Quantum game theory and the complexity of approximating quantum Nash equilibria
QQuantum game theory and the complexity ofapproximating quantum Nash equilibria
John Bostanci John Watrous
Institute for Quantum Computing and School of Computer ScienceUniversity of WaterlooWaterloo, Ontario, Canada
January 31, 2021
Abstract
This paper is concerned with complexity theoretic aspects of a general formulationof quantum game theory that models strategic interactions among rational agents thatprocess and exchange quantum information. In particular, we prove that the com-putational problem of finding an approximate Nash equilibrium in a broad class ofquantum games is, like the analogous problem for classical games, included in (andtherefore complete for) the complexity class PPAD. Our main technical contribution,which facilitates this inclusion, is an extension of prior methods in computationalgame theory to strategy spaces that are characterized by semidefinite programs.
1. Introduction
Game theory is a fascinating topic of study with connections to computer science, eco-nomics, and the social sciences, among other subjects. This paper focuses on complexitytheoretic aspects of game theory within the context of quantum information and compu-tation.Quantum game theory began with the work of David Meyer [Mey99] and Jens Eis-ert, Martin Wilkens, and Maciej Lewenstein [EWL99] in 1999. These works investigatedgames involving quantum information, highlighting examples in which quantum players Some authors argue that the origins of quantum game theory go back further. Here, however, we arereferring to the specific line of work that self-identifies as being concerned with a quantum informationtheoretic variant of game theory in the tradition of von Neumann and Morgenstern [vNM53] and Nash[Nas50a, Nas50b], as opposed to quantum information and computation research that can be associatedwith game theory as a broad umbrella term. a r X i v : . [ c s . CC ] J a n ave advantages over classical players. Many other examples of quantum games, primar-ily based on the frameworks proposed by Meyer and Eisert, Wilkens, and Lewenstein,were subsequently analyzed. (See, for instance, the survey [GZK08] for summaries andreferences.)Aspects of this line of work have been criticized for multiple reasons. A common pointof criticism of many (but certainly not all) quantum game theory papers is their poorlymotivated notion of classical behavior. In particular, classical players in quantum gametheory papers are often limited to coherent permutations of standard basis states, or simi-larly restricted classes of unitary operations, while quantum players have access to a lessrestricted set of unitary operations, possibly all of them. This notion of classicality, whichis a key ingredient in the original examples of Meyer and Eisert, Wilkens, and Lewen-stein, essentially invites exploitation by quantum players. A more standard interpretationof classical behavior in quantum information theory assumes the complete decoherenceof any quantum system a classical player manipulates.Another point of criticism, raised by van Enk and Pike [vEP02], is that comparingquantum games with their classical namesakes within the specific frameworks typicallyadopted by quantum game theory papers is akin to comparing apples with oranges. Al-though one may argue that these games offer faithful representations of classical gameswhen players’ actions are restricted to permutations of standard basis states, their quan-tum reformulations are, simply put, different games. It is therefore not surprising that lessrestricted quantum players may find advantages, leading to new Nash equilibria and soon. However, although it was not their primary focus, Meyer and Eisert, Wilkens, andLewenstein did both clearly suggest more general definitions of quantum games in whicha wide range of interactions could be considered, including ones in which the criticismsjust raised no longer have relevance. In particular, Meyer mentions a convex form of hismodel of quantum games, in which classical players could be modeled by completelydecohered operations. And, Eisert, Wilkens, and Lewenstein, in a footnote of their paper,describe a model in which players’ actions correspond not just to unitary operations, butto arbitrary quantum channels (as modeled by completely positive and trace preservinglinear maps). In either case, more general strategic interactions may be considered, andone need not restrict their attention to analogues of classical games or in identifying a“quantum advantage.”For example, quantum interactive proof systems of various sorts, as well as manyquantum cryptographic scenarios and primitives, can be viewed as quantum games. An-other example is quantum communication, which can be modeled as a game in whichone player attempts to transmit a quantum state to another, while a third player repre-senting an adversarial noise model attempts to disrupt the transmission. We do not offerany specific suggestions in this paper, but it is not unreasonable to imagine that quantumgames having social or economic applications could be discovered.We will now summarize the definition of quantum games we adopt, beginning withthe comparatively simple non-interactive setting and then moving on to the more general2nteractive setting. For the sake of simplicity and exposition in this introduction, we willrestrict our attention to games in which there are just two players: Alice and Bob. Thedefinitions are easily extended to any finite number of players, as is done in the maintext. Non-interactive quantum games
In a (two-player) non-interactive quantum game , the players Alice and Bob each hold aquantum system, represented by a register of a predetermined size: Alice holds X andBob holds Y . They must each independently prepare in the register they hold a quantumstate: Alice prepares a quantum state represented by a density operator ρ and Bob pre-pares a state represented by σ . Just like in the standard non-cooperative setting of classicalgame theory, Alice and Bob are assumed to be unable to correlate their state preparationswith one another. The registers X and Y are sent to a referee, who performs a joint mea-surement on the pair ( X , Y ) . Here, when we refer to a measurement, we mean a generalquantum measurement, often called a POVM (positive operator valued measure), havingany finite and non-empty set of measurement outcomes. The outcome of the measure-ment is assumed to determine a real number payoff for each player. We note explicitlythat Jinshan Wu [Wu04a, Wu04b] has proposed and analyzed an equivalent definition ofnon-interactive quantum games to this one.In order to formally describe a non-interactive quantum game, one must specify thereferee’s measurement together with the payoff functions for each player. As will be ex-plained later, when it suffices to specify each player’s expected payoff, given any choice ofstates the players may select, the referee may be described by a collection of Hermitianmatrices, one for each player.One may observe that the standard notion of a classical game in normal form is easilyrepresented within this framework by defining the referee so that it first measures theregisters X and Y with respect to the standard bases of the associated spaces, and thenassigns payoffs in a completely classical manner.A non-interactive quantum game can, up to a discretization, also be viewed as a classi-cal game, where the players send the referee classical descriptions of their chosen densityoperators and the referee performs the required calculation to determine their payoffs, butthe normal form description of this new classical game will, naturally, be exponentiallylarger than the description of the original quantum game. Interactive quantum games
Quantum games in which players can process and exchange quantum information witha referee over the course of multiple rounds of interaction may also be considered. It is interesting to consider meaningful ways in which this assumption may be relaxed or dropped, butthe simplest and most direct quantum extension of classical game theory beings with this assumption ofindependence. lice’sturn 1 Alice’sturn 2Bob’sturn 1 Bob’sturn 2Referee’spreparation Referee’sintermediateaction Referee’sfinalmeasurementAlice’smemoryBob’smemory Figure 1: An illustration of a game between Alice and Bob, run by a referee, in whichAlice and Bob each receive and transmit quantum information twice, potentially keepinga quantum memory between the two turns. Each arrow represents a quantum register,which could be of any fixed size (including the possibility of trivial registers, which areequivalent to nothing being transmitted). Games involving any finite number of roundsof interaction may be considered.For example, the referee could prepare registers X and Y in a joint quantum state, send X to Alice and Y to Bob, allowing them to transform these registers as they choose, andthen measure the pair ( X , Y ) upon receiving them back from Alice and Bob. In such agame, Alice and Bob therefore each play a quantum channel, with their payoffs againbeing determined by the outcome of the referee’s measurement. The framework of Eisert,Wilkens, and Lewenstein falls into this category, provided that the players are permittedto play channels and not just unitary operations.More generally, an interactive quantum game may involve an interaction between thereferee and the players over the course of multiple rounds, as suggested by Figure 1.In this setting it is natural to assume that Alice and Bob each have their own privatequantum memory, which they utilize if it is to their advantage.The actions of the players in such a game may be represented through the frameworkalternatively known as the quantum strategies framework [GW07] and the quantum combs framework [CDP08, CDP09]. This framework, which will be described in greater detail inthe next section, allows the actions of any one player over the course of multiple roundsof interaction, accounting for the possibility of a quantum memory, to be faithfully rep-resented by a single positive semidefinite matrix that satisfies a finite collection of affinelinear constraints. Thus, the sets of strategies available to the players are convex and com-pact, and one may efficiently optimize real-valued linear functions defined on these setsthrough the paradigm of semidefinite programming.4imilar to non-interactive quantum games, interactive quantum games are formallyexpressed by specifying the referee’s actions, including state preparations, channels, andmeasurements, along with payoff functions of the possible measurement outcomes corre-sponding to each player. Once again, when it is sufficient to describe the expected payofffor each player, given a specification of their strategies, a referee in an interactive quan-tum game may be specified by a list of Hermitian matrices, one for each player, as will beexplained. Our results and contributions
