Relation between quantum walks with tails and quantum walks with sinks on finite graphs

We connect the Grover walk with sinks to the Grover walk with tails. The survival probability of the Grover walk with sinks in the long time limit is characterized by the centered generalized eigenspace of the Grover walk with tails. The centered eigenspace of the Grover walk is the attractor eigenspace of the Grover walk with sinks. It is described by the persistent eigenspace of the underlying random walk whose support has no overlap to the boundaries of the graph and combinatorial flow in the graph theory.


Introduction
A simple random walker on a finite and connected graph starting from any vertex hits an arbitrary vertex in a finite time. This fact implies that if we consider a subset of the vertices of this graph as sinks, where the random walker is absorbed, then the survival probability of the random walk in the long time limit converges to zero. However, for quantum walks (QW) [1] the situation is more complicated and the survival probability depends in general on the graph, coin operator and the initial state of the walk. For a two-state quantum walk on a finite line with sinks on both ends and a non-trivial coin the survival probability is also zero, as shown by the studies of the corresponding absorption problem [2,3,4,5]. However, for a three-state quantum walk with the Grover coin [6] the survival probability on a finite line is non-vanishing [7] due to the existence of trapped states. These are the eigenstates of the unitary evolution operator which do not have a support on the sinks. Trapped states crucially affect the efficiency of quantum transport [8] and lead to counter-intuitive effects, e.g. the transport efficiency can be improved by increasing the distance between the initial vertex and the sink [9,10]. We find a similar phenomena to this quantum walk model in the experiment on the energy transfer of the dressed photon [11] through the nanoparticles distributed in a finite three dimensional grid [12]. The output signal intensity increases when the depth direction is larger. Although when the depth is deeper, a lot of "detours" newly appear to reach to the position of the output from the classical point of view, the output signal intensity of the dressed photon becomes stronger. The existence of trapped states also results in infinite hitting times [13,14].
In this paper we analyse such counter-intuitive phenomena for the Grover walk on general connected graph using the spectral analysis. The Grover walk is an induced quantum walk of the random walk from the view point of the spectral mapping theorem [15].
To this end, first we connect the Grover walk with sink to the Grover walk with tails. The tails are the semi-infinite paths attached to a finite and connected graph. We call the set of vertices connecting to the tails the boundary. The Grover walk with tail is introduced by [16,17] in terms of the scattering theory. If we set some appropriate bounded initial state so that the support is included in the tail, the existence of the fixed point of the dynamical system induced by the Grover walk with tails is shown, and the stable generalized eigenspace H s , in which the dynamical system lives, is orthogonal to the centered generalized eigenspace H c [26] at every time step [19]. The centered generalized eigenspace is generated by the generalized eigenvectors of the principal submatrix of the time evolution operator of the Grover walk with respect to the internal graph, and all the corresponding absolute values of the eigenvalues are 1. This eigenstate is equivalent to the attractor space [8] of the Grover walk with sink. Indeed, we show that the stationary state of the Grover walk with sink is attracted to this centered generalized eigenstate. Secondly, we characterize this centered generalized eigenspace using the persistent eigenspace of the underlying random walk whose supports have no overlaps to the boundary and also using the concept of "flow" from the graph theory. From this result, we see that the existence of the persistent eigenspace of the underlying random walk influences significantly the asymptotic behavior of the corresponding Grover walk, although it has little effect on the asymptotic behavior of the random walk itself. Moreover, we clarify that the graph structure which constructs the symmetric or antisymmetric flow satisfying the Kirchhoff's law contributes to the non-zero survival probability of the Grover walk as suggested by [15,8].
This paper is organized as follows. In section 2, we prepare the notations of graphs and give the definition of the Grover walk and the boundary operators which are related to the chain. In section 3, we give the definition of the Grover walk on a graph with sinks. In section 4, a necessary and sufficient condition for the surviving of the Grover walk are described. In section 5, we give an example. Section 6 is devoted to the relation between the Grover walk with sink and the Grover walk with tail. In section 7, we partially characterize the centered generalized eigenspace using the concept of flow from the graph theory.

