controllability of quantum walks on graphs

Math. Control Signals Syst. (2012) 24:321–349DOI 10.1007/s00498-012-0084-0

ORIGINAL ARTICLE

Controllability of quantum walks on graphs

Francesca Albertini · Domenico D’Alessandro

Received: 17 June 2010 / Accepted: 6 March 2012 / Published online: 23 March 2012© Springer-Verlag London Limited 2012

Abstract In this paper, we consider discrete time quantum walks on graphs withcoin, focusing on the decentralized model, where the coin operation is allowed tochange with the vertex of the graph. When the coin operations can be modified atevery time step, these systems can be looked at as control systems and techniques ofgeometric control theory can be applied. In particular, the set of states that one canachieve can be described by studying controllability. Extending previous results, wegive a characterization of the set of reachable states in terms of an appropriate Liealgebra. Controllability is verified when any unitary operation between two states canbe implemented as a result of the evolution of the quantum walk. We prove generalresults and criteria relating controllability to the combinatorial and topological prop-erties of the walk. In particular, controllability is verified if and only if the underlyinggraph is not a bipartite graph and therefore it depends only on the graph and not on theparticular quantum walk defined on it. We also provide explicit algorithms for controland quantify the number of steps needed for an arbitrary state transfer. The results ofthe paper are of interest in quantum information theory where quantum walks are usedand analyzed in the development of quantum algorithms.

Keywords Control theory methods in quantum information · Quantum walks ·Lie algebras and lie groups

F. Albertini (B)Dipartimento di Matematica Pura ed Applicata, Universitá di Padova,Via Trieste 63, 35121 Padua, Italye-mail: [email protected]

D. D’AlessandroDepartment of Mathematics, Iowa State University,440 Carver Hall, Ames, 5001 IA, USAe-mail: [email protected]

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322 F. Albertini, D. D’Alessandro

1 Introduction

In recent years, quantum walks on graphs have emerged as one of the most useful pro-tocols to design quantum algorithms. This concerns, in particular, problems that arenaturally formulated on graphs, such as search problems where one is allowed to visitone location at a time moving between neighboring vertices (see, e.g., [4,19]). Thestudy of these systems has now developed in a new, rich, area of quantum informationand mathematics. There are several aspects that are worth studying, all interconnected:the design of quantum algorithms with better performances than the classical ones,and, in particular, than the randomized algorithms based on classical random walks;the complexity theory of these algorithms; the dynamics of these systems; their phys-ical implementation. Reviews on quantum walks and their algorithmic applicationscan be found in [3,12,13]. Moreover, quantum walks are often used as models in thestudy of natural phenomena (see, e.g., [14] for an application to energy transfer inphotosynthesis).

There are two different versions of quantum walks: continuous and discrete time.In its simplest form, a continuous time quantum walk on a graph is a quantum systemwith state |ψ〉 varying in a finite dimensional Hilbert space. We will follow Diracnotation of quantum mechanics in denoting by |ψ〉 a general vector, so that 〈ψ |φ〉denotes the inner product of two vectors |ψ〉 and |φ〉 and |ψ〉〈φ| the outer product(column by row). The state |ψ〉 evolves according to the Schrödinger equation

i ˙|ψ〉 = H |ψ〉, (1)

where the linear operator H, called the Hamiltonian, is constrained by the underlyinggraph, i.e., Hjk �= 0 if and only if there is an edge connecting the j th and kth vertexof the graph. One important case is when H is the adjacency matrix of the graph. Dis-crete time quantum walks come in different forms. One may use a quantum system,whose basis states represent the edges of the graph and define the evolution on thecorresponding Hilbert space (see, e.g., [11] and the references therein) or one mayuse two quantum systems, called the coin and the walker, the coin having dimensionequal to the degree d of the graph, assumed regular (see definitions at the beginningof Sect. 2), and the walker having dimension equal to the number of vertices N . Thissecond model, although restricted to regular graphs, has the advantage of making therole of the coin more transparent and intuitive and requiring a Hilbert space whosedimension (d N ) may be significantly smaller than the one (N 2) for the walk definedon the edges of the graph. There are some known relations among the various types ofquantum walks. Some of them are discussed in [5,7]. Related models are consideredin the context of quantum cellular automata and quantum robots (see, e.g., [21,22]and references therein).