We prove, as our main technical result, that the problem of computing an approximateNash equilibrium in any interactive game of the form described above, given an explicitmatrix representation of the referee, is contained in the complexity class PPAD. As thisproblem includes non-interactive classical games as a special case, it follows that thisproblem is complete for PPAD [DGP09, CDT09].There is a sense in which this result is not unexpected; prior work on the complex-ity of computing approximate Nash equilibria, and more generally on the complexity ofcomputing fixed points of different classes of continuous maps, suggests that approx-imations of Nash equilibria in a wide variety of games should be contained in PPAD[Pap94, DGP09, CDT09, EY10]. The principal challenge that arises in the setting of in-teractive quantum games is that, although one may efficiently optimize over individualplayer’s strategies through semidefinite programming, closed form expressions of theseoptimizations are not known to exist.To confront this challenge, we consider a fairly general convex form of Nash’s no-tion of a gain function —and then we fight fire with fire, so to speak, using semidefiniteprogramming to approximate continuous functions that arise through this formulation.Possibly our methodology for handling this issue will be of independent interest.Although it is not an essential aspect of our proof, we also make use of the elegantnotion of a discrete Wigner representation of a quantum state, which is convenient withinthe proof. Although discrete Wigner representations have been investigated in the theoryof quantum information (see, for instance, [GHW04] and [Gro06]), we are not aware thatthey have been used previously in quantum complexity theory, and we feel that they offera convenient tool that might be useful in other contexts.Beyond this main technical result, we hope that this paper may serve as a suggestionthat quantum game theory is worthy of a second look. We believe that the general defi-nition of quantum games we have reiterated is well motivated by the theory of quantuminformation, and can provide a basic foundation through which quantum game theoryand its complexity theoretic aspects may be investigated. In the conclusion of this paperwe mention several open problems and research directions concerning quantum gametheory that may be of interest. 5 . Technical preliminaries
This section summarizes technical concepts required later in the paper. The first threesubsection that follow discuss aspects of the barycentric subdivision of a simplex, the dis-crete Wigner representation of quantum states, and the quantum strategies/combs framework ,respectively. In the last subsection we present the standard definition of the complexityclass PPAD and reference a theorem of Etessami and Yannakakis [EY10] that establishesthe containment of a certain computational fixed-point problem in PPAD, to which wewill reduce the problem of finding an approximate Nash equilibrium of a quantum game.It will be assumed throughout the paper that the reader is familiar with basic notionsof computational complexity [AB09] and quantum information [NC00, Wil17, Wat18].Hereafter we will take Σ = {
0, 1 } to denote the binary alphabet. Suppose that a positive integer n is given, and consider a simplex in an n -dimensionalspace having vertices { u , . . . , u n } . The barycentric subdivision of such a simplex is a divi-sion of it into n ! simplices in the following way.First, with each nonempty subset A ⊆ {
1, . . . , n } we define a point v A = | A | ∑ k ∈ A u k , (1)which is the uniform convex combination, or barycenter , of the vertices of the originalsimplex labeled by elements of A . For example, v { k } = u k for each k ∈ {
1, . . . , n } , while v { n } is the true barycenter of the original simplex. Figure 2 illustrates the barycentricsubdivision for a simplex when n = v A and v B if and only if A ⊂ B or B ⊂ A (propercontainments), we obtain a division of the original simplex into n ! new simplices, one foreach possible chain A ⊂ · · · ⊂ A n (2)of subsets of {
1, . . . , n } . There are n ! such chains and they may be placed in correspon-dence with the symmetric group S n . To be precise, for any fixed ordering ( k , . . . , k n ) ofthe set {
1, . . . , n } , we associate the chain (2) with the permutation π ∈ S n satisfying A j = { k π ( ) , . . . , k π ( j ) } (3)for every j ∈ {
1, . . . , n } . Thus, the simplices in the subdivision are identified with ele-ments of S n .The barycentric subdivision may naturally be applied iteratively within the simplicesconstructed by the subdivision. For example, Figure 3 illustrates the barycentric subdi-vision of just the shaded simplex illustrated in Figure 2. Hereafter we shall assume that6
1, 2, 3 ) (
1, 3, 2 )(
2, 1, 3 ) (
3, 1, 2 )(
2, 3, 1 ) (
3, 2, 1 ) v { } = u v { } = u v { } = u v { } v { } v { } v { } Figure 2: The barycentric subdivision of a simplex with three vertices u , u , and u . Theshaded region indicates one of the 3! = { v { } , v { } , v { } } , which is naturally identified with the identitypermutation π = (
1, 2, 3 ) .the initial simplex is the standard simplex ∆ n , so that u , . . . , u n are elementary unit vec-tors (or, equivalently, standard basis vectors). With this assumption in place, we define asequence of finite subsets of the standard unit simplex B n ⊂ B n ⊂ B n ⊂ · · · ⊂ ∆ n ; (4) B n contains the n vertices of the simplex ∆ n , B n contains the 2 n − {
1, . . . , n } when the barycentric subdivision has been applieda single time, and in general B rn denotes the set of vertices after the r -th level subdivisionhas been performed.For any function f : ∆ n → ∆ n , and any nonnegative integer r , the r-th level barycentricapproximation to f is the function g r : ∆ n → ∆ n defined in the following way. Each point v ∈ ∆ n may be expressed uniquely as a convex combination v = λ v + · · · + λ m v m (5)of distinct vertices v , . . . , v m ∈ B rn , all contained in the same r -th level simplex (whichtherefore implies m ≤ n ). One then defines g r ( v ) = λ f ( v ) + · · · + λ m f ( v m ) . (6)Various computations involving the r -th level barycentric subdivision may be per-formed efficiently and exactly through rational number computations for r being polyno-mial in the size of the problem being considered, but we will not have a need to explicitly7igure 3: The barycentric subdivision applied to the simplex shaded gray in Figure 2.refer to these computations. We will make use of the well known fact that any two points u and v contained in the same simplex constructed at the r -th level of the barycentricsubdivision must satisfy (cid:107) u − v (cid:107) ≤ (cid:16) − n + (cid:17) r , (7)with the norm being the Euclidean norm on R n . A proof of this fact may be found inmany texts on algebraic topology, including in [Bre93] where it appears as Lemma 17.3. Throughout this subsection we will take n to be an odd positive integer, and let us renamethe elements of the standard basis {| (cid:105) , . . . , | n (cid:105)} as {| a (cid:105) : a ∈ Z n } by taking each indexmodulo n . In general, we shall interpret expressions inside of bras and kets as referringto modulo n arithmetic.It is helpful to begin with the definition of the discrete Weyl operators . First define X = ∑ a ∈ Z n | a + (cid:105)(cid:104) a | and Z = ∑ a ∈ Z n ω an | a (cid:105)(cid:104) a | (8)for ω n = exp ( π i / n ) denoting the first principal n -th root of unity. The discrete Weyloperators { W a , b : a , b ∈ Z n } ⊂ U ( C n ) , (9)as we will define them, are then given by W a , b = X a Z b (10)for every a , b ∈ Z n . The discrete Weyl operators form an orthogonal basis for the vectorspace L ( C n ) .Next, define an operator T ∈ L ( C n ) as T = ∑ a ∈ Z n |− a (cid:105)(cid:104) a | . (11)8or example, in dimension n = T = . (12)We then define V a , b = W a , b TW ∗ a , b (13)for every a , b ∈ Z n , and consider the collection (cid:8) V a , b : a , b ∈ Z n (cid:9) . (14)One may observe that T is unitary, Hermitian, and, by the assumption that n is odd, hasunit trace, and therefore the same is true for every operator in the collection (14).Next let us verify that the collection (14) is orthogonal, with respect to the usual(Hilbert–Schmidt) inner product on operators. For any choice of a , b , c , d ∈ Z n , one mayverify directly that (cid:10) V a , b , V c , d (cid:11) = (cid:10) T , V c − a , d − b (cid:11) . (15)Also observe the following expression for the diagonal entries of the operator TV a , b : (cid:10) c (cid:12)(cid:12) TV a , b (cid:12)(cid:12) c (cid:11) = (cid:10) c (cid:12)(cid:12) TW a , b TW ∗ a , b (cid:12)(cid:12) c (cid:11) = (cid:40) a (cid:54) = ω − bcn a =
0. (16)Noting the expression ∑ c ∈ Z n ω − bcn = (cid:40) n b = b (cid:54) =
0, (17)where again we have used the assumption that n is odd, we conclude that (cid:10) T , V a , b (cid:11) = (cid:40) n ( a , b ) = (
0, 0 ) ( a , b ) (cid:54) = (
