A New Perspective on the Average Mixing Matrix

We consider the continuous-time quantum walk defined on the adjacency matrix of a graph. At each instant, the walk defines a mixing matrix which is doubly-stochastic. The average of the mixing matrices contains relevant information about the quantum walk and about the graph. We show that it is the matrix of transformation of the orthogonal projection onto the commutant algebra of the adjacency matrix, restricted to diagonal matrices. Using this formulation of the average mixing matrix, we find connections between its rank and automorphisms of the graph.

The continuous-time quantum walk is an important object of study in quantum computing since it is an universal computational primitive [4]. They were first studied in [6]. Since then, many aspects of quantum walks have been studied, including state transfer [11,7,5,13], uniform mixing [3,9,1] and average mixing [8,2]. We note that U(t) is unitary and symmetric. If the initial state of the system is given by a density matrix D, then the state of the system at time t is U(t)DU(−t).
We note that we do not have direct access to states, all that experiment provides is the value of expressions of the form tr(DP r ) where P 1 , . . . , P n are positive semidefinite matrices such that r P r = I. Because of this we find ourselves using the trace inner product on Mat m×n (C):

M, N = tr(M T N).
Unlike classical random walks on a connected graph, a continuous quantum walk does not reach a steady state. Thus, if the eigenvalues of X are integers (e.g., if X is the complete graph K n ) then U(t) is a periodic function of t. The focus of this paper, the mixing matrix M of X provides a useful substitute for a steady state.
There are two convenient ways to define M . The first is to introduce what we call the mixing matrices M(t) of the walk, given by (Here we use M • N to denote the Schur or element-wise product of two matrices of the same order.) Since U(t) is unitary and symmetric, we see that M(t) is symmetric non-negative matrix with each row and column summing to one. We can now define To see that this definition makes sense, we recall that since A is real and symmetric, it has a spectral decomposition where θ r runs over the distinct eigenvalues of A and E r is the matrix that represents orthogonal projection onto the θ r -eigenspace of A. Given this we also have U(t) = r e itθr E r and, as U(t) = U(−t), It follows that the limit on the right side of (1) is exists and is equal to that is, the limit is the sum of the Schur squares of the spectral idempotents of A.
We consider one example. The spectral idempotents of the complete graph K n are 1 n J, I − 1 n J, (with corresponding eigenvalues n − 1 and −1) and hence its average mixing matrix is For large n, this is very close to I. This is unexpected, but we are dealing with quantum physics, where the unexpected is not uncommon. If A ∈ Mat n×n (R), its commutant Comm(A) is the set of matrices that commute with A. This is an object of some combinatorial interest-the permutation matrices in Comm(A) are the automorphisms of A. The first main result of this paper is that, relative to a natural basis, M is the matrix that represents the restriction to the diagonal matrices of the orthogonal projection from Mat n×n (R) onto Comm(A). We then use this connection to investigate the relations between properties of the graph X and the rank of M .

Projection onto the commutant of the adjacency matrix
Let A be a n × n real symmetric matrix with columns and rows indexed by elements of a set V . We denote by Mat n×n (R) the set of n × n matrices with entries in R. Let the spectral decomposition of A be given as follows: where θ 1 , . . . , θ m are the distinct eigenvalues of A and E r is the projection matrix onto the θ r eigenspace of A. We denote by Comm(A) the set of all real matrices which commute with A. Proof. When A is diagonal, this is immediate. Since A is similar to a diagonal matrix, the lemma follows.
We consider a map Ψ : R n×n → Comm(A) such that for M ∈ R n×n . We see that Ψ is an endomorphism of R n×n .
2.2 Lemma. The following are true: Hence each element of Comm(A) lies in the image of Ψ and therefore im(Ψ) = Comm(A). Finally from which it follows that Ψ is self-adjoint.

Corollary. The map Ψ is the orthogonal projection of the n × n real matrices onto Comm(A).
Proof. This is immediate from the fact that Ψ is idempotent and selfadjoint-but we simply note that for any two n × n matrices M and N The following is a standard fact in linear algebra, see e.g., [10, Lemma 4.3.1].

Lemma.
For B, C, N ∈ R n×n , we have that From this we see that, relative to the standard basis of Mat n×n (R), the matrix that represents Ψ is r E r ⊗ E r . (Since tr(E r ) = m r it follows that tr(Ψ) = r m 2 r ; since P is idempotent tr(P ) = rk(P ) and thus we have a second proof of Lemma 2.1.)

