Quantum and non-signalling graph isomorphisms
We introduce a two-player nonlocal game, called the -isomorphism game, where classical players can win with certainty if and only if the graphs and are isomorphic. We then define the notions of quantum and non-signalling isomorphism, by considering perfect quantum and non-signalling strategies for the -isomorphism game, respectively. In the quantum case, we consider both the tensor product and commuting frameworks for nonlocal games. We prove that non-signalling isomorphism coincides with the well-studied notion of fractional isomorphism, thus giving the latter an operational interpretation. Second, we show that, in the tensor product framework, quantum isomorphism is equivalent to the feasibility of two polynomial systems in non-commuting variables, obtained by relaxing the standard integer programming formulations for graph isomorphism to Hermitian variables. On the basis of this correspondence, we show that quantum isomorphic graphs are necessarily cospectral. Finally, we provide a construction for reducing linear binary constraint system games to isomorphism games. This allows us to produce quantum isomorphic graphs that are nevertheless not isomorphic. Furthermore, it allows us to show that our two notions of quantum isomorphism, from the tensor product and commuting frameworks, are in fact distinct relations, and that the latter is undecidable. Our construction is related to the FGLSS reduction from inapproximability literature, as well as the CFI construction.
Given graphs and , an isomorphism from to is a bijection such that is adjacent to if and only if is adjacent to . When such an isomorphism exists, we say that and are isomorphic and write . The notion of isomorphism is central to a broad area of mathematical research encompassing algebraic and structural graph theory, but also combinatorial optimization, parameterized complexity, and logic. The graph isomorphism (GI) problem consists of deciding whether two graphs are isomorphic. It is a question with fundamental practical interest due to the number of problems that can be reduced to it. Additionally, the GI problem has a central role in theoretical computer science as it is one of the few naturally defined problems in NP which is not known to be polynomial-time solvable or NP-complete. While there is a deterministic quasipolynomial algorithm for the GI problem [B15], regardless of its worst case behavior, the problem can be solved with reasonable efficiency in practice (e.g. see [MP13]). In relation to the context of this paper, it is valuable to notice that the discussion around graph isomorphism has branched into the analysis of many equivalence relations that form hierarchical structures. Prominent instances are, for example, cospectrality, fractional isomorphism, etc. [B96, G13, VH03].
We remark here that, though we will touch on algorithmic aspects of the relations we define, this is not a paper about algorithms, and we make no claims that this work is useful for developing algorithms for the graph isomorphism problem. This work is concerned with theoretical aspects of some new and old relaxations of graph isomorphism.
Integer programming formulations.
As is the case for all constraint satisfaction problems, the GI problem can be formulated as an integer programming problem. Our next goal is to give two of these formulations as they are relevant to this work. The first one is an integer quadratic program (IQP) and the second one an integer linear program (ILP). We note that several recent developments concerning the graph isomorphism problem are based on hierarchies of linear programming relaxations of the ILP formulation for the GI problem we give below (e.g. see [atserias, GO12]).
Consider two graphs and with adjacency matrices and respectively. Recall that the adjacency matrix, , of a graph is a symmetric matrix whose rows and columns are indexed by , and such that if is adjacent to , and , otherwise. Throughout this work we will only consider undirected loopless simple graphs. In the IQP below, and throughout this work, we will use to denote the relationship of and , i.e., whether they are equal, adjacent, or distinct and non-adjacent. It is easy to verify that if and only if there exist real scalar variables for all such that following IQP is feasible:
The second integer programming formulation for the GI problem is based on permutation matrices, i.e., square -matrices with a single 1 in every row and column. Again, it is straightforward to verify that if and only if there exists an permutation matrix such that , or equivalently when the following ILP is feasible:
By the Birkhoff-von Neumann theorem, the convex hull of the set of permutation matrices is equal to the set of doubly stochastic matrices, i.e., entrywise nonnegative matrices where the sum of the entries in each row and column is equal to . This naturally suggests the following linear relaxation of the GI problem. We say that and are fractional isomorphic, and write , if there exists a doubly stochastic matrix such that . This defines an equivalence relation on graphs that has been studied in detail and characterized in multiple ways [fraciso].
In this work we focus on two natural matrix relaxations of (IQP) and (ILP). First, we consider (IQP), where we relax the scalar variables to Hermitian indeterminates . This leads to the following quadratic polynomial system in Hermitian variables:
For the matrix relaxation of (ILP), we replace the permutation matrix with a block matrix where each block is a orthogonal projector. Thus we consider the following program in Hermitian indeterminates :
Note that for the matrix is exactly a permutation matrix.