0, 0 ) . (18)The collection (14) is therefore orthogonal.At this point we have no further need to refer to modulo n arithmetic, so let us assumethat the elements of the collection (14) have been renamed as { V , . . . , V n } , with respectto any sensible way of doing this. The key property of this collection is that each V k isunitary, Hermitian, and has trace equal to 1, and that the collection is orthogonal.Finally, define an affine linear map of the form ψ : Herm ( C n ) → R n as ψ ( H ) = n ( n + ) n ∑ k = ( (cid:104) V k , H (cid:105) + ) | k (cid:105) (19)9 Φ Φ Z Z Y Y Y X X X Figure 4: The actions of an agent, or a strategy , in a three-round interaction may be de-scribed by a network of three channels. Time goes from left to right: first the register X isreceived and fed into the channel Φ , which produces Y and Z as output, with Y beingsent to another agent and Z representing a memory register that is retained by the agentbeing described. Then X is received, both X and the memory register Z are fed into thesecond channel Φ , and so on.for every H ∈ Herm ( C n ) . The inverse of this mapping is given by ψ − ( v ) = ( n + ) n ∑ k = v k V k − n (20)for every v ∈ R n . This mapping defines a discrete Wigner representation of quantum states;each density operator ρ ∈ D ( C n ) is represented by the vector v = ψ ( ρ ) ∈ ∆ n . (21)The inclusion of the vector v = ψ ( ρ ) in the unit simplex follows from two observations,the first being that Tr ( ψ − ( v )) = v + · · · + v n =
1, and the secondbeing that (cid:104) V k , ρ (cid:105) ∈ [ −
1, 1 ] by virtue of the fact that V k is unitary and Hermitian, for each k ∈ {
1, . . . , n } .It should be noted that, except in the trivial case n =
1, the inclusion ψ ( D ( C n )) ⊂ ∆ n is proper; only a subset of the vectors in the standard simplex represent a valid densityoperator, others represent unit-trace Hermitian operators having negative eigenvalues. We now summarize the aspects of the quantum strategies/combs framework [GW07,CDP08, CDP09], hereafter be referred to as the quantum strategies framework in this pa-per, that are required for our main result. This framework provides a convenient way ofdescribing and characterizing the actions of agents that interact and exchange quantuminformation with one another over the course of multiple rounds.Consider an agent that engages in an interaction involving the exchange of quantuminformation with one or more other agents. Let us suppose, in particular, that the agentbeing considered first receives a register X , then sends a register Y , then receives X ,then sends Y , and so on, with its role in the hypothetical interaction concluding after10t receives X r and then sends Y r . It is to be assumed that the agent may store quantuminformation between the rounds of interaction. Figure 4 depicts the actions of an agent ofthis sort in the case r =
3. Hereafter we will refer to a network of this form as a strategy for the agent being described.In the quantum strategies framework, strategies of this sort are represented by the
Choi representation of the network. To be more precise, the network is considered as asingle quantum channel Φ ∈ C ( X ⊗ · · · ⊗ X r , Y ⊗ · · · ⊗ Y r ) , (22)with ( X , . . . , X r ) collectively forming the input to this channel and ( Y , . . . , Y r ) formingthe output, and the Choi representation J ( Φ ) is taken as a representation of the strategy.In general, the Choi representation of a channel taking the form Φ ∈ C ( X , Y ) is given by J ( Φ ) = ∑ a , b Φ (cid:0) | a (cid:105)(cid:104) b | (cid:1) ⊗ | a (cid:105)(cid:104) b | , (23)where a and b range over all classical states (or, equivalently, standard basis elements) ofthe input space X , and therefore for Φ taking the form (22) we have J ( Φ ) ∈ L ( Y ⊗ · · · ⊗ Y r ⊗ X ⊗ · · · ⊗ X r ) . (24)Not every channel of the form (22) can be obtained by composing channels Φ , . . . , Φ r in the manner just described; a given channel might not respect the “time ordering” inwhich each register Y k is produced prior to the registers X k + , . . . , X r being received. Anecessary and sufficient condition for a channel of the form (22) to decompose into anetwork of channels Φ , . . . , Φ r is that its Choi representation is positive semidefinite, J ( Φ ) ∈ Pos ( Y ⊗ · · · ⊗ Y r ⊗ X ⊗ · · · ⊗ X r ) , (25)and satisfies a collection of affine linear constraints:Tr Y r (cid:0) J ( Φ ) (cid:1) = X r − ⊗ X r Tr Y r − ( X r − ) = X r − ⊗ X r − ...Tr Y ( X ) = X ⊗ X Tr Y ( X ) = X (26)for X , . . . , X r − being operators having sizes required by the equalities. By the assump-tion that J ( Φ ) is positive semidefinite, the operators X , . . . , X r − (if they exist) must bepositive semidefinite, and so we may write X r − ∈ Pos ( Y ⊗ · · · ⊗ Y r − ⊗ X ⊗ · · · ⊗ X r − ) ... X ∈ Pos ( Y ⊗ X ) . (27)11 Φ Φ Ψ Ψ Ψ Ψ Z Z Y Y Y X X X W W W Figure 5: A strategy of the form depicted in Figure 4 may be interfaced with the actionsof one or more other agents. In the interaction pictured, the second agent produces ameasurement outcome at the conclusion of the interaction.(The operators X r − , . . . , X happen to be the Choi representations of the strategies ob-tained by “truncating” the strategy described by the channels Φ , . . . , Φ r , assuming thefinal memory register is tracing out in each case.) It may be noted that, in the case r = player and the new agent will be calledthe referee . Through the quantum strategies framework, for each possible outcome a thatmay be produced by the referee’s measurement, one may compute a positive semidefiniteoperator P a ∈ Pos ( Y ⊗ · · · ⊗ Y r ⊗ X ⊗ · · · ⊗ X r ) (28)with the property that, when the referee and player interact, the probability for each mea-surement outcome to appear is given byPr ( measurement outcome equals a ) = (cid:10) P a , Q (cid:11) , (29)for Q = J ( Φ ) being the representation of the player’s strategy. It is not necessary forthe purposes of this paper to explain precisely how each operator P a is obtained, exceptto say that this operator may be computed efficiently given descriptions of the channels Φ , . . . , Φ r + and the final measurement. Finally, the quantum strategies framework extends to interactions involving multipleagents in a fairly straightforward way. In the context of quantum games, we are interestedin interactions in which a referee, who produces a final measurement outcome at the con-clusion of the interaction, interacts not just with a single player, but with multiple players. The process for obtaining these operators is again based on the Choi representation, but one mustaccount for the measurement, the spaces must be ordered in a way that matches with the representation J ( Ξ ) , and an entry-wise complex conjugation is required to ensure that the expression (cid:10) P a , Q (cid:11) correctlyrepresents the probability associated with the outcome a . Ψ , . . . , Ψ along with the box suggesting a measurement, interacts with two players,each designated by a superscript 1 or 2 on their respective channels and the registers theytouch. In general, such an interaction may involve any number of players m . Followingthe standard assumption in non-cooperative game theory, the m players are assumed tonot directly interact with one another: all interactions are between a player and the ref-eree. (The referee could choose to pass information from one player to another, but suchan action must be understood as being in accordance with the referee’s specification.)Also, although Figure 6 might suggest a symmetry between the players, this is notrequired—the registers being exchanged can have arbitrary size, including the possibil-ity of trivial (dimension 1) registers that effectively represent the absence of informationbeing sent or received. Equivalently, the referee may interleave the messages exchangedwith different players in an arbitrary way, and the number of exchanges may be differentwith different players.In any case of this sort, similar to the single-player case just discussed, there will al-ways exist an efficiently computable positive semidefinite operator P a , for each possibleoutcome of the referee’s measurement, for which the probability associated with that mea-surement outcome is given byPr ( measurement outcome equals a ) = (cid:10) P a , Q ⊗ · · · ⊗ Q m (cid:11) , (30)assuming that the m players play strategies represented by matrices Q , . . . , Q m . Indeed,aside from a permutation of tensor factors, one need not see this as being an extensionof the single-player case at all, for if the m players do not directly interact, they may becollectively viewed as a single player, whose representation (again, up to a permutationof tensor factors) is given by the tensor product Q ⊗ · · · ⊗ Q m . We now recall the definition of the complexity class PPAD, which was first defined byPapadimitriou [Pap94] to capture the complexity of certain total functions, including ap-proximate fixed-point problems when a fixed point is guaranteed to exist. We also statea result due to Etessami and Yannakakis [EY10] concerning the containment of a specificfixed-point problem in PPAD to which we will later reduce the problem of computingapproximate fixed points of functions defined on density operators.Before proceeding to these definitions, let us remark that all computational problemsin this paper involving real or complex scalars, vectors, matrices, and so on, are assumedto refer to rational and/or Gaussian rational inputs and outputs in which the number a / b is encoded as a pair (cid:104) a , b (cid:105) and a / b + i c / d is encoded as a 4-tuples (cid:104) a , b , c , d (cid:105) , for integers a , b , c , and d represented in signed binary notation. The length of any such number thenrefers to the length of the encoding. 13 Φ Φ Φ Φ Φ Ψ Ψ Ψ Ψ Z Z Y Y Y X X X Z Z Y Y Y X X X W W W Figure 6: An interaction between a referee (represented by channels Ψ , . . . , Ψ along witha measurement) and two players, both having a form similar to the strategy pictured inFigure 4. Total search problems and the complexity class PPAD
