Hybrid cluster state proposal for a quantum game

We propose an experimental implementation of a quantum game algorithm in a hybrid scheme combining the quantum circuit approach and the cluster state model. An economical cluster configuration is suggested to embody a quantum version of the Prisoners’ Dilemma. Our proposal is shown to be within the experimental state of the art and can be realized with existing technology.The effects of relevant experimental imperfections are also carefully examined.


Introduction
The study of the role played in a physical process by the quantumness of nature is a central issue for many branches of modern physics, leading to important and stimulating intellectual speculations [1]. We have witnessed the possibilities offered by quantum mechanics as an exploitable resource which allows the accomplishment of tasks that are prohibitively difficult, if not impossible, in the classical domain. This point of view is particularly interesting in quantum information processing (QIP) where intrinsic quantum features such as entanglement are seen as valuable 2 Institute of Physics ⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT tools in applications like quantum cryptography, communication and computation. QIP can also be used in order to show the quantum behaviour of nature and several studies along these lines have been performed. Non-classical states of electromagnetic fields [2], entangled states of internal and external degrees of freedom of trapped ions [3] and the entanglement between atoms and light fields [4] are outstanding examples of this research. Recently, there has been considerable interest in showing the quantum behaviour of macroscopic objects, paving the way towards a study of the boundary between classical and quantum physics [5].
It is of the utmost importance to design simple experiments implementing quantum algorithms from which the specific role of quantum resources can be gathered. In this paper, we propose a readily available experimental protocol for quantum generalizations of classical games. We address how to implement a quantum version of the two-player Prisoners'Dilemma as an example of the distinctiveness of the intervention of quantum mechanics. Through quantum preparations and strategies [6], the players achieve goals which classically were impossible [7]. Our proposal is based on the use of small multipartite entangled states, recently realized in all-optical setups [8,9]. The choice of an optical scenario for the implementation of our proposal is incomparably suitable in studying the influences of quantum entanglement to the game. Indeed, even if the quantum game in [6] has been implemented in a nuclear magnetic resonance (NMR) system [10], the density matrices of the highly mixed states involved in NMR can always be described as disentangled [11]: the observation of ensemble-averaged pseudo-pure states renders the observation of the effects of the entanglement ambiguous [11]. An all-optical implementation is not affected by this ambiguity. The multipartite entangled resource used in our proposal is given by cluster states [12] constructed through a double-pass scheme generating a four-photon entangled state via parametric down-conversion [8] (the information is encoded in orthogonal photonic polarizations). Cluster states are a subclass of more general graph states and represent the quantum resource used in one-way computation, where one-and two-qubit gates are simulated through a proper pattern of single-qubit measurements performed on a (sufficiently large) cluster state [12]. The one-way model has triggered significant theoretical and experimental interest, and schemes to improve the efficiency of linear optics computation have been proposed based on this model [13,14]. Cluster states naturally and economically simulate some operations which are central to our scheme [15,16]. We discuss how our proposal combines the standard quantum circuit model and cluster state-based QIP, enhancing the one-way model. This hybrid model makes our scheme for a quantum game within the current state of the art. It is also one of the first immediately realizable protocols for quantum algorithms designed for small cluster configurations (for a two-qubit quantum search algorithm, see [8]).
The rest of the paper is organized as follows. In section 2 we introduce the model for the game we play and the formal scheme to be used in order to describe it quantum mechanically. We introduce the cluster state configuration to be considered as a natural implementation of the Prisoners' Dilemma. We show that, by relying on a cluster state architecture, the entangling steps introduced in the quantized version of the game are automatically embedded in the cluster structure, representing a major experimental simplification in the realization of the algorithm. On the other hand, external local operations have to be introduced in order to realize the most relevant strategies in the game. Moreover, we investigate the freedom in spanning the payoff of the players offered by an extended six-qubit graph-state realization of the game, outlining that, with this configuration, the entire functionals can be explored. Section 3 provides an analysis of the role played by imperfect entangled resources in the performances of the game. We show that, by degrading the entanglement through imperfect strategic moves or a mixed entangled strategy, the Dilemma can no longer be reconciliated. This should highlight the role of the entanglement as an indispensable resource in this specific problem.