In this paper, we consider discrete time quantum walks with coin on regular graphs.The evolution of these systems at every step is the sequence of two operations; oneoperation on the coin system, called coin tossing, and one operation on the walkersystem, called the conditional shift, which changes the state of the walker accordingto the state of the coin. We assume that, at every step, one can change the coin tossingtransformation and we adopt a decentralized model where the coin transformation

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Controllability of quantum walks on graphs 323

may depend on the current state of the walker system. The main topic of this paper isto characterize the set of states that can be obtained with these models, that is, theircontrollability.

The paper is organized as follows. In Sect. 2, we describe in mathematical termsthe models we want to study. In Sect. 3, we define the controllability of these modelsand give criteria to describe the set of reachable states. In particular, by modifyingthe proof that was given in [2,8] we extend and strengthen a result which describesthe set of admissible evolutions of these systems as a Lie group (Theorems 1 and 2).This Lie group might have one or more connected components and its Lie algebra isgenerated by an appropriate set of matrices. An important problem, in this context, isto characterize explicitly this Lie algebra for various quantum walks. In Sect. 4, werelate the Lie algebraic controllability criterion described in Sect. 3 with the orbits ofthe permutations associated with the walk. This correspondence will allow us to inferfurther controllability properties of these systems and, in particular, to solve the Liealgebra characterization problem above mentioned (Theorem 6). As a consequence ofthis general result, we obtain several strong statements in special cases. In particular,quantum walks with a graph of degree d greater than N

2 , where N is the number ofvertices, are always completely controllable (Proposition 4.1). Here, complete con-trollability means that every unitary evolution can be obtained with the dynamics ofthe system. In Sect. 5, we adopt a more direct approach to the study of controllability,by giving explicit constructive algorithms for state transfer. In doing so, we obtain anupper bound on the worst case number of steps needed for an arbitrary state transfer.A byproduct of this method of control is another condition of controllability whichis expressed in terms of the properties of the graph underlying the quantum walk(Theorem 7). Sect. 6 relates the results obtained with this constructive approach, withthe ones obtained in the previous sections. We do this in Theorem 8 and we add apurely graph theoretic criterion of controllability (point 4 of Theorem 8). In particular,controllability is verified if and only if the underlying graph is not a bipartite graph.We notice that controllability only depends on the graph and not on the walk definedon it and that even purely graph theoretic questions can be answered using the conceptof quantum walk (cf. Theorem 9). Section 7 contains some examples including a fulltreatment for graphs of degree two (i.e., cycles).

2 Model definition

Let G := {V, E} be an undirected graph, where V denotes the associated set of verti-ces and E the associated set of edges. We shall denote by N the number of elementsin V . We assume that

H1) G is a regular graph, that is, all the vertices have the same number d of adjacentvertices. The number d is referred to as the degree of the graph G.

H2) G is connected, without self loops (i.e., edges connecting a vertex to itself) andwithout multiple edges (i.e., two or more edges connecting the same pair ofvertices).

Given the graph G, a quantum walk on G is a quantum mechanical system definedas follows. The state of this system varies on a Hilbert space C ⊗ W . The space W is

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Fig. 1 Quantum walk on acycle with four vertices

called the walker space and an orthonormal basis for W is given by {|0〉, . . . , |N −1〉},where {0, . . . , N − 1} are the labels of the vertices in V . The space C is called thecoin space and an orthonormal basis for C is given by |c1〉, . . . , |cd〉, where d is thedegree of the graph. The states {|c1〉, . . . , |cd〉} are in one to one correspondence withpossible ‘directions’ of motion on the graph. More precisely, with every value cl itis associated a permutation πl of the vertices in V . The permutation πl is such thatπl i = j implies that there is an edge in E connecting the vertices i and j and for anedge connecting i and j there exists a unique value cl and associated permutation πl

such that πl i = j . A quantum mechanical system with state varying on W will bereferred to as a walker system and a system varying on C will be referred to as a coinsystem.