The complexity class PPAD contains total search problems . In general, a total search prob-lem in the complexity class TFNP is represented by a collection of sets { A x : x ∈ Σ ∗ } ,with A x ⊆ Σ ∗ for each x ∈ Σ ∗ , satisfying these properties:1. There exists a polynomial p such that | y | ≤ p ( | x | ) for every x ∈ Σ ∗ and y ∈ A x .2. There exists a polynomial-time computable predicate R such that R ( x , y ) = y ∈ A x , for every choice of x , y ∈ Σ ∗ .3. For every x ∈ Σ ∗ , the set A x is non-empty.On a given input string x ∈ Σ ∗ , the goal of the associated problem is to find any string y ∈ A x . Such search problems are deemed total because an acceptable solution is alwaysguaranteed to exist.In the context of total search problems in TFNP, it is said that a problem { A x : x ∈ Σ ∗ } is polynomial-time reducible to another problem { B x : x ∈ Σ ∗ } if there exist polynomial-time computable functions f and g with the property that for y ∈ B f ( x ) ⇒ g ( y ) ∈ A x (31)for every string x ∈ Σ ∗ . In words, any input to the problem A = { A x : x ∈ Σ ∗ } can betransformed in polynomial time to an instance f ( x ) of the problem B = { B x : x ∈ Σ ∗ } in such a way that any acceptable solution y to B on input f ( x ) can be transformed inpolynomial time back to an acceptable solution g ( y ) to A on input x .Next, to state the definition of the class PPAD, which is contained in TFNP, we beginwith one specific problem in this class called the end-of-the-line problem .14 nd-of-the-line problem Input: Boolean circuits P and S , both having n input bits and n output bits, satisfying P ( n ) = n (cid:54) = S ( n ) .Output: Any string z ∈ Σ n such that S ( P ( z )) (cid:54) = z (cid:54) = n or P ( S ( z )) (cid:54) = z .(Formally speaking, if one is given an input string that does not encode Boolean circuits P and S with the properties indicated, then the associated set of acceptable solutions isdefined as the singleton set containing the empty string. Alternatively, one may modifythe definition of TFNP so that problems may be defined only on a subset of the possiblestrings.)The intuition behind this problem is that the circuits P and S allegedly represent prede-cessor and successor functions on the set Σ n . We envision a graph having vertex set Σ n witha directed edge from x to y , for distinct vertices x and y , if and only if both y = S ( x ) and x = P ( y ) . The vertex 0 n must have in-degree 0 by the assumption P ( n ) = n , meaningthat it is a source . The goal is to find either a sink , meaning a vertex with out-degree 0,which must necessarily exist, or a source different from 0 n . If P ( S ( z )) (cid:54) = z , then z is a sink(which could include the possibility z = n if 0 n happens to have out-degree 0), while if S ( P ( z )) (cid:54) = z (cid:54) = n then z is a source different from 0 n . It is important that a source differ-ent from 0 n is an acceptable answer; the variant of this problem that demands a sink asan output might potentially be a more difficult computational problem.Finally, PPAD is defined as the class of all total search problems that are polynomial-time reducible to the end-of-the-line problem. Fixed points of barycentric approximations in PPAD
Suppose that { f x : x ∈ Σ ∗ } is a collection of functions having the form f x : ∆ n → ∆ n , (32)for n = n ( x ) being polynomially bounded and polynomial-time computable. We say that { f x : x ∈ Σ ∗ } is a polynomial-time computable family if there exists a polynomial-timecomputable function F so that F ( x , (cid:104) v (cid:105) ) = (cid:104) f x ( v ) (cid:105) (33)for every rational vector v ∈ ∆ n , where angled brackets indicate the encoding of anyrational element of ∆ n .The following theorem follows from a more general result due to Etessami and Yan-nakakis [EY10]. Theorem 1.
Suppose that { f x : x ∈ Σ ∗ } is a polynomial-time computable family of functionshaving the form f x : ∆ n → ∆ n , and for each x ∈ Σ ∗ and each positive integer r letg x , r : ∆ n → ∆ n (34) be the r-th order barycentric approximation to f x . The problem of computing an exact fixed pointof g x , r on the input (cid:104) x , 0 r (cid:105) is contained in the class PPAD . . Definitions of quantum games We will now define a general class of quantum games and state the computational prob-lem upon which the remainder of the paper focuses.In the class of games we consider, a referee exchanges quantum registers with m play-ers over the course of r rounds in a way that generalizes Figure 6 (in which m = r = Ψ , . . . , Ψ r + along with a finalmeasurement, with these objects taking the following forms. For j ∈ {
2, . . . , r } , the chan-nel Ψ j takes input registers (cid:0) W j − , Y j − , . . . , Y mj − (cid:1) (35)and outputs registers (cid:0) W j , X j , . . . , X mj (cid:1) . (36)The channels Ψ and Ψ r + have a similar form except that Ψ takes no input and Ψ r + produces a single register W r + as output. We note that the registers need not all havethe same size, and some may be trivial, effectively representing the absence of a messagetransmission. Finally, the measurement is performed on the register W r + and has set ofoutcomes Γ . In addition to the referee’s actions, as just described, it is to be assumed thata payoff function v k : Γ → R has been selected for each player k ∈ {
1, . . . , m } .A referee of this form determines a non-cooperative game, in which an m -tuple ofindependent strategies for the players is interfaced with the referee in the most naturalway, leading to a distribution over payoffs for the m players.Suppose that, by means of the quantum strategies framework, a selection of the m players’ strategies Q , . . . , Q m has been made, with each being represented by an operator Q k ∈ Pos (cid:0) Y k ⊗ · · · ⊗ Y kr ⊗ X k ⊗ · · · ⊗ X kr (cid:1) , (37)and suppose moreover that the referee has been represented by a collection of operators { P a : a ∈ Γ } , as was described in the previous section. We then have that each outcome a ∈ Γ is produced by the referee with probability (cid:104) P a , Q ⊗ · · · ⊗ Q m (cid:105) , and the payoffs arethen determined accordingly. The expected payoff for player k is therefore given by ∑ a ∈ Γ v k ( a ) (cid:104) P a , Q ⊗ · · · ⊗ Q m (cid:105) = (cid:104) H k , Q ⊗ · · · ⊗ Q m (cid:105) (38)for H k = ∑ a ∈ Γ v k ( a ) P a . (39)When it is convenient, we will refer to the operators H , . . . , H m as payoff operators . Weobserve that the payoff operators of an interactive quantum game can be efficiently com-puted given the description of a referee’s actions.Hereafter let us write S k ⊂ Pos (cid:0) Y k ⊗ · · · ⊗ Y kr ⊗ X k ⊗ · · · ⊗ X kr (cid:1) (40)16o denote the set of strategy representations available to player k , for each k ∈ {
1, . . . , m } .Each of these sets is bounded and characterized by a finite collection of affine linear con-straints on the positive semidefinite cone acting on the corresponding spaces. In particu-lar, the sets S , . . . , S m are convex and compact.A Nash equilibrium of a quantum game of the form being considered is an m -tuple ( Q , . . . , Q m ) ∈ S × · · · × S m for which the equality (cid:104) H k , Q ⊗ · · · ⊗ Q m (cid:105) = sup R ∈ S k (cid:104) H k , Q ⊗ · · · ⊗ Q k − ⊗ R ⊗ Q k + ⊗ · · · ⊗ Q m (cid:105) (41)holds for every k ∈ {
1, . . . , m } . Thus, no player can increase their expected payoff byunilaterally deviating from a Nash equilibrium ( Q , . . . , Q m ) . The existence of a Nashequilibrium in every interactive quantum game follows from Glicksberg’s generalizationof Nash’s theorem [Gli52]. It is also straightforward to prove the existence of a Nash equi-librium in an interactive quantum game more directly, through the Kakutani fixed-pointtheorem (upon which Glicksberg’s generalization is also based), following the same rea-soning as in Nash’s proof in [Nas50a] for the existence of an equilibrium point in classicalgames.For any choice of ε >
0, an m -tuple of strategies ( Q , . . . , Q m ) ∈ S × · · · × S m is an ε -approximate Nash equilibrium if it is the case that (cid:104) H k , Q ⊗ · · · ⊗ Q m (cid:105) ≥ sup R ∈ S k (cid:104) H k , Q ⊗ · · · ⊗ Q k − ⊗ R ⊗ Q k + ⊗ · · · ⊗ Q m (cid:105) − ε (42)for every k ∈ {
1, . . . , m } . In words, no player can increase their expected payoff by morethan ε by deviating from an ε -approximate Nash equilibrium ( Q , . . . , Q m ) .For the sake of efficiency, it is prudent at this point to introduce some additional nota-tion. For each k ∈ {
1, . . . , m } we define V k = Y k ⊗ · · · ⊗ Y kr ⊗ X k ⊗ · · · ⊗ X kr , (43)so that S k ⊂ Pos ( V k ) , as well as V − k = V ⊗ · · · ⊗ V k − ⊗ V k + ⊗ · · · ⊗ V m . (44)For a given choice of strategies ( Q , . . . , Q m ) ∈ S × · · · × S m we define Q − k = Q ⊗ · · · ⊗ Q k − ⊗ Q k + ⊗ · · · ⊗ Q m , (45)so that Q − k ∈ Pos ( V − k ) . We stress that this is a tensor product—a similar notation is oftenused for Cartesian products.Observe that, for each k ∈ {
1, . . . , m } , there exists a Hermitian-preserving linear maptaking the form Ξ k : L ( V − k ) → L ( V k ) and having the property that (cid:104) H k , Q ⊗ · · · ⊗ Q m (cid:105) = (cid:104) Ξ k ( Q − k ) , Q k (cid:105) (46)17or every choice of ( Q , . . . , Q m ) ∈ S × · · · × S m (or indeed for any choice of Hermitianoperators Q , . . . , Q m , not just strategies). Explicitly, Ξ k ( Q − k ) = Tr V − k (cid:0) ( Q ⊗ · · · ⊗ Q k − ⊗ V k ⊗ Q k + ⊗ · · · ⊗ Q m ) H k (cid:1) . (47)An equivalent condition to (41) is then that (cid:104) Ξ k ( Q − k ) , Q k (cid:105) = sup R ∈ S k (cid:104) Ξ k ( Q − k ) , R (cid:105) , (48)while (42) is equivalent to (cid:104) Ξ k ( Q − k ) , Q k (cid:105) ≥ sup R ∈ S k (cid:104) Ξ k ( Q − k ) , R (cid:105) − ε . (49)We may now define the computational problem of approximating a Nash equilibriumof a quantum game. We assume that the input to the problem consists of the payoff op-erators of a given game, along with positive real number ε , although as noted above onecould alternatively describe a quantum game in terms of the referee’s actions, from whichthe payoff operators may be computed. Approximate quantum Nash equilibrium
Input: Hermitian operators H , . . . , H m ∈ Herm ( V ⊗ · · · ⊗ V m ) , for each V k takingthe form (43), along with a positive real number ε .Output: An ε -approximate Nash equilibrium ( Q , . . . , Q m ) of the interactive quantumgame described by H , . . . , H m .The following theorem, which is proved in the next section, represents the main resultof this paper. Theorem 2.