The model
Let us consider the players A and B involved in the classical Prisoners' Dilemma, which is a nonzero sum game. The strategy space of each player is S j = {c j , d j } (j = A, B). The game is non-cooperative and selfish, as the players aim to maximize their own payoff $ j (s), where s is the strategy profile s = (s A , s B ) and s j ∈ S j is the strategy chosen by player j = A, B [17]. We When both players carry out the same strategy, the payoff is equally shared. They obtain the is found to be a dominant profile. In fact, choosing d j and regardless of the strategy adopted by the adversary, player j maximizes his payoff. The profile (d A , d B ) has the property that neither player can improve their payoff by a unilateral change of strategy, making it a Nash equilibrium [7]. The rationality of the players and the non-cooperative nature of the game prevents A and B playing (c A , c B ) which is the Pareto optimum [7]: no player can increase their payoff (which is the CP), by changing strategy, without reducing the payoff of the adversary. The Dilemma is in the dichotomy between the best choice for both and the highest payoff available individually.
This Dilemma cannot be solved without some cooperativity. This is introduced in the quantum version of the game in [6], where the strategies which the players can use are embodied by a qubit whose states are |c = 1 0 and |d = 0 1 . Entangling stages P and M are introduced before and after the players perform their strategies. The strategy space is now and c j = U j (0, 0), d j = U j (π, 0). In [6,17], the choice of U A,B and its consequences on the performances of the game are discussed. The entanglement gives A and B a degree of cooperativity. If their strategy profile s = (U A , U B ) is such that this cooperativity is preserved, a reconciliation between CP and EP can occur. 1 We stress that the procedure in [6] is just one of the ways in which the game can be extended to the quantum realm. The choice in [6] implies that the payoffs associated with c j and d j will be the classical values and a Pareto optimal point is sought from the additional strategies provided by the quantum strategic space. In general, the less restrictive constraint imposed on the quantum version of a protocol is that it reproduces the classical process, in the proper limiting case. Here, this means that the description of the Prisoners' Dilemma when P and M are removed must match the classical one. At the same time, we look for a generalized game where the payoffs are affected by the entanglement so as to provide a Pareto optimal point lying within the strategy space of a separable game. The structure of the entangling steps is dictated by the interaction naturally realized by the setup considered. In our case, P and M must be related to the two-qubit gates simulated by a particular cluster configuration. In this respect, it is important to notice that a simple two-qubit cluster state results in the 'effective simulation' of a controlled π-phase gate (CP π ) [16]. Indeed, the construction of a two-qubit cluster state gives rise to a state which can be straightforwardly reinterpreted in terms of the application of a CP π gate onto a two-qubit register initially prepared in |+ 0 ⊗ |+ 0 with |± a = |c ± e ia |d . The same holds true when arbitrary information is encoded in the initial state of the register and the cluster is created. Throughout this paper, the use of the word 'simulation' has to be intended as the reinterpretation of the state of the register in terms of effective equivalent unitaries. The effective simulation of two-qubit gates in cluster states is the key advantage with respect to non-cluster-based standard quantum circuit schemes. Our proposal is able to naturally embody nearly the entire quantum steps P and M, which otherwise, have to be implemented by two independent two-qubit operations. Networking these operations to obtain the scheme in figure 1(a) is, in general, a difficult task. The use of a cluster state in our proposal represents a major simplification in this respect. In addition, in the same two-qubit cluster, the measurement of a qubit in the basis {|± a } simulates the application of R −a x H on a logical qubit, where R −a x is a rotation by an angle −a around the x-axis of the Bloch sphere and H the Hadamard gate [16]. The full quantum circuit we propose is shown in figure 1(a), with the part prior to the dotted box being simulated by the cluster in figure 1(b), as it is easy to see by concatenating the simulated unitary operations discussed above, according to the rules established in [16]. 2 The state corresponding to this box cluster, introduced by Walther et al in [8], can be put into the form where |ghz = (1/ √ 2)(|ccc + |ddd ) 234 is a GHZ state of qubits 2, 3 and 4. We now exploit the naturally simulated CP π gate and H P j 's (implicit in the preparation of a cluster state [12]) to obtain P = CP π (H P A ⊗H P B ). The H P j 's allow us to generate a maximally entangled strategic state and to combine superpositions of orthogonal strategies and entanglement [19]. Despite the Quantitatively, we need to calculate the expression which gives the probability that the evolved strategy profile, after the operations by the players, is s = (ξ A , χ B ). With equation (3) it is easy to evaluate with the numerical coefficients taken according to the payoffs associated with the strategy profiles in the classical game. Each payoff is a functional depending on equation (1). The results are shown in figures 2(a) and (b). The strategic sector [c j , d j ] ([q j , c j )) corresponds to φ j = 0 (θ j = 0) with θ j ∈ [0, π] (φ j ∈ [0, π/2]). We take this parametrization as it reveals the relevant features of the game. From figure 2(a), we see that for B choosing d B or q B , the best strategy by A is d A with payoffs $ A (d A , d B ) = 3 or $ A (d A , q B ) = 5 respectively. Analogous considerations can be made mutatis mutandis about $ B (U A , U B ) ( figure 2(b)). It can be seen that the profile (d A , d B ) is the only Nash equilibrium. The players' payoff for this profile is exactly the CP, which shows that (d A , d B ) is Pareto optimal. This result is quantum mechanical, as the payoff corresponding to (d A , d B ), in a game without P and M, is EP. In the separable quantum game resulting fromthe removal of CP π 's in P and M (keeping H