As we will see in Section LABEL:sec:quantum, the system () is feasible if and only if () is feasible. Thus the feasibility of () (or equivalently ()) corresponds to a “robust” relaxation of the notion of graph isomorphism which we call quantum isomorphism (see Definition 1.1).
Although the term “quantum isomorphism” might seem unmotivated at this point, as we will see in the next section, feasibility of () corresponds to a relaxation of graph isomorphism based on the existence of winning quantum strategies for a certain type of game. The relaxation makes use of the mathematical formalism of quantum theory and its definition requires physical resources available in quantum mechanics (see Theorem 2).
1.1 Nonlocal games
A two-party nonlocal game includes a verifier and two players, Alice and Bob, that devise a cooperative strategy. The game is defined in terms of finite input sets and finite output sets for Alice and Bob respectively, a Boolean predicate and a distribution on .
In the game, the verifier samples a pair using the distribution and sends to Alice and to Bob. The players respond with and respectively. We say the players win the game if .
The goal of Alice and Bob is to maximize their winning probability. In the setting of nonlocal games, the players are allowed to agree on a strategy beforehand, but they cannot communicate after they receive their questions. The parties only play one round of this game, but we will be concerned with strategies that win with certainty, i.e., probability equal to 1. We will refer to such a strategy as a winning or perfect strategy. Lastly, note that as we only consider perfect strategies we may assume without loss of generality that the distribution has full support.
Strategies for nonlocal games.
A deterministic classical strategy for a nonlocal game is one in which Alice’s response is determined by her input, and similarly for Bob. In a general classical strategy, the players may use shared randomness to determine their responses. They may also use local randomness, but this can be incorporated into the shared randomness without loss of generality.
In this paper we focus on another family of strategies where the players are allowed to use quantum resources to determine their answers. Specifically, a quantum strategy for a nonlocal game allows the players to determine their answers by performing joint measurements on a shared quantum state. A driving force behind the emerging field of quantum computing is that quantum nonlocal effects can lead to advantages for various distributed tasks, e.g. see [bellnonlocality]. We will introduce the mathematical formalism of quantum strategies for nonlocal games in Section 3.2.
In Section 3.3, we consider the family of strategies satisfying the non-signalling constraints (see Equation (8)). Intuitively, the non-signalling property says that Alice’s local marginal distributions are independent of Bob’s choice of measurement and, symmetrically, Bob’s local marginal distributions are independent of Alice’s choice of measurement. Thus, Alice cannot obtain any information about Bob’s input based on her input and output, and vice versa. This is the most general class of strategies we consider in this work.
For any of the above classes of strategies, the typical goal is to determine the maximum (or supremum) probability of winning a given nonlocal game. This is known as the classical/quantum/non-signalling value of the game.
The -isomorphism game.
Given two graphs and , we now define a nonlocal game which we call the -isomorphism game, with the intent of capturing and extending the notion of graph isomorphism. The -isomorphism game is played as follows: The verifier selects uniformly at random a pair of vertices and sends to Alice and to Bob respectively. The players respond with vertices . Throughout, we assume that and are disjoint so that players know which graph their vertex is from.
The first winning condition is that each player must respond with a vertex from the graph that the vertex they received was not from. In other words we require that:
If condition (1) is not met, the players lose. Assuming (1) holds we define to be the unique vertex of among and , and we define , and similarly. In order to win, the answers of the players must also satisfy the following conditions:
In other words, if Alice and Bob are given the same vertex, then they must respond with the same vertex. If they receive (non-)adjacent vertices they must return (non-)adjacent vertices. Also, assuming that Alice receives and Bob , if Alice outputs then in order to win we require that Bob returns . Note that we do not explicitly require that and have the same number of vertices.
In this work we use the -isomorphism game in order to capture and extend the notion of graph isomorphism. First, we show that there exists a perfect classical strategy for the -isomorphism game if and only if and are indeed isomorphic graphs. This suggests that by considering perfect quantum and non-signalling strategies for the -isomorphism game we may define the notions of quantum and non-signalling isomorphisms of graphs. Note that we will actually consider two different classes of quantum strategies: those from the tensor product framework of joint measurements and those from the commuting operator framework. However, we will mainly focus on the former class, and when we refer to quantum strategies we will be referring to these. We will use “quantum commuting strategies” to refer to the latter class of strategies. The detailed formalism of quantum strategies will be given in Section 2, and quantum commuting strategies will be introduced in Section LABEL:subsec:qcisos.