The problem of computing an approximate quantum Nash equilibrium is containedin the complexity class PPAD.
4. Approximate quantum Nash equilibria in PPAD
The purpose of this section is to prove Theorem 2. We shall begin with an overview of theproof, followed by three subsections that address specific aspects of it.The proofs of the existence of Nash equilibria in interactive quantum games sug-gested above are both based on the Kakutani fixed-point theorem. Toward the goal ofestablishing that the problem of approximating Nash equilibria in quantum games is inthe complexity class PPAD, however, it is instructive to consider a different path, basedon an extension of Nash’s 1951 proof [Nas51] of the existence of equilibria in classicalgames, which makes use of the Brouwer fixed-point theorem together with the notionof a gain function. This is a familiar path to analogous results in the classical setting[Pap94, DGP09, CDT09, EY10]. 18ur first step is to prove that the problem of approximating fixed points of a certainclass of continuous functions defined on density operators is contained in PPAD. This isdone by means of a reduction, based on the discrete Wigner representation defined inSection 2.2, to the fixed-point problem on the simplex established to be in PPAD by The-orem 1.The second step is to consider an interactive quantum generalization of Nash’s gainfunction. Intuitively speaking, this is a function defined on m -tuples of strategies that im-proves each player’s strategy, relative to the other players’ strategies being considered,so that the fixed points of this function are equilibrium points. This allows for the reduc-tion of the problem of finding an approximate Nash equilibrium in a quantum game tofinding an approximate fixed point of this gain function, which may be expressed as afunction on density operators.The computations required by both of the steps just described cannot be performedexactly using rational number computations. To control the precision required by ratio-nal number approximations to these computations, we must bound the Lipschitz moduli of various functions that are composed to obtain the reduction. This includes functionsexpressible as semidefinite programs but not known to have closed form expressions.The subsections that follow address these aspects of the proof. The first subsection isconcerned entirely with the Lipschitz moduli of various function that will be needed inthe remaining subsections, establishing bounds that allow the proof to go through. Thesecond subsection establishes that the problem of computing approximate fixed pointsof continuous functions defined on density operators (or Cartesian products of densityoperators) is contained in PPAD. And finally, the third subsection reduces the problem ofcomputing approximate Nash equilibria of interactive quantum games to the problem ofcomputing fixed points of continuous functions on density operators.
This subsection simply lists several functions relevant to the proof together with boundson their Lipschitz moduli.Whenever we refer to the Lipschitz condition for any function, defined for vectors oroperators, we will always use the 2-norm, meaning the standard Euclidean norm for R n and the Frobenius norm for the n × n complex Hermitian operators Herm ( C n ) . That is, afunction f is K -Lipschitz if (cid:107) f ( u ) − f ( v ) (cid:107) ≤ K (cid:107) u − v (cid:107) (50)for all vectors u and v on which it is defined, and likewise for functions defined on op-erators rather than vectors. We refer to K as the Lipschitz modulus of f , as opposed to themore standard Lipschitz constant , as K will generally not be constant (as a function of theinput length) for the functions we will encounter.19 he discrete Wigner representation. The function ψ : Herm ( C n ) → R n associated with the discrete Wigner representationwe have defined is ( K ) -Lipschitz, while ψ − is K -Lipschitz, for K = √ n ( n + ) . Moreprecisely, by the orthogonality of the operators { V , . . . , V n } , we have the equality condi-tions (cid:107) ψ ( H ) − ψ ( K ) (cid:107) = √ n ( n + ) (cid:107) H − K (cid:107) (51)and (cid:13)(cid:13) ψ − ( u ) − ψ − ( v ) (cid:13)(cid:13) = √ n ( n + ) (cid:107) u − v (cid:107) . (52)These two moduli will cancel one another in the analysis to follow in the next subsection. Tensor products of density operators.
The tensor product mapping ( ρ , . . . , ρ m ) (cid:55)→ ρ ⊗ · · · ⊗ ρ m , (53)from D ( C n ) × · · · × D ( C n m ) to D ( C n ⊗ · · · ⊗ C n m ) , is √ m -Lipschitz: (cid:107) ρ ⊗ · · · ⊗ ρ m − σ ⊗ · · · ⊗ σ m (cid:107) ≤ (cid:107) ρ ⊗ ρ ⊗ · · · ⊗ ρ m − σ ⊗ ρ ⊗ · · · ⊗ ρ m (cid:107) + (cid:107) σ ⊗ ρ ⊗ · · · ⊗ ρ m − σ ⊗ σ ⊗ · · · ⊗ σ m (cid:107) ≤ (cid:107) ρ − σ (cid:107) (cid:107) ρ ⊗ · · · ⊗ ρ m (cid:107) + (cid:107) σ (cid:107) (cid:107) ρ ⊗ · · · ⊗ ρ m − σ ⊗ · · · ⊗ σ m (cid:107) ≤ (cid:107) ρ − σ (cid:107) + (cid:107) ρ ⊗ · · · ⊗ ρ m − σ ⊗ · · · ⊗ σ m (cid:107) , (54)and by iterating, (cid:107) ρ ⊗ · · · ⊗ ρ m − σ ⊗ · · · ⊗ σ m (cid:107) ≤ (cid:107) ρ − σ (cid:107) + · · · + (cid:107) ρ m − σ m (cid:107) ≤ √ m (cid:13)(cid:13) ( ρ , . . . , ρ m ) − ( σ , . . . , σ m ) (cid:13)(cid:13) . (55) The maps Ξ k . Recall the maps Ξ k : L ( V − k ) → L ( V k ) defined in the previous section, which satisfy (cid:104) H k , Q ⊗ · · · ⊗ Q m (cid:105) = (cid:104) Ξ k ( Q − k ) , Q k (cid:105) (56)for all Hermitian operators Q , . . . , Q m , where H k is considered to be fixed. Explicitly, Ξ k ( Q − k ) = Tr V − k (cid:0) ( Q ⊗ · · · ⊗ Q k − ⊗ V k ⊗ Q k + ⊗ · · · ⊗ Q m ) H k (cid:1) . (57)Let us write n k = dim ( V k ) and n = n · · · n m .20irst, the mapping Q − k (cid:55)→ Q ⊗ · · · ⊗ Q k − ⊗ V k ⊗ Q k + ⊗ · · · ⊗ Q m (58)is √ n k -Lipschitz, while right-multiplication by H k is (cid:107) H k (cid:107) -Lipschitz, (cid:107) H k (cid:107) denoting thespectral norm of H k .Next, every quantum channel Ψ : L ( X ) → L ( Y ) , including the trace, is 1-Lipschitzwith respect to the trace norm, and therefore (cid:107) Ψ ( X ) − Ψ ( Y ) (cid:107) ≤ (cid:107) Ψ ( X ) − Ψ ( Y ) (cid:107) ≤ (cid:107) X − Y (cid:107) ≤ (cid:113) dim ( X ) (cid:107) X − Y (cid:107) . (59)That is, every channel is (cid:112) dim ( X ) -Lipschitz with respect to the Frobenius norm, for X being the input space of the channel. We may also note that tensoring any linear map(whether a channel or not) with the identity channel does not change its Lipschitz modu-lus. It follows that the partial trace over the space V − k has Lipschitz modulus √ n / n k .Composing these functions, we find that the mapping Ξ k is ( (cid:107) H k (cid:107)√ n ) -Lipschitz. Projections onto closed and convex sets.
