We present a quantum implementation of Parrondo's game with randomly switched strategies using (1) a quantum walk as a source of ‘randomness’ and (2) a completely positive (CP) map as a randomized evolution. The game exhibits the same paradox as in the classical setting where a combination of two losing strategies might result in a winning strategy. We show that the CP-map scheme leads to significantly lower net gain than the quantum-walk scheme.
We present a quantum implementation of Parrondo's game with randomly switched strategies using (1) a quantum walk as a source of ‘randomness’ and (2) a completely positive (CP) map as a randomized evolution. The game exhibits the same paradox as in the classical setting where a combination of two losing strategies might result in a winning strategy. We show that the CP-map scheme leads to significantly lower net gain than the quantum-walk scheme.
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.
We construct a Parrondo's game using discrete-time quantum walks (DTQWs). Two losing games are represented by two different coin operators. By mixing the two coin operators UA(αA, βA, γA) and UB(αB, βB, γB), we may win the game. Here, we mix the two games in position instead of time. With a number of selections of the parameters, we can win the game with sequences ABB, ABBB, etc. If we set βA = 45°, γA = 0, αB = 0, βB = 88°, we find game 1 with [Formula: see text], [Formula: see text] will win and get the most profit. If we set αA = 0, βA = 45°, αB = 0, βB = 88° and game 2 with [Formula: see text], [Formula: see text] will win most. Game 1 is equivalent to game 2 with changes in sequences and steps. But at large enough steps, the game will lose at last. Parrondo's paradox does not exist in classical situation with our model.
Parrondo's paradox is ubiquitous in games, ratchets and random walks.The apparent paradox, devised by J. M. R. Parrondo, that two losing games A and B 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.
Game A + Game B = Game C. Parrondo's games follow this basic structure where A and B are losing games and C is a winning game-a phenomenon called Parrondo's paradox. These games can take on a wider class of definitions and exhibit these paradoxical results. In this paper, we show three paradoxical cases. (i) The successive "tossing" of a single fair quantum coin gives a biased result, a previously known result. (ii) The random tossing of two quantum coins, each with successive biased expectations, gives an average random walk position of approximately zero. (iii) The sequential periodic tossing of two quantum coins, each with successive negative biased expectations, gives an average random walk with positive expectation. Using these results, we then propose a protocol for identifying and classifying quantum operations that span the same Hilbert space for a two-level quantum system.
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.
Abstract The Parrondo effect is a well‐known apparent paradox where a combination of biased random walks displays a counterintuitive reversal in direction. These random walks can be expressed in terms of classical coin tossing games, leading to the surprising result that a combination of losing games can result in a winning game. There is now a large body of literature on quantum walks theoretically analyzing the quantum version of this effect, but to date, there have been no experimental observations of quantum Parrondo walks. Here, the first experimental verification of a quantum Parrondo walk within a quantum optics scenario is demonstrated. Based on the compact large‐scale experimental quantum‐walk platform, two rotation operators are implemented to realize the quantum Parrondo effect. The effect of quantum coherence in a quantum Parrondo walk is also investigated based on a delayed‐choice scheme that cannot be realized with classical light. It is demonstrated that the Parrondo effect vanishes when the quantum walk has a completely decoherent initial state in a delayed‐choice setting. Quantum walks are fundamental to multiple quantum algorithms, and this research provides motivation to expand the results to further explore quantum Parrondo walks.