We say that two graphs and are quantum isomorphic/quantum commuting isomorphic/non-signalling isomorphic, and write / / , whenever there exists a perfect quantum/quantum commuting/non-signalling strategy for the -isomorphism game.
This idea of associating a nonlocal game to a constraint satisfaction problem corresponding to a certain graph property and studying its quantum and non-signalling value is not new. This was first done for the chromatic number of a graph in [Cameron07] and generalized to graph homomorphisms in [qhomos].
Since every classical strategy can be trivially considered as a quantum strategy and any quantum strategy is necessarily non-signalling (see Equation (7)), we have that
As we will see, neither of these implications can be reversed.
In Section 4 we focus on non-signalling isomorphism. Based on a result of Ramana, Scheinerman, and Ullman [fraciso] which relates fractional isomorphism to the existence of a common equitable partition, in Theorem LABEL:thm:ns1 we show the following:
For any graphs and we have that if and only if
It is worth noting that there is a polynomial time algorithm for determining if two graphs are fractionally isomorphic [fraciso]. Combined with Result 1 this implies that non-signalling isomorphism is also polynomial-time decidable. Furthermore, it is known that fractional isomorphism distinguishes almost all graphs [randiso], and so it follows that the same holds for non-signalling and quantum isomorphism, since the latter is a more restrictive relation.
In Section LABEL:sec:quantum we focus on quantum isomorphism. We show that perfect quantum strategies for the isomorphism game must take a special form. This allows us to reformulate quantum isomorphism in terms of the existence of a set of projectors satisfying certain orthogonality constraints. Based on this we can show the following:
Specifically, we prove the equivalence in Theorem LABEL:thm:qreform and in Theorem LABEL:lem:projperm. As a consequence of Result 2 it follows that quantum isomorphic graphs must be cospectral with cospectral complements. This allows us to conclude that quantum and non-signalling isomorphism are different relations, since there are many examples of graphs that are fractionally isomorphic but not cospectral (e.g. any pair of -regular graphs is fractionally isomorphic).
Lastly, in Section LABEL:sec:separation we consider the question of whether isomorphism and quantum isomorphism are different relations. In Theorem LABEL:thm:result3 we show that they are indeed distinct notions:
There exist graphs that are quantum isomorphic but not isomorphic.
The main ingredient in the proof of Result 3 is a reduction from linear binary constraint system (BCS) games, introduced by Cleve and Mittal [clevemittal], to isomorphism games. Specifically, we show that a linear BCS game has a perfect classical (quantum) strategy if and only if a pair of graphs constructed from the BCS are (quantum) isomorphic. Since there exist linear BCS games that have perfect quantum strategies but no perfect classical strategies, this allows us to produce pairs of graphs that are quantum isomorphic but not isomorphic. The smallest example of such a pair we are able to construct uses the Mermin magic square game, which produces two graphs on vertices each that are quantum isomorphic but not isomorphic (see Figures LABEL:fig:qiso1 and LABEL:fig:qiso2).
The same reduction as above can be used in the quantum commuting case, and thus we obtain that a given linear BCS game has a perfect quantum commuting strategy if and only if the corresponding pair of graphs are quantum commuting isomorphic. Using this reduction and two recent results of Slofstra [slofstra16], we are able to prove the following:
There exist graphs that are quantum commuting isomorphic but not quantum isomorphic. Furthermore, determining if two graphs are quantum commuting isomorphic is undecidable.
It will also follow from the above that quantum commuting isomorphism and non-signalling isomorphism are distinct relations, since the latter is polynomial time decidable by its equivalence with fractional isomorphism.
The standard basis of is denoted by , where . For a matrix we denote by its conjugate transpose and by its transpose. We denote the set of Hermitian operators by . Throughout this work we equip with the Hilbert-Schmidt inner product . A matrix is called positive semidefinite (psd) if for all . The set of psd matrices is denoted by . We use the fact that for two psd matrices we have that if and only if .
A matrix is called an (orthogonal) projector if it satisfies . We typically omit the term “orthogonal” because we will often refer to two projectors and being orthogonal (to each other) whenever they satisfy . We use the fact that for any family of projectors satisfying we have that , for all .