For any closed and convex set C , we define proj ( X | C ) to be the projection of X onto theset C , meaning the unique point contained in C that is closest to X with respect to the2-norm (or Frobenius norm). The function X (cid:55)→ proj ( X | C ) is, as is well known, always1-Lipschitz. Normalizing positive semidefinite operators.
Next, define a function that normalizes any positive semidefinite operator P ∈ Pos ( C n ) inthe following way: normalize ( P ) = P Tr ( P ) Tr ( P ) ≥ P + ( − Tr ( P )) n n Tr ( P ) <
1. (60)Strictly speaking this may not really be a normalization in the case that Tr ( P ) <
1, butthis function serves our purposes nevertheless.The function normalize : Pos ( C n ) → D ( C n ) is ( n ) -Lipschitz. This is perhaps easiestto prove by expressing the function as normalize = g ◦ f where f and g are defined asfollows: f ( P ) = P Tr ( P ) ≥ P + ( − Tr ( P )) n n Tr ( P ) < g ( P ) = P Tr ( P ) Tr ( P ) ≥ P Tr ( P ) <
1. (61)21he function f is ( √ n ) -Lipschitz, which may be established by considering threecases. If Tr ( P ) ≥ ( Q ) ≥
1, then (cid:107) f ( P ) − f ( Q ) (cid:107) = (cid:107) P − Q (cid:107) , trivially. If Tr ( P ) ≥ ( Q ) <
1, then (cid:107) f ( P ) − f ( Q ) (cid:107) = (cid:13)(cid:13)(cid:13) P − Q − ( − Tr ( Q )) n n (cid:13)(cid:13)(cid:13) ≤ (cid:107) P − Q (cid:107) + ( − Tr ( Q )) ≤ (cid:107) P − Q (cid:107) + ( Tr ( P ) − Tr ( Q )) ≤ (cid:107) P − Q (cid:107) (62)and therefore (cid:107) f ( P ) − f ( Q ) (cid:107) ≤ (cid:107) P − Q (cid:107) ≤ √ n (cid:107) P − Q (cid:107) . (63)If Tr ( P ) < ( Q ) <
1, then (cid:107) f ( P ) − f ( Q ) (cid:107) = (cid:13)(cid:13)(cid:13) P − Q − ( Tr ( P ) − Tr ( Q )) n n (cid:13)(cid:13)(cid:13) ≤ (cid:107) P − Q (cid:107) + | Tr ( P ) − Tr ( Q ) | ≤ (cid:107) P − Q (cid:107) , (64)and so again (cid:107) f ( P ) − f ( Q ) (cid:107) ≤ (cid:107) P − Q (cid:107) ≤ √ n (cid:107) P − Q (cid:107) . (65)On the set of positive semidefinite operators having trace at least one, the function g is ( + √ n ) -Lipschitz; supposing that P , Q ∈ Pos ( C n ) satisfy Tr ( Q ) ≥ ( P ) ≥ (cid:13)(cid:13)(cid:13)(cid:13) P Tr ( P ) − Q Tr ( Q ) (cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13)(cid:13) P Tr ( P ) − Q Tr ( P ) (cid:13)(cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13)(cid:13) Q Tr ( P ) − Q Tr ( Q ) (cid:13)(cid:13)(cid:13)(cid:13) = (cid:107) P − Q (cid:107) Tr ( P ) + (cid:18) Tr ( Q ) − Tr ( P ) Tr ( P ) Tr ( Q ) (cid:19) (cid:107) Q (cid:107) ≤ ( + √ n ) (cid:107) P − Q (cid:107) . (66)Using 2 √ n ( √ n + ) ≤ n we obtain that normalize is ( n ) -Lipschitz. We now consider the computational problem of approximating fixed points of continuousfunctions defined on density operators, proving that this problem is in PPAD for functionshaving exponentially bounded Lipschitz moduli.To state this fact more precisely, we require a few definitions. First, a density operator ρ ∈ D ( C n ) is an ε -approximate fixed point of a function f : D ( C n ) → D ( C n ) provided that (cid:107) f ( ρ ) − ρ (cid:107) ≤ ε . (67)Next, suppose that { f x : x ∈ Σ ∗ } is a collection of functions having the form f x : D ( C n ) → D ( C n ) (68)22or n = n ( x ) being polynomially bounded. Mirroring a definition from Section 2.4 forfunctions defined on the unit simplex, we shall say that { f x : x ∈ Σ ∗ } is a polynomial-timecomputable family if there exists a polynomial-time computable function F so that F ( x , (cid:104) ρ (cid:105) ) = (cid:104) f x ( ρ ) (cid:105) (69)for every rational density operator ρ ∈ D ( C n ) , with angled brackets representing encod-ings of rational density operators. We must also define an approximate variant of thisnotion: { f x : x ∈ Σ ∗ } is a polynomial-time approximable family if there exists a polynomial-time computable family { g x , ε } satisfying (cid:13)(cid:13) f x − g x , ε (cid:13)(cid:13) ≤ ε (70)for every x ∈ Σ ∗ and every positive rational number ε .Finally, the problem of computing ε -approximate fixed points of the family { f x } is tooutput the encoding of any ε -approximate fixed point of the function f x on input ( x , ε ) . Theorem 3.
Let { f x } be a polynomial-time approximable family of functions on density operators,let p be a polynomial, and assume that each function f x is K x -Lipschitz, for K x = p ( | x | ) . Theproblem of computing ε -approximate fixed points of the family { f x } is in PPAD .Proof.
Let us first observe that there is no loss of generality in assuming that, for everyinput x , the dimension n is odd and at least 3. The case n = n is even,one may substitute the function f x by h x : D ( C n + ) → D ( C n + ) defined as h x (cid:18) P uu ∗ λ (cid:19) = (cid:18) f x (cid:0) P + λ n (cid:1)
00 0 (cid:19) . (71)The Lipschitz modulus of h x is at most √ f x , and every fixed point of h x takes the form σ = (cid:18) ρ
00 0 (cid:19) (72)for ρ a fixed point of f x . If (cid:18) P uu ∗ λ (cid:19) (73)is an ε -approximate fixed point of h x , then it follows that2 (cid:107) u (cid:107) + λ = (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) uu ∗ λ (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13)(cid:13) h x (cid:18) P uu ∗ λ (cid:19) − (cid:18) P uu ∗ λ (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) ≤ ε , (74)from which it follows that (cid:13)(cid:13)(cid:13)(cid:13) f x (cid:18) P + λ n (cid:19) − (cid:16) P + λ n (cid:17)(cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13)(cid:13) h x (cid:18) P uu ∗ λ (cid:19) − (cid:18) P uu ∗ λ (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) P uu ∗ λ (cid:19) − (cid:18) P + λ n
00 0 (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:16) + √ (cid:17) ε . (75)23hus, an ε -approximate fixed point of f x is easily obtained from an ( ε /3 ) -approximatefixed point of h x .Assuming now that n is odd for each x , we define a function g x : ∆ n → ∆ n as g x ( v ) = ψ (cid:0) f x (cid:0) proj (cid:0) ψ − ( v ) (cid:12)(cid:12) D ( C n ) (cid:1)(cid:1)(cid:1) (76)Here, the projection function is as defined in the previous subsection and ψ is the mappingassociated with the discrete Wigner representation defined in Section 2.2. Given that f x is K x -Lipschitz, it follows that g x is K x -Lipschitz as well, as the projection is 1-Lipschitz andthe Lipschitz moduli of the discrete Wigner mappings cancel.Given that f x is polynomial-time approximable, it is possible to compute, in polyno-mial time, an approximation (cid:101) g x to g x satisfying (cid:107) (cid:101) g x ( u ) − g x ( u ) (cid:107) ≤ ε n K x (77)for every rational vector u ∈ ∆ n . We note, in particular, that the projection onto D ( C n ) may be approximated by first approximating a spectral decomposition of the operator ψ − ( v ) and then projecting its eigenvalues onto the unit simplex ∆ n . Alternatively, thisprojection arises as a special case of one discussed in the next subsection, where the ellip-soid method provides a polynomial-time algorithm to approximate the projection.Next, set r = ( n + ) (cid:6) log ( ε ) + p ( | x | ) + ( n ) + (cid:7) . (78)This number is polynomial in | x | and log ( ε ) , and has been selected so that (cid:16) − n + (cid:17) r < exp (cid:0) − log ( ε ) − p ( | x | ) − ( n ) − ) < ε n K x . (79)By Theorem 1, one may therefore compute an exact fixed point v ∈ ∆ n of the r -th levelbarycentric approximation to (cid:101) g x in PPAD.Supposing that such a fixed point v is expressed as a convex combination v = q v + · · · + q n v n (80)for v , . . . , v n ∈ B rn denoting vertices in any one of the simplices constructed at the r -level of the barycentric subdivision, we find that (cid:107) g x ( v ) − v (cid:107) = (cid:13)(cid:13) g x ( v ) − (cid:0) q (cid:101) g x ( v ) + · · · q n (cid:101) g x ( v n ) (cid:1)(cid:13)(cid:13) ≤ n ∑ j = q j (cid:13)(cid:13) g x ( v ) − (cid:101) g x ( v j ) (cid:13)(cid:13) ≤ n ∑ j = q j (cid:16)(cid:13)(cid:13) g x ( v ) − g x ( v j ) (cid:13)(cid:13) + (cid:13)(cid:13) g x ( v j ) − (cid:101) g x ( v j ) (cid:13)(cid:13) (cid:17) ≤ K x (cid:16) − n + (cid:17) r + ε n K x ≤ ε n K x . (81)24hus, v is an ( ε / ( n K x )) -approximate fixed point of g x .Now consider the Hermitian operator ψ − ( v ) . We have (cid:13)(cid:13) ψ − ( v ) − ψ − ( g ( v )) (cid:13)(cid:13) = √ n ( n + ) (cid:13)(cid:13) v − g ( v ) (cid:13)(cid:13) ≤ ε K x , (82)and given that ψ − ( g ( v )) is necessarily a density operator, the operator ψ − ( v ) thereforehas distance at most ε / ( K x ) from the set of density operators. By computing a densityoperator ρ satisfying (cid:13)(cid:13) ρ − proj (cid:0) ψ − ( v ) (cid:12)(cid:12) D ( C n ) (cid:1)(cid:13)(cid:13) ≤ ε K x (83)as suggested above, we therefore have (cid:13)(cid:13) ρ − ψ − ( v ) (cid:13)(cid:13) ≤ ε K x . (84)Consequently, noting that f (cid:0) proj (cid:0) ψ − ( v ) (cid:12)(cid:12) D ( C n ) (cid:1) = ψ − ( g ( v )) , (85)we find, by the triangle inequality, that (cid:13)(cid:13) f x ( ρ ) − ρ (cid:13)(cid:13) ≤ (cid:13)(cid:13) f ( ρ ) − f (cid:0) proj (cid:0) ψ − ( v ) (cid:12)(cid:12) D ( C n ) (cid:1)(cid:1)(cid:13)(cid:13) + (cid:13)(cid:13) ψ − ( g ( v )) − ψ − ( v ) (cid:13)(cid:13) + (cid:13)(cid:13) ψ − ( v ) − ρ (cid:13)(cid:13) ≤ ε + ε K x + ε K x ≤ ε . (86)Thus, ρ is an ε -approximate fixed point of f x . As ρ has been computed in polynomial timefrom v , the theorem is proved. Corollary 4.