Average states
Recall that a state D is a positive semidefinite matrix with trace 1. If D is the initial state of a continuous quantum walk, then the state D(t) at time t is Using the spectral decomposition of U(t), we have We call this limit (or sum) an average state and denote it by Ψ(D). We see at once that Ψ(D) is equal to the orthogonal projection of D onto Comm(A).
Since D is positive semidefinite and the idempotents E r are symmetric, we see that Ψ(D) is a positive semidefinite matrix. As we also see that Ψ(D) is a density matrix.
In the context of quantum walks on graphs, there is a natural class of density matrices we will focus on. If a ∈ V (X), let e a denote the standard basis vector of C V (X) indexed by a and define D a = e a e T . Then certainly D a is a density matrix, moreover rk(D a ) = 1 and D 2 a = D a . (Physicists refer to a density matrix with rank 1 as a pure state.) The following theorem gives one reason why average states are of interest.
3.1 Theorem. Let X be a graph. The average mixing matrix is the Gram matrix of the average states Ψ(D a ) for a in V (X).
Proof. Our claim is that, if a, b ∈ V (X), then and the theorem follows.
3.2 Corollary. The dimension of the space spanned by the average state Ψ(D a ) for a ∈ V (X) is equal to rk( M).
3.3 Corollary. The average mixing matrix is completely positive semidefinite.

Diagonal matrices in the commutant
In this section, we consider the restriction of Ψ to the set of n × n diagonal matrices. Let D be the set of n × n diagonal matrices. The standard basis of D is given by {D a } a∈V where D a := e a e T a . The map that sends a matrix M to Ψ(M) • I is a linear map from Mat n×n (R) into the space D of diagonal matrices. We denote the restriction of this map to D by Φ; it is evidently an endomorphism of D. Proof. The entries of the matrix representing Ψ are given by the the inner products D a , Ψ(D b ) for vertices a and b of X. We have Hence Observe that A 0 is always non-empty, since A ∈ A 0 . Also I ∈ Ψ(D), and therefore the direct sum decomposition in our next result is always nontrivial.

The rank of some average mixing matrices
There are a number of graph invariants that can be constructed from the average mixing matrix. In this section we focus on rk( M). From [8] we know that if rk( M ) = 1, then X is K 1 or K 2 .
Our first two results concern graphs with only simple eigenvalues; we note that almost all graphs have this property [12]. Proof. Since the eigenvalues of X are simple, dim(Comm(A)) = n and since A ∈ A 0 , it follows from Lemma 4.3 that dim(Ψ(D)) ≤ n − 1.
Suppose X is regular with valency k. Then the matrices A, J − I and A 2 − kI all lie in A 0 . If these matrices are linearly dependent, the minimal polynomial of A has degree at most two and hence A has at most two eigenvalues. Since X has n distinct eigenvalues, the minimal polynomial of A over 1 ⊥ has degree n − 1. Thus n − 1 ≤ 2, which contradicts our assumption that n ≥ 4. Table 1 shows the number of graphs on n vertices whose mixing matrices have given ranks. The only regular graph with simple eigenvalues on up to 8 vertices is K 2 , so the data for regular graphs is not included in this table. Instead, we count cubic graphs with simple eigenvalues on 10 to 18 vertices, as presented in Table 2.

Lemma.
Assume X is a graph on n vertices with only simple eigenvalues. If X is bipartite, then rk( M ) ≤ ⌊(n + 1)/2⌋.
Proof. Note that A k ∈ A 0 if and only if k is odd, and for all k < n, these matrices are independent. Hence dim A 0 ≥ ⌊n/2⌋, and the result follows from Lemmas 4.1 and 4.3. Table 3 shows the number of bipartite graphs on n vertices whose mixing matrices have given ranks. Proof. We identify the automorphism group of X with the set of permutation matrices that lie in Comm(A). If a ∈ S, let P a be the automorphism of X with a as its only fixed point. From Lemma 4.3, for each P a , there is a matrix in Ψ(D) with the same diagonal. These matrices and the identity matrix form a set of linearly independent matrices in Ψ(D).
Some of our numerical data indicates that for most graphs on n vertices with simple eigenvalues, the average mixing matrix has rank n − 1. In view of Theorem 4.3, this implies that, for these graphs, dim A 0 = 1. Therefore any matrix that commutes with A and has zero diagonal must be a scalar multiple of A. The following corollary is an immediate consequence.

Corollary.
Suppose X is a connected graph with at least three vertices. If all eigenvalues of X are simple and rk( M ) = n−1, then any automorphism of X has fixed points.

Open problems
As the theory of the average mixing matrix is relatively new, there are many interesting problems one can ask. We discuss a few below.
One interesting question about the average mixing matrix concerns the non-negative rank of the average mixing matrix. The non-negative rank of a non-negative n × n matrix A is the least number k, such that there are k matrices {M r } k r=1 of rank 1 with non-negative entries, such that A = k r=1 M r . If X is a graph on n > 1 vertices with simple eigenvalues, Lemma 5.1 gives that M has rank at most n − 1. One can ask when the non-negative rank of M is equal to n.
We may interpret the (v, v) diagonal entry of M as the average probability of measuring at vertex v, after starting at vertex v. Thus questions about the trace of M are very natural. In particular, it is interesting to ask how graph invariants correspond to the trace of M. Following in the vein of the questions about rank and trace, one can ask if the spectrum of M determines any graph properties. This gives rise to many natural questions about the average mixing matrix.