We denote by the set of block matrices whose blocks are matrices in . For any family of matrices we denote by the element of whose -block is equal to . The -block of a matrix is denoted by .
In this section we briefly review some basic concepts from the theory of quantum information. For additional details we refer the reader to [NC] and references therein.
To any quantum system one can associate a Hilbert space . The state of the system is described by a unit vector . Note that states that can be described in this way are actually known as pure states, and more generally the state of a quantum system is described by a Hermitian positive semidefinite matrix with trace equal to one. However, for quantum strategies for nonlocal games it suffices to consider only pure states, so we restrict our attention to this case.
One can obtain classical information from a quantum system by measuring it. For the purposes of this paper, the most relevant mathematical formalism of the concept of measurement is given by a Positive Operator-Valued Measure (POVM). A POVM consists of a family of Hermitian psd matrices such that , where is some integer and is the identity matrix. According to quantum mechanics, if the measurement is performed on a quantum system in state , then the probability that outcome occurs is . We say that a measurement is projective if all of the POVM elements are projectors. Note that for any set of projectors the condition implies that the ’s are mutually orthogonal. Therefore the POVM elements of any projective measurement are orthogonal to each other.
Consider two quantum systems and with corresponding state spaces and respectively. The state space of the joint system is given by the tensor product . Moreover, if the system is in (pure) state and is in (pure) state then the joint system is in state . Not every state in the joint system space can be written as a tensor product. States that cannot be written as a tensor product are known as entangled states. It is the existence of entangled states that allows for quantum advantage in nonlocal games and many other scenarios.
If and define measurements on the individual systems and then the family of operators defines a product measurement on the joint system . The probability of getting outcome , when measuring the quantum state , is equal to .
It is often convenient to use the fact that any quantum state admits a so-called Schmidt decomposition: where and are orthonormal bases of and for all . The bases and are known as the Schmidt bases of , and the are its Schmidt coefficients. We say that has full Schmidt rank if its Schmidt coefficients are all positive. Note that one can also consider a Schmidt decomposition of states in where , but for us it suffices to consider .
We say that a state is maximally entangled if all of its Schmidt coefficients are the same. The canonical maximally entangled state in is the state , where is the standard basis vector. We will make use of the fact that
3 Strategies for the -isomorphism game
In this section we introduce three families of strategies for the -isomorphism game (classical, quantum, and non-signalling) and study how they relate to each other.
Given a fixed strategy for the -isomorphism game, we denote by the joint conditional probability of Alice and Bob responding with and upon receiving inputs and respectively. We call such a joint conditional probability distribution a correlation. Let be a strategy for a nonlocal game and let be the corresponding correlation. An easy but important observation is that is a perfect strategy if and only if whenever do not meet the winning conditions of the game, i.e.,
In particular, if we specialize (4) to the -isomorphism game we have that the correlation corresponds to a perfect strategy if and only if
As a consequence we have that any winning strategy for the -isomorphism game is also a winning strategy for the -isomorphism game, as well as the -isomorphism game.
3.1 Classical Strategies
In a classical strategy, Alice and Bob are allowed to make use of shared randomness to determine how they respond. Note that this does not allow them to communicate. They may also use local randomness, but this can be incorporated into the shared randomness without loss of generality. Formally, this means that the correlation associated to a classical strategy has the form where the ’s encode the shared randomness and satisfy , and , and the are correlations arising from deterministic classical strategies, i.e., for each , for all . Since whether a correlation corresponds to a winning strategy is determined by its zeros, the correlation arises from a winning strategy if and only if is winning for all . Thus we can consider the deterministic strategy corresponding to . A deterministic classical strategy amounts to a pair of functions, , which map inputs to outputs for each of Alice and Bob respectively. Assuming the strategy is winning, we have that , and that for all . Since , we will refer to both of them as simply . For , the winning conditions of the -isomorphism game require that . This implies that the restriction of to is an isomorphism from to an induced subgraph of . Similarly, the restriction of to is an isomorphism of to an induced subgraph of . This is only possible if and are isomorphic and the above two restrictions of are isomorphisms. Finally, for , let . The case where Alice is sent and Bob is sent allows us to conclude that . Since , the relationship between these vertices is ‘equality’, thus . In other words, the restriction is the inverse of the restriction .
