Parrondoʼs game using a discrete-time quantum walk
Highlights
► Novel form of Parrondoʼs game on a single particle discrete-time quantum walk. ► Strategies for players to emerge as individual winners or as joint winners. ► General framework for controlling and using quantum walk with multiple coins. ► Strategies can be used in algorithms and situations involving directed motion.
Introduction
Game theory gives an account of how the involved parties decide strategies in their personal interest in a given situation, on rational grounds. It has found to have relevance to social sciences, biology, and economics [1], [2], [3], [4], [5], among others. By replacing classical probabilities with quantum amplitudes and allowing the players to employ superposition, entanglement and interference, quantum game theory has lead to new interesting effects and has become an active area of research. Quantum game can be quantified as any quantum system which can be manipulated by two or more parties according to their personal interest [6], [7].
The system in which we propose a quantum game in this Letter is a quantized version of the classical random walk (CRW), that is, the quantum walk system. Quantum walk (QW) evolution on a particle involves the quantum features of interference and superposition, resulting in the quadratically faster spread in position space in comparison to its classical counterpart, CRW [8], [9], [10], [11], [12]. QWs are studied in two forms: continuous-time QW (CTQW) [12] and discrete-time QW (DTQW) [10], [11], [13], [14] and are found to be very useful from the perspective of quantum algorithms [15], [16], [17], [18], [19]; to demonstrate coherent quantum control over atoms; quantum phase transition [20]; to explain phenomena such as the breakdown of an electric-field driven system [21] and direct experimental evidence for wavelike energy transfer within photosynthetic systems [22], [23]; to generate entanglement between spatially separated systems [24]; to induce dynamic localization in Bose–Einstein condensate in an optical lattice [25]. Experimental implementation of QWs has been made in an NMR system [26], [27], [28]; in the continuous tunneling of light fields through waveguide lattices [29]; in the phase space of trapped ions [30], [31]; with single optically trapped neutral atoms [32]; and with single photons [33], [34]. Various other schemes have been proposed for their realization in different physical systems [35], [36], [37].
In this Letter we present a quantum game, in the form of Parrondoʼs game, using a single-particle QW system. Parrondoʼs game involves two games, which when played individually, produce a losing expectation and when played in any alternating order, winning expectation is produced [38], [39], [40]. Parrondoʼs games were devised, originally, to provide a mechanism to harness Brownian motion and convert it to directed motion, or more generally, a Brownian motor, without the use of macroscopic forces or gradients [41]. They have since found applications in many areas. Their applications have also been made in the quantum regime. Groverʼs algorithm [42] has been analyzed in the context of quantum Parrondoʼs paradox [43]. A quantum implementation of a capital-dependent Parrondoʼs paradox, using an economical number of qubits, has been presented in [44]. Different forms of Parrondoʼs game using a QW with a space and time-dependent potential [45], (with noise [46]), using a QW with history dependence [47], using a QW as a source of randomness for randomly switched strategies [48] and using cooperative QW in which interaction between multiple participants replaces position-dependence as an opportunity for the Parrondo effect to occur [49], have been introduced earlier. We present a simple form of Parrondoʼs game using two players A and B with different quantum coins as quantum coin operators to evolve single-particle DTQW on a line. If the player A (B), using his coin evolves the DTQW to t steps such that the probability on the right (left) hand side of the origin () of the particle subjected to QW is greater than that on the left (right), the player A (B) is declared as winner. In the game form we present, the players A and B individually losing the game using their coins can develop a strategy to maintain equilibrium (equal probability on both the sides of the origin) and emerge as joint winners using their coins alternatively, or in combination for each step of the DTQW evolution. We also present a situation where player A (B) can have a winning probability more than a player B (A) and emerge as a solo winner. Recently, using the chirality distribution of the DTQW, a coin flipping game has been presented [50]. We believe that the manipulation of DTQW in the form of game using multiple players will give a general framework for application of QW, using multiple coins, to algorithms and various physical processes.
In Sections 2 Discrete-time quantum walk, 3 Parrondoʼs game, we present the standard form of DTQW and Parrondoʼs game. In Section 4, a game using DTQW, in the form of Parrondoʼs game, is presented. Different strategies for different situations is discussed in Section 5. In Section 6, we conclude by discussing the significance of game strategy in information theory and physical applications.
Section snippets
Discrete-time quantum walk
The DTQW in one-dimension is modelled as a system, that is, a two-level system (a qubit) in the Hilbert space , spanned by |0〉 and |1〉, and a K level position system, a position degree of freedom in the Hilbert space , spanned by , where , the set of integers. A t-step DTQW, with unit time required for each step of walk, is generated by iteratively applying a unitary operation W which acts on the Hilbert space : Here , is
Parrondoʼs game
Standard form of Parrondoʼs game involves games of chance. Two games, A and B, when played individually, produce a losing expectation. An apparently paradoxical situation arises when the two games are played in an alternating order, a winning expectation is produced [39], [38], [45], [40]. The apparent paradox that two losing games A and B can produce a winning outcome when played in an alternating sequence was devised by Parrondo as a pedagogical illustration of the Brownian ratchet [55].
Parrondoʼs game using DTQW
Here we present a novel scheme of a game in the setting of a DTQW, viz., one in which two players A and B use different quantum coin operators to walk on a single particle. Multiple coin DTQWs have been studied before [61] and the effect of various coin parameters has been studied by [62]. Both these features have been examined together in the form of Parrondoʼs game for the first time here, to the best of our knowledge. This could have impact on the studies of directed phenomena such as
Winning strategy
Here we discuss our simple scheme of the winning strategies of the game on a DTQW. The winning strategies are developed making use of the different coin operators. The parameter ξ, θ and ζ are physically realizable rotations on the two state system [51] and hence restrictions on the rotational degree of freedom lead to situations presented here. We believe this to be a new and simple way of implementing a game on a DTQW.
For the game presented in Section 4, players emerging as joint winners,
Conclusion
Superior performance of quantum strategies is usually seen, if entanglement is present. In the context of DTQW, this is natural as the walk evolves by entangling the coin and position degrees of freedom. Therefore, making use of the above, and the fact that the walk can be manipulated by varying the parameters in the quantum coin operation, we presented a scheme for a quantum game using DTQW in the form of Parrondoʼs game. Our system involves two players A and B as quantum coin operators to
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