approach unity. In contrast, the continue-regime yields an asymptotic value of the Polya-number of 2/π, which is reached in good approximation after only four steps and then remains almost constant.
We analyze the recurrence probability (Pólya number) for d-dimensional unbiased quantum walks. A sufficient condition for a quantum walk to be recurrent is derived. As a by-product we find a simple criterion for localization of quantum walks. In contrast with classical walks, where the Pólya number is characteristic for the given dimension, the recurrence probability of a quantum walk depends in general on the topology of the walk, choice of the coin and the initial state. This allows us to change the character of the quantum walk from recurrent to transient by altering the initial state.