Let { f x } be a polynomial-time approximable family of functions having the formf x : D (cid:0) C n (cid:1) × · · · × D (cid:0) C n m (cid:1) → D (cid:0) C n (cid:1) × · · · × D (cid:0) C n m (cid:1) , (87) for positive integers n , . . . , n m , let p be a polynomial, and assume that each function f x is K x -Lipschitz, for K x = p ( | x | ) . The problem of computing ε -approximate fixed points of the family { f x } is in PPAD .Proof.
Let n = n + · · · + n m and define a mapping h : D ( C n ) → D ( C n ) as follows: h X · · · X m ... . . . ... X m ,1 · · · X m , m = m normalize ( mX )
0. . .0 normalize ( mX m , m ) , (88)where it is to be understood that each X i , j has n i rows and n j columns. The mapping h is ( n ) -Lipschitz and projects onto operators having the form1 m ρ
0. . .0 ρ m . (89)25y composing f x with h in the natural way, one obtains a function g x : D ( C n ) → D ( C n ) such that g x X · · · X m ... . . . ... X m ,1 · · · X m , m = m σ
0. . .0 σ m (90)for ( σ , . . . , σ m ) = f x ( normalize ( mX ) , . . . , normalize ( mX m , m )) . (91)Finally, from any approximate fixed point of the family { g x } , an ε -approximate fixedpoint for { f x } is obtained by applying to it the function h and reading off the diagonaloperators. The problem of approximating fixed points of { f x } therefore reduces in poly-nomial time to that of { g x } , which is in the class PPAD. The final step of the proof of Theorem 2 is to reduce the problem of computing approxi-mate Nash equilibria of interactive quantum games to the approximate fixed-point prob-lem on Cartesian products of density operators established by Corollary 4 to be in PPAD.To do this, we will consider an extension of Nash’s gain function to quantum strategies,as they are represented within the quantum strategies framework.For a quantum game of the general form described in Section 3, the set of strategiesavailable each player k ∈ {
1, . . . , m } is represented by the set S k ⊂ Pos (cid:0) Y k ⊗ · · · ⊗ Y kr ⊗ X k ⊗ · · · ⊗ X kr (cid:1) , (92)and we observe that for every choice of Q k ∈ S k we haveTr ( Q k ) = d k def = dim (cid:0) X k ⊗ · · · ⊗ X kr (cid:1) . (93)Define the set C k = d k S k ⊆ D ( V k ) , (94)as well as the cone K k = cone ( C k ) = (cid:8) λρ : λ ≥ ρ ∈ C k (cid:9) . (95)Now, for a given m -tuple ( ρ , . . . , ρ m ) of density operators, we define G ( ρ , . . . , ρ m ) inthe following way. First, for each k ∈ {
1, . . . , m } , define σ k = proj ( ρ k | C k ) , α k = (cid:10) Ξ k ( σ − k ) , σ k (cid:11) , P k = proj (cid:0) Ξ k ( σ − k ) − α k V k | K k ) , (96)and G k ( ρ , . . . , ρ m ) = normalize (cid:0) σ k + P k ) = σ k + P k + Tr ( P k ) . (97)26hen define G ( ρ , . . . , ρ m ) = (cid:0) G ( ρ , . . . , ρ m ) , . . . , G m ( ρ , . . . , ρ m ) (cid:1) . (98)By combining the Lipschitz moduli for the functions from which G is formed, over-estimating for the sake of a simple expression, we have that G is K -Lipschitz for K = n mM , M = max {(cid:107) H (cid:107) , . . . , (cid:107) H m (cid:107)} +
1, and n = n · · · n m (99)for n k = dim ( V k ) . The following lemma establishes that G can be efficiently approxi-mated. Lemma 5.
There exists a deterministic, polynomial-time algorithm that, on input H , . . . , H m , ρ , . . . , ρ m , and δ > , outputs ( ξ , . . . , ξ m ) ∈ C × · · · × C m satisfying (cid:13)(cid:13) G ( ρ , . . . , ρ m ) − ( ξ , . . . , ξ m ) (cid:13)(cid:13) < δ . (100) Proof.
Let us begin with the approximation of the projections σ k = proj ( ρ k | C k ) for each k ∈ {
1, . . . , m } . The projection σ k is given by the unique optimal solution to the followingsemidefinite program: minimize : Tr ( Z k ) subject to : (cid:18) Z k ρ k − Y k ρ k − Y k V k (cid:19) ≥ Y k ∈ C k Z k ∈ Pos ( V k ) . (101)Specifically, the unique optimal solution ( Y k , Z k ) to this semidefinite program satisfies Y k = σ k = proj ( ρ k | C k ) and Tr ( Z k ) = (cid:107) ρ k − σ k (cid:107) .Through the use of the ellipsoid method, as presented by [GLS88] for instance, onemay compute in time polynomial in the input length and log ( η ) , for any given positivereal number η , a feasible solution ( Z k , Y k ) to this semidefinite program that is within η ofits optimal value. That is, in polynomial time one may compute ξ k ∈ C k such that (cid:107) ρ k − ξ k (cid:107) ≤ (cid:107) ρ k − σ k (cid:107) + η . (102)This requires an examination of specific aspects of the semidefinite program that are re-flected (up to a scalar multiple in the last constraint) by the equations (26) in Section 2.3along with a recognition that the feasible region may be bounded. The analysis is straight-forward and we omit it here.Now, if it is the case that ρ k ∈ C k , then σ k = ρ k , and we conclude immediately that (cid:107) ξ k − σ k (cid:107) ≤ √ η . (103)If ρ k (cid:54)∈ C k , then it follows that (cid:104) ξ k − σ k , ρ k − σ k (cid:105) ≤
0; that this inequality holds for everychoice of ξ k ∈ C k is, in fact, a well known necessary and sufficient condition for σ k to bethe projection of ρ k into C k . By the law of cosines we have (cid:107) ρ k − ξ k (cid:107) = (cid:107) ρ k − σ k (cid:107) + (cid:107) ξ k − σ k (cid:107) − (cid:104) ξ k − σ k , ρ k − σ k (cid:105) , (104)27nd so we conclude that (cid:107) ρ k − ξ k (cid:107) ≥ (cid:107) ρ k − σ k (cid:107) + (cid:107) ξ k − σ k (cid:107) , (105)which again implies (cid:107) ξ k − σ k (cid:107) ≤ √ η . (106)The computation of each P k = proj (cid:0) Ξ k ( σ − k ) − α k V k | K k ) may be performed in almostexactly the same manner, through almost exactly the same semidefinite program. We notein particular that the optimal value is no larger than (cid:107) Ξ k ( σ − k ) − α k V k (cid:107) , as the projectionof Ξ k ( σ − k ) − α k V k onto K k is no further away from this operator than the zero operator,which is contained in K k , and so once again the feasible region may be bounded. Thus wemay compute, again in polynomial time, R k ∈ K k satisfying (cid:107) P k − R k (cid:107) ≤ √ η .All of the other computations required to approximate G can be performed exactly.The lemma follows by choosing η to be sufficiently small while polynomial in δ and theinput length to the problem.It therefore follows from Corollary 4 that, on input H , . . . , H m and δ >
0, the prob-lem of computing a δ -approximate fixed point of G is contained in PPAD. It remains toprove that from such an approximate fixed point of G , we obtain an approximate Nashequilibrium for a game described by H , . . . , H m .At this point we face a minor inconvenience: an approximate fixed point ( ρ , . . . , ρ m ) of G provided by the PPAD computation whose existence is implied by Corollary 4 mightnot be contained in C × · · · × C m , although by necessity it will be close. Because we re-quire an approximate Nash equilibrium to consist of strategies and not “near strategies,”we must project these density operators onto the sets C , . . . , C m . Specifically, suppose that ( ρ , . . . , ρ m ) is an ( η /2 ) -approximate fixed point of G , for η = ε ( nM ) . (107)Writing σ k = proj ( ρ k | C k ) for each k ∈ {
1, . . . , m } as before, we find by the definition of G that G ( ρ , . . . , ρ m ) = G ( σ , . . . , σ m ) ∈ C × · · · × C m , (108)and combining this observation with the fact that projections are 1-Lipschitz, it followsthat ( σ , . . . , σ m ) is also an ( η /2 ) -approximate fixed point of G . Although the density op-erators ( σ , . . . , σ m ) cannot be computed exactly from ( ρ , . . . , ρ m ) , the analysis used inthe proof of the previous lemma implies that, in polynomial time, one may compute from ( ρ , . . . , ρ m ) an m -tuple ( ξ , . . . , ξ m ) ∈ C × · · · × C m (with this containment guaranteedby the ellipsoid method) satisfying (cid:107) ( σ , . . . , σ m ) − ( ξ , . . . , ξ m ) (cid:107) < η K . (109)28t follows that (cid:107) G ( ξ , . . . , ξ m ) − ( ξ , . . . , ξ m ) (cid:107) ≤ (cid:107) G ( ξ , . . . , ξ m ) − G ( σ , . . . , σ m ) (cid:107) + (cid:107) G ( σ , . . . , σ m ) − ( σ , . . . , σ m ) (cid:107) + (cid:107) ( σ , . . . , σ m ) − ( ξ , . . . , ξ m ) (cid:107) ≤ η . (110)Thus, ( ξ , . . . , ξ m ) is an η -approximate fixed point of G .One more lemma is needed, which will imply that by scaling the density operators ( ξ , . . . , ξ m ) , an ε -approximate Nash equilibrium is obtained. Lemma 6.