Graph notation
Let G = (V, A) be a connected and symmetric digraph such that an arc a ∈ A if and only if its inverse arc a ∈ A. The origin and terminal vertices of a ∈ A are denoted by o(a) ∈ V and t(a) ∈ V , respectively. Assume that G has no multiple arcs. If t(a) = o(a), we call such an arc a the self-loop. In this paper, we regard a = a for any self-loops. We denote A σ as the set of all the self-loops. The degree of v ∈ V is defined by The support edge of a ∈ A \ A σ is denoted by |a| with |a| = |a|. The set of (non-directed) edges is A walk in G is a sequence of arcs such that p = (a 0 , a 1 , . . . , a r−1 ) with t(a j ) = o(a j+1 ) for any j = 0, . . . , r − 2, which may have the same arcs in p. The cycle in G is a subgraph of G which is isomorphic to a sequence of arcs (a 0 , a 1 , . . . , a r−1 ) (r ≥ 3) satisfying t(a j ) = o(a j+1 ) with a j = a j+1 for any j = 0, . . . , r − 1, where the subscript is the modulus of r. We identify (a k , a k+1 , . . . , a k+r−1 ) with (a 0 , a 1 , . . . , a r−1 ) for k ∈ Z. The spannig tree of G is a connected subtree of G covering all vertices of G. A fundamental cycle induced by the spanning tree is the cycle in G generated by recovering an arc which is outside of the spanning tree to the spanning tree. There are two choices of orientations for each support of the fundamental cycle, but we choose only one of them as the representative. Fixing a spanning tree, we denote the set of fundamental cycles by Γ. Then the cardinality of Γ is |E| − |V | + 1 =: b 1 . We call b 1 the first Betti number.

Definition of the Grover walk
Let Ω be a discrete set. The vector space whose standard basis is labeled by each element of Ω is denoted by C Ω . The standard basis is denoted by δ Throughout this paper, the inner product is standard, i.e., for any ψ, φ ∈ C Ω , and the norm is defined by For any ψ ∈ C Ω , the support of ψ is defined by For subspaces M, N ⊂ C Ω , the relation means that M and N are complementary spaces in C Ω , i.e., for any f ∈ C Ω , g ∈ M and h ∈ N are uniquely determined such that f = g + h; which means if u ′ + u ′′ = 0 for some u ′ ∈ Ω ′ and u ′′ ∈ Ω ′′ , then u ′ and u ′′ must be u ′ = u ′′ = 0. Note that g, h Ω = 0 in general, i.e., M and N are not necessarily orthogonal subspaces. Especially in this paper, we treat an operator which is a submatrix of a unitary operator, and we are not ensured that it is a normal operator. The vector space describing the whole system of the Grover walk is C A . The time evolution operator of the Grover walk on G is defined by for any ψ ∈ C A and a ∈ A. Note that since U G is a unitary operator on C A , U G preserves the ℓ 2 norm, i.e., ||U G ψ|| 2 A = ||ψ|| 2 A . Let ψ n ∈ C A be the n-th iteration of the Grover walk ψ n = U G ψ n−1 (n ≥ 1) with the initial state ψ 0 . Then the probability distribution at time n, µ n : V → [0, 1], can be defined by if the norm of the initial state is unity. Our interest is the asymptotic behavior of the sequence of probabilities µ n and also of amplitudes ψ n on the graph comparing with the behavior of the corresponding random walk.

Boundary operators
Let G = (V, A) be the original graph. The set of sinks is denoted by V s ⊂ V . The subgraph of G; G 0 = (V 0 , A 0 ), is defined by The set of self-loops in G 0 is denoted by A 0,σ ⊂ A 0 . See Fig 1. The set of the fundamental cycles in G 0 is denoted by Γ hereafter. The set of boundary vertices of G 0 is defined by Under the above settings of graphs, let us now prepare some notations to show our main theorem.
Definition 1. Letd(u) be the degree of u in the original graph G. Let G 0 = (V 0 , A 0 ) be the subgraph as above. Then the boundary operators d 1 : respectively, for any ψ ∈ C A , Ψ ∈ C Γ and v ∈ V 0 , a ∈ A 0 . Here A(c) is the set of arcs of c ∈ Γ.
Note thatd(u) is the degree of G, so if u ∈ δG 0 , thend(u) is greater than the degree in G 0 . The adjoint operators of d 1 and ∂ 2 are defined by Let S : C A 0 → C A 0 be a unitary operator defined by (Sψ)(a) = ψ(a). We prove that the composition of d 1 (I − S) • ∂ 2 is identically equal to zero as follows.
