On Quantum Walks and Iterated Quantum Games G. Abal, R. Donangelo, H. Fort Universidad de la República, Montevideo, Uruguay UFRJ, RJ, Brazil

0. THE MAIN IDEA Quantum Walks Quantum Games Random Walks Classical Games QUANTIZATION

1.QUANTUM WALKS AND QUANTUM GAMES Quantum walks (QWs) are expected to have potential for the development of new quantum algorithms. When two quantum walks are considered, the joint state of both walkers may be entangled in several ways this opens new possibilities for quantum information manipulation. Quantum walks have been realized using technologies ranging from NMR to linear optics. 1.1 Quantum Walks

Classical Game Theory constitutes a powerful tool for strategic analysis and optimization. Bipartite quantum games (QGs), in which players can resort to quantum operations, open new possibilities for information processing. It was shown that in QG, given a sufficient amount of entanglement, the players can achieve results not available to classical players 1.2 Quantum Games

2. FROM QW TO ITERATED QG The Hilbert space of a quantum walk on the line is composed of two parts, H = H x  H c. H x :  || x   integer x = 0, ±1,±2... associated to discrete positions on the line. H c is spanned by the 2 orthonormal kets {|0 , |1  }. The quantum walk with two walkers A,B takes place in a Hilbert space H AB = H A  H B. The evolution operator 2.1 Discrete-time QW on the Line

The well known case of a Hadamard walk is obtained if U c = H  H, where H is the one-qubit Hadamard gate defined by H|0> = (|0>+|1>)/ √2 and H|1> = (|0>−|1>) √2. Here we shall be concerned with more general coin operations U c which cannot be written as products of local operations. The conditional shift operation can be represented as

00  R=+1, R=+101  S=-2, T=+2 01  T=+2, S=-200  P=-1, P=-1 Let’s see how works  or: But, wait a minute, 2.2 QW as a QG exchanging 1 by C & 0 by D, this is eq. to a well known game: The PRISONER’S DILEMMA GAME ! R=1, S= -2, T= -S = 2 & P= - R = -1

T > R > P > S Silent Confess SHORT SENTENCE SHORT SENTENCE Prisoner’s Dilemma Game in Matricial Form FREE. LONG SENTENCE LONG SENT. FREE INTERM. SENTENCE INTERM. SENTENCE C C D D (R, R)(S, T) (T, S)(P, P) Non Optimal Situation !

Let’s specify a strategy by a 4-tuple : [p R, p S, p T, p P ] where p X is the conditional probability of cooperation of an agent after he got the payoff X in the previous round. Examples: [1/2, 1/2, 1/2, 1/2] = “RANDOM” [1, 0, 0, 1] = “win-stay, lose-shift” or PAVLOV Repeated games differ from “one-shot” games because the actions of the agents can produce retaliation or reward. Agents need a strategy (that is, a rule to update their behavior), and, some strategies favor cooperation. 2.3 Escape from the Prisoner’s Dilemma: Repeated Games

2.4 Implementation of Iterated QG 1. The coin states of the QW are interpreted as |0 >  C (cooperation) |1 >  D (defection) 2. Each agent can alter his/her own “coin” qubit by applying a unitary operation (a strategy) U A or U B in Hc  Hc 3. The position corresponds to the accumulated payoff. If X A is the position operator for Alice, X A | x A >= x A, X A | x A > her average payoff is = trace (  X A ) (idem for Bob). Consider 2 agents A (Alice) and B (Bob), players in an iterated QG. Connection with the QW is made by 3 simple rules :

The first qubit from the left is Alice’s and the second is Bob’s: |,  The possible strategies available to Alice are represented by the set of unitary 2-qubit operations that don’t alter the second qubit: The coeff. a i are expressed in terms of the conditional prob p X as: And similar expresions for Bob. p R +p T =p S +p P = 1

The joint coin operation is constructed as U C = U B · U A, assuming Alice moves 1st or U C = U A · U B, otherwise. For instance, the quantum version of Pavlov’s [1, 0, 0, 1], played by A may be implemented through an operator: If the 3 phases are chosen = 0, a CNOT operation results in which Bob’s coin is the control qubit: 00     01

2.4-A Example: Pavlov vs. Random Alice plays randomly and Bob responds with Pavlov. The operation transforms a product state into a maximally entangled (Bell) state. Schematic circuits representing the coin operation of a Pavlov vs. Random quantum game. Alice plays Pavlov and Bob plays random, an operation which disentangles a Bell state.

50 iterations, start |00> 2.4-B Pavlov vs. Random: Results

Let’s consider now the results for strategies that interpolate between Random and Pavlov. For both p R + p S = 1,  neglecting phases, each player’s strategy depends on a single real parameter: 3. PARAMETERIZED QUANTUM STRATEGIES For A: √p R = cos  and √p S = sin  For B:

Assuming Alice plays first, the joint coin operation is Alice’s (red) and Bob’s (blue) payoffs after 50 steps as a function of the strategic choice. The initial coin is the unbiased Bell state

Or more illuminating perhaps: Optimal situation for both players.

A connection between iterated bipartite quantum games and discrete-time quantum walk on the line was established. 4. CONCLUSIONS Examples of this: Pavlov ↔ CNOT Random ↔ Hadamard In particular, conditional strategies, depending on the previous state of both players, are naturally formulated within this scheme. As a by-product of this correspondence, popular strategies in Game Theory can be mapped into elementary quantum gates.

● An example of a QG in which both agents are allowed to choose a strategy that interpolates continuously between Pavlov and Random has been analyzed in detail using two unbiased initial coin states. ● Within this limited strategic choice, in the case of initial coin state (|00> + |11>)/√2 there is a Pareto optimal Nash equilibrium when Alice plays Pavlov,  =0, and Bob responds using  =  /20. ● In one-shot quantum games, the initial state must include a minimum amount of entanglement so that truly quantum features emerge. In the iterated QG based on the QW, entanglement is dynamically generated, so that entangled initial states are not a requirement.

● Obviously, this scheme for quantizing the iterated PD game also works for 2×2 games with arbitrary payoff matrix. There are several popular games that seem interesting to analyze within this framework. ● This connection introduces an entire new set of coins and shift operators that may be useful for quantum information processing tasks and opens the possibility to experimental tests using the facilities that are being developed for the QW.