P,M
A,B ), no reconciliation is attained. In addition, the game played by preparing the initial strategy as |c A ⊗ |c B and excluding the H

P,M
A,B (so that CP π 's in P and M are ineffective) turns out to be identical to the classical one. Thus, in this way we see that, by keeping the freedom of spanning a continuous set of local strategies, the separable version of this game is unable to attain the Pareto optimality of the dominant strategy. In turn, this implies that entanglement is genuinely necessary for the reconciliation of the Dilemma. Moreover, later we show that the Pareto optimality cannot be attained by using classical correlations shared between the players of the game, suggesting that the entanglement provided to the players favours the reconciliation of the Dilemma. While the procedure in [6] introduces a new strategy profile which is a Nash equilibrium and achieves CP, in our scheme the equilibrium strategy is the same as in the 6 Institute of Physics ⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT  .
non-entangled game. The entanglement renders (d A , d B ) the profile preserving the cooperativity introduced by P.
Parts of the game are naturally implemented by a box cluster, but the strategies U A,B must be simulated by an appropriate measurement pattern. As shown in figures 1(a) and (b), by measuring qubits 1 and 4 we can simulate just a rotation around the x axis of the single-qubit Bloch sphere [16]. Thus, we need more freedom for the players to perform their strategies. For this task, we exploit the fact that H M A,B and CP π belong to the Clifford group. We consider the operations A, B ∈ {σ x,y,z , R µ x } in the dotted box of figure 1(a), which can be imported to the dashed section of the circuit. They are seen as operations on the qubits 2 and 3 of the box cluster applied before their final measurement. Together with R −a,−b x simulated by the measurement of 1 and 4, these enlarge the strategy space of the players. In table 1, we show the measurement angles a and b and the corresponding A, B. Only two measurement bases are needed and 1 1 or σ x must be imported before the measurements are performed.
We remark that the use of local operations on the logical output qubits of a cluster is inherent to the one-way model [12]. The randomness of the measurement outcomes affects a gate simulation which has to be corrected by local decoding operators. Here, we are implicitly assuming the postselection of those events corresponding to the projection of qubits 1 and 4 onto |+ a,b 1,4 . In this case, the decoding operators are 1 1 2,3 . A and B may be seen as decoding operators selected not by the measurement outcomes but by the task to perform. The hybrid nature of our approach should be clear: we cannot rely just on the measurement-based gate simulations because we need additional rotations of the logical output qubits. Here, A and B can be easily realized in the all-optical setups in [8]. Indeed, by exploiting that H M j σ x,j = σ z,j H M j , the players only need to apply σ z to the output qubits, which is possible via phase shifters, before they are measured in the σ x eigenbasis. In this way, all the strategies in table 1 can be attained, which is sufficient to experimentally study the Pareto-optimality of the Nash equilibrium point (figure 3(a) graphically shows these strategies). However, this does not exhaust all our possibilities. For instance, the entire quadrant [q A , c A ] × [q B , c B ] can be sampled simply by taking a = b = 0, importing A = R µ x , B = R ν x and scanning the angles µ, ν. Single-qubit manipulations through linear-optical elements just prior to the detection stage make our scheme feasible [8].
However, we cannot sample the entire payoff $ A,B with the box cluster because it is not possible to obtain R θ j x ≡ U j (θ j , 0). For a complete tomography of $ A,B , the price to pay is the use of a larger number of qubits. Indeed, using the concatenation technique [16] and an analysis similar to the one relative to the box cluster, it can be seen that the wafer configuration in figure 1(c) can fully embody this quantum game. In addition, the rotation R θ y can be realized by choosing α = β = π/2, γ = θ A , δ = θ B and importing A = B = R π/2 x , which correspond to a phase shift R π/2 z applied to 3 and 4 before measuring in the σ x eigenbasis. The wafer configuration is realized by combining two four-photon entangled states [9] using the technique suggested in [13] and realized by Zhang et al in [8].