Lemma 2.1. Let d 1 and ∂ 2 be the above. Then we have Proof. For any c ∈ Γ, let δ (Γ) c ∈ C Γ be the delta function, i.e., Then it is enough to see that which is the desired conclusion.
Let us set the function ξ for all a ∈ A 0 . The adjoint χ * S : The function ξ (+) c satisfies the following properties: Proof. The following direct computation gives the consequence: Here the first equality derives from the definition of U G . In the second equality, since supp(ξ (+) c ) ⊂ A 0 ⊂ A and the summation of RHS in the first equality are essentially the same as the one over A 0 , we can apply the definition of d 1 to this. We used Lemma 2.1 in the last equality.
The self-adjoint operator Here the matrix representation of P ′ is described by for any u, v ∈ V 0 . If T f = xf and T g = yg (x = y), then we find the orthogonality such that

Definition of the Grover walk on graphs with sinks
Let G = (V, A) be a finite and connected graph with sinks For simplicity, in this paper we consider the initial state of the Grover walk φ 0 that satisfies the condition supp(φ 0 ) ⊂ A 0 . * The time evolution of the Grover walk with sinks V s with such an initial state φ 0 is defined by This means that a quantum walker at a sink falls into a pit trap. We are interested in the survival probability of the Grover walk defined by It is the probability that the quantum walker remains in the graph without falling into the sinks forever. Considering the corresponding isotropic random walk with sinks such that we find that its survival probability is zero because the first hitting time of a random walk to an arbitrary vertex for a finite graph is finite. On the other hand, in the case of the Grover walk the survival probability becomes positive, up to the initial state. In this paper, we clarify a necessary and sufficient condition for γ > 0.

Main theorem
We consider the case study on G 0 by Case A: A 0,σ = ∅ and G 0 is a bipartite graph; Case B: A 0,σ = ∅ and G 0 is a non-bipartite graph; Case C: A 0,σ = ∅ and G 0 \ A 0,σ is a bipartite graph; we can reproduce the QW with this initial state after n ≥ 1 by our setting.
For a subspace H ⊂ C A 0 , the projection operator onto H is denoted by Π H . Then we obtain the following theorem.
Theorem 4.1. Let φ n be the n-th iteration of the Grover walk on G = (V, A) with sinks. Let the survival probability at time n be defined by The subspaces A, B, C, D of C A 0 are defined in (7.7),...,(7.10), respectively. Then we have (1) lim n→∞ γ n = γ exists; (2) The survival probability γ is expressed by From this Theorem, we obtain useful sufficient conditions for non-zero survival probability as follows.
Corollary 4.1. Assume G 0 is a finite and connected graph. If G 0 is not a tree or G 0 has more than 2 self-loops, then γ > 0.

Example
Let us consider a simple example in Fig. 1 This graph fits into Case C. So let q be the closed walk by q = (a 1 , a 2 , a 3 , a 4 ) and q ′ be the walk between two selfloops by (b 1 , a 1 , a 2 , b 2 ). Then The matrix representation of the self adjoint operator T is expressed by Then we have It holds that Eϕ ± = λ ± ϕ ± . We obtain After the Gram Schmidt procedure to C, we have Fig. 2; we express the functions ϕ ± , ξ Then the time evolution of the asymptotic dynamics of this quantum walk is described by Finally, for example, if the initial state is ϕ 0 = δ b 1 , then, the survival probability can be computed by The second equality derives from the fact that the orthonormalized eigenvectors in the centered generalized eigenspace which have an overlap with the self-loop b 1 are given by 6 Relation between Grover walk with sinks and Grover walk with tails 6.1 Grover walk on graphs with tails Let G = (V, A) be a finite and connected graph with the set of sinks V s ⊂ V . We introduce the infinite graphG = (Ṽ ,Ã) by adding the semi-infinite paths to each vertex of δV = {v 1 , . . . , v r }, that is, Here P i 's is the semi-infinite paths named the tail whose origin vertex is identified with v i (i = 1, . . . , r). See Fig. 1. Recall that G 0 = (V 0 , A 0 ) is the subgraph of G eliminating the sinks V s . Recall also that χ S : for all a ∈ A 0 . In the same way, we newly introduce χ T : CÃ → C A 0 by (χ T φ)(a) = φ(a) Figure 2: The centered eigenspace of the example: The centered eigenspace to which Grover walk with sinks asymptotically belongs in this example is T ⊕ K ⊕ C. Each weighted sub-digraph represents a function in C A 0 ; the complex value at each arc is the returned value of the function. Each eigenspace; T , K and C, is spanned by the functions represented by these weighted sub-digraphs.
for all a ∈ A 0 . The adjoint χ * T : The following theorem was proven in [19].
