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arXiv:2407.16558 · Physical Review A

Parrondo's paradox in quantum walks with inhomogeneous coins

Vikash Mittal, Yi-Ping Huang

Published Nov 27, 2024• 7 citations
arXivHTMLPDF

Abstract

Parrondo's paradox, a counterintuitive phenomenon in which two losing strategies combine to produce a winning outcome, has been a subject of interest across various scientific fields, including quantum mechanics. In this study, we investigate the manifestation of Parrondo's paradox in discrete-time quantum walks. We demonstrate the existence of Parrondo's paradox using site- and time-dependent coins without the need for a higher-dimensional coin or adding decoherence to the system. Our results enhance the feasibility of practical implementations and provide deeper insights into the underlying quantum dynamics, specifically the propagation constrained by the interference pattern of quantum walks. The implications of our results suggest the potential for more accessible and efficient designs in quantum transport, broadening the scope and application of Parrondo's paradox beyond conventional frameworks.

DOI
10.1103/physreva.110.052440

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Parrondo's paradox, a counterintuitive phenomenon in which two losing strategies combine to produce a winning outcome, has been a subject of interest across various scientific fields, including quantum mechanics. In this study, we investigate the manifestation of Parrondo's paradox in discrete-time quantum walks. We demonstrate the existence of Parrondo's paradox using site- and time-dependent coins without the need for a higher-dimensional coin or adding decoherence to the system. Our results enhance the feasibility of practical implementations and provide deeper insights into the underlying quantum dynamics, specifically the propagation constrained by the interference pattern of quantum walks. The implications of our results suggest the potential for more accessible and efficient designs in quantum transport, broadening the scope and application of Parrondo's paradox beyond conventional frameworks.
Parrondo's paradox is a well-known counterintuitive phenomenon, where the combination of unfavorable situations can establish favorable ones. In this paper, we study one-dimensional discrete-time quantum walks, manipulating two different coins (two-state) operators representing two losing games A and B, respectively, to create the Parrondo effect in the quantum domain. We exhibit that games A and B are losing games when played individually but could produce a winning expectation when played alternatively for a particular sequence of different periods for distinct choices of the relative phase. Furthermore, we investigate the regimes of the relative phase of the initial state of coins where Parrondo games exist. Moreover, we also analyze the relationships between Parrondo's game and quantum entanglement and show regimes where the Parrondo sequence may generate a maximal entangler state in our scheme. Along with the applications of different kinds of quantum walks, our outcomes potentially encourage the development of new quantum algorithms.
The well-known counterintuitive phenomenon, where the combination of unfavorable situations can establish favorable ones, is called Parrondo's paradox. Here, we study one-dimensional discrete-time quantum walks, manipulating two different coins (two-state) operators representing two losing games A and B, respectively, to create the Parrondo effect in the quantum domain. We exhibit that games A and B are losing games when played individually but could produce a winning expectation when played alternatively for a particular sequence of the different periods. Moreover, we also analyze the relationships between Parrondo's games and quantum entanglement in our scheme. Along with the applications of different kinds of quantum walks, our outcomes potentially encourage the development of new quantum algorithms.
Parrondo's paradox shows that alternating two losing strategies can yield a net positive outcome—a counterintuitive effect relevant to probabilistic systems and control. We demonstrate a quantum analogue using single-qubit inhomogeneous quantum walk—unlike prior work that required multi-coin or position-dependent constructions. Instead, we apply a local phase gate at the origin and alternate two fixedSU(2)\text{SU}(2)SU(2)coins, with parameters from recent inhomogeneous quantum walks. Sequence ABB yields positive displacement despite no individual bias. This establishes that coherent interference alone can reproduce the Parrondo effect in single-qubit systems, with implications for low-cost quantum algorithms and transport protocols.
We study Parrondo's paradox -- the phenomenon where combining losing strategies yields a winning one -- in a minimal discrete-time quantum walk. We introduce the transport vector, encoded in the coin's steady state, whose inner product with the initial coin state gives the walker's asymptotic velocity. More generally, the paradox emerges exactly when the transport vector of the combined strategy falls outside the cone spanned by the individual ones -- a geometric criterion valid for any combination of strategies. It explains, for instance, why composing two coin operators within a single step can produce the paradox while simple alternation between them cannot, since alternation keeps the combined vector confined to the cone. The paradoxical set has nonzero measure, and we compute its probability explicitly in representative cases. This casts the paradox as one instance of designing reachable transport in quantum walks.
We propose a quantum game based on coin-based quantum walks. Given a quantum walk and a Hermitian operator on the coin-position composite space, winning this game involves choosing an initial coin state such that the given quantum walk leads to a composite state in which the expectation value of the given Hermitian operator is greater than a certain value. Parrondo's paradox is a phenomenon where a combination of losing strategies becomes a winning strategy. We give a deterministic scheme for identifying Parrondo's paradox in our game, in the sense that, given a collection of distinct quantum steps, we identify initial coin states which happen to be losing states for all quantum walks comprised solely of these steps individually, but turn out to be winning states for a quantum walk comprised of all the given steps taken in a sequence. Unlike traditional quantum steps that allow for equal magnitude forward and backward strides based on the outcome of the coin toss, the steps of the quantum walks employed here, though still contingent upon coin toss, permit the strides to be of unequal magnitude, and not necessarily in opposite directions. We believe the results presented here will contribute to a deeper understanding of evolution of expectation values of observables in quantum walks, and facilitate the development of novel quantum algorithms.
Parrondo's paradox is ubiquitous in games, ratchets and random walks.The apparent paradox, devised by J. M. R. Parrondo, that two losing games AAA and BBB can produce an winning outcome has been adapted in many physical and biological systems to explain their working. However, proposals on demonstrating Parrondo's paradox using quantum walks failed in the asymptotic limits. In this work, we show that instead of a single coin if we consider a two coin initial state which may or may not be entangled, we can observe a genuine Parrondo's paradox with quantum walks. Further we focus on reasons for this and pin down the asymmetry in initial two-coin state or asymmetry in shift operator, either of which are necessary for observing a genuine Parrondo's paradox. We extend our work to a 3-coin initial state too with similar results. The implications of our work for observing quantum ratchet like behavior using quantum walks is also discussed.