Let C ⊆ D ( C n ) be a nonempty, convex, and compact set of density operators, let K = cone ( C ) be the cone generated by C , and let A ∈ Herm ( C n ) be a Hermitian operator. For agiven density operator σ ∈ C , defineP = proj ( A − (cid:104) A , σ (cid:105) | K ) , (111) and assume that (cid:13)(cid:13)(cid:13)(cid:13) σ + P + Tr ( P ) − σ (cid:13)(cid:13)(cid:13)(cid:13) ≤ η (112) for η > . It is the case that (cid:104) A , σ (cid:105) ≥ sup ξ ∈ C (cid:104) A , ξ (cid:105) − δ (113) for δ = ( + n (cid:107) A (cid:107) ) √ η . (114) Proof.
The operator P is defined to be the closest element of the cone K to the operator A − (cid:104) A , σ (cid:105) with respect to the Frobenius norm, which is to say that (cid:13)(cid:13)(cid:0) A − (cid:104) A , σ (cid:105) (cid:1) − P (cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:0) A − (cid:104) A , σ (cid:105) (cid:1) − λξ (cid:13)(cid:13) (115)for every choice of λ ≥ ξ ∈ C . We may first consider the case that λ =
0, from whichthe bound (cid:107) P (cid:107) ≤ (cid:13)(cid:13) A − (cid:104) A , σ (cid:105) (cid:13)(cid:13) ≤ √ n (cid:107) A (cid:107) , (116)is obtained, implying that Tr ( P ) ≤ n (cid:107) A (cid:107) . It follows that (cid:107) P − Tr ( P ) σ (cid:107) = ( + Tr ( P )) (cid:13)(cid:13)(cid:13)(cid:13) σ + P + Tr ( P ) − σ (cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:0) + n (cid:107) A (cid:107) (cid:1) η . (117)Next, by squaring both sides of the inequality (115) and simplifying, one obtains λ (cid:10) A − (cid:104) A , σ (cid:105) , ξ (cid:11) ≤ (cid:10) A − (cid:104) A , σ (cid:105) , P (cid:11) + λ (cid:107) ξ (cid:107) − (cid:107) P (cid:107) . (118)Disregarding the negative final term and observing the inequality (cid:107) ξ (cid:107) ≤ (cid:10) A − (cid:104) A , σ (cid:105) , σ (cid:11) =
0, we find that λ (cid:10) A − (cid:104) A , σ (cid:105) , ξ (cid:11) ≤ (cid:10) A − (cid:104) A , σ (cid:105) , P − Tr ( P ) σ (cid:11) + λ λ ≥ ξ ∈ C . Setting λ = √ η and applying the Cauchy–Schwarzinequality yields (cid:104) A , ξ (cid:105) − (cid:104) A , σ (cid:105) = (cid:10) A − (cid:104) A , σ (cid:105) , ξ (cid:11) ≤ √ η (cid:13)(cid:13) A − (cid:104) A , σ (cid:105) (cid:13)(cid:13) (cid:13)(cid:13) P − Tr ( P ) σ (cid:13)(cid:13) + √ η ≤ (cid:16) √ n (cid:107) A (cid:107) ( + n (cid:107) A (cid:107) ) + (cid:17) √ η ≤ ( + n (cid:107) A (cid:107) ) √ η . (120)As this bound holds for every ξ ∈ C , the lemma is proved.We conclude from this lemma that (cid:10) Ξ k ( ξ − k ) , ξ k (cid:11) ≥ sup τ ∈ C k (cid:10) Ξ k ( ξ − k ) , τ (cid:11) − (cid:0) + n k (cid:107) Ξ k ( ξ − k ) (cid:107) (cid:1) √ η . (121)Define Q k = d k ξ k for each k ∈ {
1, . . . , m } so that ( Q , . . . , Q m ) ∈ S × · · · × S m . (122)By (121) it follows that (cid:10) Ξ k ( Q − k ) , Q k (cid:11) ≥ sup R ∈ S k (cid:10) Ξ k ( Q − k ) , R (cid:11) − d · · · d m (cid:0) + n k (cid:107) Ξ k ( σ − k ) (cid:107) (cid:1) √ η ≥ sup R ∈ S k (cid:10) Ξ k ( Q − k ) , R (cid:11) − ε , (123)and therefore ( Q , . . . , Q m ) is an ε -approximate Nash equilibrium of the interactive quan-tum game having associated payoff operators H , . . . , H m . As ( Q , . . . , Q m ) has been ob-tained from ε together with the approximate fixed point ( ρ , . . . , ρ m ) of G by a polynomial-time computation, Theorem 2 is proved.
5. Discussion of directions for further research
We conclude the paper with a collection of open problems and suggestions of topics thatwe hope might inspire further work on quantum game theory and its connections totheoretical computer science.1. Is there a quantum extension or variant of the Lemke–Howson algorithm [LH64]for computing or approximating a Nash equilibrium in a non-interactive two-playerquantum game?2. It is interesting to consider quantum players having different restrictions placed ontheir strategies. For example, we might insist that players process quantum informa-tion using limited resources, or restrict player’s actions so that they represent adver-sarial models of noise. What can be said about quantum games in contexts such asthese? 30. We have limited our focus to a non-cooperative setting, in which players must playindependently, representing an inability for the players to form collusions. The con-sideration of collusions, and more generally the study of cooperative quantum gametheory , is an interesting research direction.For instance, let us imagine that there exists a shared quantum state that allows play-ers to implement a strategy in a quantum game that is good by some measure. Non-local games, for instance, may naturally be viewed as non-interactive games in apurely cooperative setting where shared quantum states can lead to improved strate-gies. In the general, not completely cooperative setting, such a shared state could beprovided by a trusted non-participant in the game, as in a correlated (or entangled)equilibrium—but an alternative is a setting in which such a state must arise froman unmediated interaction between colluding players, in which case players coulddeviate from any prescribed protocol that produces this state.4. Closely related to the notion of an unmediated interaction, one may consider gamesin which there is no referee. Coin-flipping may be cast as an example, and its evi-dently complicated structure suggests nothing less in a setting in which the goal is,perhaps, to produce a quantum state of interest.5. Generally speaking, can quantum game theory provide a foundation through whichone may discover quantum protocols having either theoretical or practical utility?
Acknowledgments
This research was undertaken thanks in part to funding from the Canada First ResearchExcellence Fund. We thank Sanketh Menda for helpful suggestions at an early stage ofthis work.
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