Here j(·) is the electric current flow on the electric circuit assigned the resistance value 1 at each edge, that is, j(·) satisfies the following properties: with the boundary conditions j(e i ) = α i − α 1 + · · · + α r r (6.5) for any e i (i = 1, . . . , r) such that t(e i ) = v j and o(e i ) ∈ V (P i ).
Remark 6.2. The function ξ and Kirchhoff 's current and voltage laws if the internal graph G 0 is not a tree, while it does not satisfy the boundary condition (6.5) because the support of this function χ * T ξ (+) c has no overlaps to the tails but is included in the fundamental cycle c in the internal graph G 0 .

Relation between Grover walk with sinks and Grover walk with tails
Let us consider the Grover walk on G with sinks V s and with the initial state ψ (S) 0 ∈ C A . We describe U G as the time evolution operator of Grover walk on G. The n-th iteration of this walk following (3.3) is denoted by ψ (S) n . Let us also consider the Grover walk onG with the tails and with the "same" initial state n−1 . Then we obtain a simple but important relation between QW with sinks and QW with tails. Lemma 6.1. Let the setting of the QW with sinks and QW with tails be as the above. Then for any time step n, we have because of the setting. Note that χ * J χ J is the projection operator onto C A 0 while χ J χ * J is the identity operator on C A 0 (J ∈ {S, T }). Since ψ Since the support of the initial state is included in the internal graph, the inflow never come into the internal graph from the tail for any time n, which implies n , in the same way as ψ (S) n , we have Therefore χ S ψ (S) n and χ T ψ (T ) n follow the same recurrence and have the same initial state which means χ S ψ (S) n = χ T ψ (T ) n for any n ∈ N. Corollary 6.1. Let the initial state for the Grover walk with sinks be φ 0 with supp(φ 0 ) ⊂ A 0 . The survival probability γ can be expressed by where τ n is the outflow of the QW with tails from the internal graph G 0 , i.e., Remark 6.3. The time evolution for φ (T ) n is given by 0 . In this case, the inflow is ρ = 0. On the other hand, in the setting of Theorem 6.1, ρ is given by a nonzero constant vector.
Let us now consider a QW with tails with a general initial state Ψ 0 ∈ CÃ onG. We denote ν = χ T Ψ 0 and ρ = χ T UG(1 − χ * χ)Ψ 0 . We summarize the relation between a QW with sinks and a QW for the setting of Theorem 6.1 in the table from the view point of a QW with tails. From the above discussion, we see the importance of the spectral decomposition E = χ S U G χ * S = χ T UGχ * T , to obtain both limit behaviors. The operator E is no longer a unitary operator and, moreover, it is not ensured that it is diagonalizable. The centered generalized eigenspace of E is defined by Let H s be defined by Here "⊕" means H c and H s are complementary spaces, that is, if u c + u v = 0 for some u c ∈ H c and u v ∈ H s , then u c and u v must be u c = u v = 0. Note that since E is not a normal operator on a vector space H c ⊕ H s , it seems that in general u c , u v = 0 for u ∈ H c and H s ∈ N. However, we can see some important properties of the spectrum of E in the following proposition.

Proposition 7.1 ([19]).
(1) For any λ ∈ Spec(E), it holds that |λ| ≤ 1, i.e., (2) Let P c be the projection operator on H c along with H s ; that is, P c E = EP c and P 2 c = P c . Then P c is the orthogonal projection onto H c , i.e., P c = P * c . (3) The operator E acts as a unitary operator on H c , that is, H c = ⊕ |λ|=1 ker(λ − E) and U G χ * S ϕ = λχ * S ϕ for any ϕ ∈ ker(λ − E) with |λ| = 1. We call H c and H s the centered eingenspace and the stable eigenspace [26], respectively. (1) The state χ T ψ n in Theorem 6.1 belongs to H s for any time step n ∈ N.