The above shows that any winning deterministic strategy for the -isomorphism game corresponds to Alice and Bob responding according to a fixed isomorphism between and . Moreover, any classical strategy can be decomposed as a probabilistic (or convex) combination of deterministic strategies. Therefore, classical players can win the -isomorphism game only if and are indeed isomorphic.
Conversely, suppose that is an isomorphism of graphs and . It is easy to see that if both players respond with upon receiving and respond with upon receiving , then they will win the -isomorphism game. So we see that there exists a winning classical strategy for the -isomorphism game if and only if and are indeed isomorphic graphs.
3.2 Quantum Strategies
A quantum strategy for the -isomorphism game consists of a shared entangled state , and POVMs for each for Alice, and for each for Bob. Upon receiving Alice performs measurement and obtains some outcome . Similarly, upon receiving Bob measures and obtains some . The probability of Alice and Bob outputting vertices and upon receiving and respectively is given by
Any correlation that can be realized as in (6) is known as a quantum correlation.
Therefore, it follows by (5) that a quantum strategy as described above is a winning strategy for the -isomorphism game if and only if
It is important to note that any classical correlation is also a quantum correlation. Indeed, any deterministic strategy can be produced by using measurements in which all but one of the POVM elements is the zero matrix. The remaining POVM element will be the identity and performing this measurement will always result in the outcome corresponding to the identity. Since any classical shared randomness can also be replicated by measurements on a shared state, this shows that any classical correlation can be produced by some quantum strategy.
3.3 Non-signalling Strategies
Suppose that Alice and Bob are playing a nonlocal game with a quantum strategy as described in the previous section. If Alice is given input , and Bob is given input , the probability that Alice obtains outcome when she performs measurement is given by:
and we see that this does not depend on Bob’s input . Similarly, the probability of Bob obtaining a particular outcome will not be dependent on Alice’s input. This property of quantum correlations is known as the non-signalling property. Formally, a correlation is non-signalling if
In other words, a non-signalling correlation does not allow the two parties to send information between themselves. If it is the case that nothing, including information, can travel faster than the speed of light, then any correlation produced by sufficiently distant parties must be non-signalling. More specifically, if Alice and Bob are separated by a large enough distance, and they are required to respond to the verifier quickly enough, then we can be certain that their correlation is non-signalling.
As we have seen, all quantum correlations are non-signalling. However, the converse is not true. For instance, for input and output sets equal to , the PR box [PR] is the correlation given by:
One can check that this correlation is non-signalling, but it is well known [PR] that it cannot be implemented by any quantum strategy.
A general non-signalling correlation may not be physically realizable, so when we refer to non-signalling strategies, we can think of this as Alice and Bob each simply having some black box where they enter their inputs into and which gives them their outputs. We only require that the resulting correlation produced by these boxes obeys the non-signalling condition.
Any correlation that is not non-signalling allows Alice and Bob to communicate some information. However, this violates the definition of a nonlocal game, since one of the requirements is that the players are not allowed to communicate during the game. Thus, one of the reasons for considering non-signalling correlations is that they represent the largest class of admissible correlations for nonlocal games. More practically, since the non-signalling condition is linear, these correlations often provide tractable upper bounds on the power of quantum correlations. Indeed, in the next section we will see that we can completely characterize when two graphs are non-signalling isomorphic.
4 Non-signalling Isomorphism
Our goal in this section is to show Result 1, i.e., that fractional isomorphism and non-signalling isomorphism are equivalent relations.
4.1 Non-signalling isomorphism implies fractional isomorphism
To show that non-signalling isomorphism implies fractional isomorphism we show that one can use a non-signalling correlation that wins the -isomorphism game to construct a doubly stochastic matrix satisfying .
First, if is a winning non-signalling correlation for the -isomorphism game, then we must have that whenever , and similarly when we replace by and/or switch Alice and Bob’s positions. Furthermore, for all we have that if , and similarly with replaced by . Therefore, we have the following observation:
Our goal is to use (9) to construct the desired doubly stochastic matrix. Specifically, the above sums will correspond to its row and column sums. We need the following intermediate result.
Let be a winning non-signalling correlation for the -isomorphism game. Then,
for all , .
Set . For all and we have that
Lastly, by the symmetry of and we also have that . Putting everything together the lemma follows.∎
which was not obvious even for quantum strategies.
We can now show that two graphs which are non-signalling isomorphic are necessarily fractionally isomorphic.
If and are non-signalling isomorphic, then they are fractionally isomorphic.