Effects of imperfections
We now address the effect of realistic imperfections in this game. Non-idealities come from errors introduced at the measurements. The waveplates in front of the photodetectors used to measure the state of the cluster qubits may introduce unwanted rotations of a polarization state, leading to wrong measurement bases. In addition, imperfections at the down-conversion stage in generating a box cluster provide mixed entangled states to the players. Both these sources of error can be formally considered by the replacement θ j → θ j + j in equation (1) (analogously for φ j ) and averaging the payoffs over appropriate probability distributions, with standard deviation σ j , attached to j 's. This randomness results in a corrupted mixed entangled resource [20] whose degree of entanglement diminishes if σ is increased. In figure 3(b) we show the differences between the ideal (Pareto optimal) $ A (d A , d B ) and the average payoff £ A (d A , d B ) obtained when j 's are normally distributed around 0. The result is not affected by fluctuations in φ j as d j does not depend on this parameter. At σ 0.9 the degree of entanglement (quantified by the measure based on the Peres-Horodecki criterion [21]) is 0.01. The larger the fluctuations allowed for j , the larger the deviation of the corresponding payoff from the behaviours in figure 2. The effect of classical correlations can also be studied by considering the mixed initial state to enter P, resulting in a non-ideal entangled mixed resource which A and B use to play the described game. For x 0.29, the mixed state equation (5) is separable so that A and B only share classical correlations. In this case, by running the algorithm using ρ corrupt and considering the evolution of the strategy profile which, in the ideal situation of x = 0 solved the Dilemma, it is easy to find that CP > $ x 0.29 A,B (d A , d B ) > EP. Also, no other Pareto optimal points arise as a result of different strategy profiles. This provides an operative way to study how the Pareto optimality is lost when corrupted resources and imperfect measurements are used in our scheme. It represents a useful practical and theoretical tool in studying the performances of the quantum game.

Remarks
We have proposed an experimental implementation of the quantum Prisoners' Dilemma through an economical and experimentally realizable cluster state configuration. At the same time, we have shown that the cluster model can be complemented by simple rotations of the logical output qubits to add freedom to the gate simulation, building a hybrid model which can be realized with existing technology. This allows for an immediate experimental investigation of the role of entanglement in the search for a Pareto optimal Nash equilibrium point in a system exhibiting quantum correlations.