(2) The state of QW with sinks; χ S φ n , asymptotically belongs to H c in the long time limit n.
Proof. The inflow ρ = χ * Uψ 0 is orthogonal to H c by a direct consequence of Lemma 3.5 in [19], which implies E n ρ ∈ H s for any n ∈ N by Proposition 7.1. Since the stationary state of part 1 is described by the limit of the following recurrence χ T ψ n = Eχ T ψ n−1 + ρ, χ T ψ 0 = 0, we obtain the conclusion of part 1. On the other hand, let us consider the proof of part 2 in the following. The time evolution in G 0 obeys χ S φ n = Eχ S φ n−1 . The overlap of χ S φ n to the space H s decreases faster than polynomial times because all the absolute value of the generalized eigenvalues of H s are strictly less than 1. (See Proposition 7.3 for more detailed order of the convergence.) Then only the contribution of the centered eigenspace, whose eigenvalues lie on the unit circle in the complex plain, remains in the long time limit.
Let W = P c E = EP c = P c EP c be the operator restricted to the centered eigenspace H c . Then we have lim for any a ∈ A 0 uniformly by Proposition 7.2. This means that in the long time limit, the time evolution is reduced to W which is a unitary operator on H c . Proposition 7.3. The survival probability is re-expressed by The convergence speed † is estimated by O(n κ r n max ), where κ = dim H s , r max = max{|λ| ; λ ∈ Spec(E), |λ| < 1}.
Proof. Putting E(1 − P c ) = W ′ , we have W + W ′ = E, W W ′ = 0, by Proposition 7.1 (2). Note that the operator E n is similar to with some natural numbers k λ 's. Here J(λ; k) is the k-dimensional matrix by We obtain that the survival probability at each time n is described by In the third equality we have used the fact that U G is unitary, the last equality follows from Corollary 7.1. The second term decreases to zero by Proposition 7.1 (2) with the convergence speed at least O(n κ r n max ) because the Jordan matrix J(λ; k) can be estimated by J(λ; k) n = O(n k |λ| n ). Hence, we find for γ n γ n = ||W n−1 χ S φ 0 || 2 + O(n κ r n max ) (n >> 1) where in the second equality we have used that W = W P c and the last equality follows from Proposition 7.1 (3).
Therefore, the characterization of H c is important to obtain the asymptotic behavior of φ n .
In the following, we consider the characterization of ker(±1 − E) using some walks on graph G 0 up to the situations of the graph; cases (A)-(D). First we prepare the following notations. For each support edge e ∈ E 0 , there are two arcs a and a such that |a| = |a|. Let us choose one of the arcs from each e ∈ E 0 and denote A + as the set of selected arcs. Then |A + | = |E 0 | and a ∈ A + if and only if a / ∈ A + holds. We set A rep = A 0,σ ∪ A + . Let us introduce the map ι : C A 0 → C Arep defined by (ιψ)(a) = ψ(a) for any ψ ∈ C A 0 and a ∈ A rep .
Let us define the boundary operator ∂ + : for any ϕ ∈ C Arep and u ∈ V 0 . On the other hand, let us also define the boundary operator Proof. Note that if ψ ∈ ker(1 + S), then ψ(a) = −ψ(a) for any a ∈ A + and if ψ ∈ ker(d), then t(a)=u ψ(a) = 0 for any u ∈ V 0 . Remark that since (Sψ)(a s ) = ψ(a s ) for any a s ∈ A 0,σ , we have ψ(a s ) = 0 if ψ ∈ ker(1 + S). Therefore if ψ ∈ ker(1 + S) ∩ ker(d), then holds. Then ker(1 + S) ∩ker d is isomorphic to {ϕ ∈ ker ∂ + | supp(ϕ) ⊂ A + }. Let us consider ker ∂ + . By the definition of ∂ + , we have ∂ + δ (Arep) a = 0 for any a ∈ A s . Hence, we should eliminate the subspace of ker ∂ + induced by the self-loops. The dimension of this subspace is |A 0,σ |. The adjoint operator ∂ * + : C V 0 → C A + of ∂ + is described by for any f ∈ C V 0 and a ∈ A rep . If ∂ * + f = 0 holds, then f (t(a)) = f (o(a)) for any a ∈ A + . This means f (u) = c for any u ∈ V 0 with some non-zero constant c. Thus dim ker(∂ * + ) = 1. Therefore, the fundamental theorem of linear algebra ‡ implies Next, let us consider dim(ker(1 − S) ∩ ker d 1 ). Note that if ψ ∈ ker(1 − S), then ψ(a) = ψ ( a). Assume that ψ ∈ ker(1 − S) ∩ ker(d 1 ), then t(a)=u (ιψ)(a) = 0 for any u ∈ V 0 , which is equivalent to The adjoint of ∂ − is described by Let us consider f ∈ ker(∂ * − ) in the cases for both A 0,σ = ∅ and A 0,σ = ∅. A 0,σ = ∅ case: If G 0 is a bipartite graph, then we can decompose the vertex set V into X ∪ Y , where every edge connects a vertex in X to one in Y . Then f (x) = k for any x ∈ X and f (y) = −k for any y ∈ Y with some nonzero constant k. Hence, dim ker(∂ * ) = 1 if A 0,σ = ∅ and G 0 is bipartite. On the other hand, if G 0 is non-bipartite, then there must exist an odd length fundamental cycle c = (a 0 , a 1 , . . . , a 2m ). We have that ).
Define Γ o , Γ e ⊂ Γ as the set of odd and even length fundamental cycles. In the following, to obtain a characterization of ker(1 + E) = ker(1 − S) ∩ ker(d 1 ), we construct the function η x,y ∈ ker(1−S)∩ker(d 1 ) which is determined by x, y ∈ A 0,σ ∪Γ o . The main idea to construct such a function is as follows. By the definition of ξ (−) q for any walk q, ξ (−) q ∈ ker(1 − S). This is equivalent to assigning the symbols "+" and "−" alternatively to each edge along (3) x ∈ A σ and y ∈ Γ o case: If |A σ | ≥ 1 and G \ A σ is a non-bipartite graph, let us fix a self-loop a * and pick up an odd cycle c = (b 1 , . . . , b t ) ∈ Γ o ; if the self-loop o(a * ) ∈ V (c), we set the walk starting from a * visiting all the vertices V (c) and returning back to a * by q = (a * , b 1 , . . . , b t , a * ); while o(a * ) / ∈ V (c), let us fix a path p = (p 1 , . . . , p t ) between o(a * ) and o(b 1 ) and set the walk starting from a * visiting all the vertices V (p) ∪ V (c) and returning back to a * ; q = (a * , p 1 , . . . , p t , b 0 . . . , b t ,p t , . . . ,p 1 , a * ). Then we set ξ (−) q =: η a * ,c .
(4) x ∈ Γ o and y ∈ A σ case: Let us fix an odd length fundamental cycle c * ∈ Γ o = (b 1 , . . . , b s−1 ) and pick up a self-loop a ∈ A σ . Let us set a short length path p between o(a) and o(b 1 ). Then we consider the same walk q as in (3) and set ξ (−) q =: η c * ,a .
By the construction, we have η x,y ∈ ker(1 − S) ∩ ker(d 1 ). Using the function η x,y , we obtain the following characterization of ker(−1 − E). into subspaces T and K, corresponding to the eigenvalues λ = ±1 and λ = 1, respectively, and an additional subspace which belongs to the eigenvalue λ = −1. While the basis of T and K can be constructed using the same procedure for all finite connected graphs G 0 , for the last subspace we provided a construction based on case separation, depending on if the graph is bipartite or not and if it involves self-loops. The use of fundamental cycles have allowed us to considerably expand the results previously found in the literature, which were often limited to planar graphs. The derived construction of the attractor space enables better understanding of the quantum transport models on graphs. In addition, our results have revealed that the attractor space can contain subspaces of eigenvalues different from λ = ±1. In such a case the evolution of the Grover walk with sink will have more complex asymptotic cycle. In fact, the example we have presented in Section 5 exhibits an infinite asymptotic cycle, since the phase θ of the eigenvalues λ ± = ±1 is not a rational multiple of π. This feature is missing, e.g., in the Grover walk on dynamically percolated graphs with sinks, where the evolution converges to a steady state.