The state of the quantum walk is described by a unit vector |ψ〉 in C ⊗ W, i.e.,

|ψ〉 :=d∑

k=1

N−1∑

j=0

αk j |ck〉 ⊗ | j〉. (2)

According to the measurement postulate of quantum mechanics (see, e.g., [16]), themeaning of the state | j〉 ∈ W is that if we measure the position of the walker we findthe position j with certainty. More in general, from (2), the probability p j of findingthe walker in position j is obtained by tracing out the coin degrees of freedom, thatis, p j = ∑d

k=1 |αk j |2.We will denote by U (n)(SU (n)) the Lie group of n × n unitary matrices (n × n

unitary matrices with determinant equal to one), while u(n)(su(n)) denotes the cor-responding Lie algebra of skew-Hermitian n × n matrices (skew-Hermitian n × nmatrices with zero trace). There are several introductory books on Lie algebras andLie groups (see e.g., [10,15,17]). The book [6] presents introductory notions with aview to applications to the control of quantum systems.

As a simple example of a quantum walk on a graph G consider for G the cyclewith four vertices in Fig. 1. The coin state |c1〉 is associated with a permutationπ1 = (0 1 2 3)while the coin state |c2〉 is associated with a permutationπ2 := (0 3 2 1).

The dynamics of a quantum walk system is a sequence of two types of operationswhich we now define. A coin tossing operation on C ⊗ W is an operation of the type

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C :=N−1∑

j=0

Q j ⊗ | j〉〈 j |, (3)

with Q j ∈ U (d). This operation applies a unitary evolution to the coin state and thisoperation is allowed to depend on the current walker state. The second operator wedefine is the conditional shift, which is an operator of the form

S :=d∑

k=1

|ck〉〈ck | ⊗ Pk . (4)

This operator applies to a state in W a permutation Pk depending on the currentvalue of the coin system. The permutation operator Pk is the one corresponding to thepermutation πk in that | j〉 = Pk |l〉 ↔ j = πkl. In the basis

ek j := |ck〉 ⊗ | j〉 k = 1, . . . , d, j = 0, . . . , N − 1, (5)

S has the matrix representation

S =

⎛

⎜⎜⎜⎜⎜⎝

P1 0 · · · 00 P2 · · · 00 0 · · · 0...

......

...

0 0 · · · Pd

⎞

⎟⎟⎟⎟⎟⎠, (6)

where, with minor abuse of notation, we denote by the same symbol Pk the permutationoperator and the matrix that represents it in the standard basis.

For the example given in Fig. 1, we have:

C =3∑

j=0

Q j ⊗ | j〉〈 j |, with Q j ∈ U (2)

and

S =(

P1 00 P2

), where P1 =

⎛

⎜⎜⎝

0 0 0 11 0 0 00 1 0 00 0 1 0

⎞

⎟⎟⎠ and P2 = P−11 .

Summarizing, the action of the coin tossing operation and conditional shift on thevector space C ⊗ W is given, for k = 1, . . . , d, and j = 0, . . . , N − 1, by

Cekj = (Q j |ck〉)⊗ | j〉,Sek j = |ck〉 ⊗ (Pk | j〉) = |ck〉 ⊗ |πk j〉. (7)

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The dynamics of the quantum walk is as follows. At every step, |ψ〉 evolves as|ψ〉 → SC |ψ〉, i.e., a coin tossing operation C is followed by a conditional shiftS. The coin tossing operation may change at any time step preserving however thestructure (3). This leads to a point of view where the operations Q j in (3) are seenas control variables in the evolution of the system. In this paper, we are interested instudying the set of states that can be obtained for the quantum walks just defined byvarying in all possible ways the coin operations.

3 Controllability

The set of all possible evolutions is given by the set of all products of the form∏mk=1 SCk where Ck are arbitrary coin tossing operations of the form (3). This set

was already studied in [2,8] for the centralized case where the Q j in (3) are all equal.Improving the technique used in these references we obtain a complete characteriza-tion of this set for our case in Theorem 1. We first set up some definitions. Recallthat S being a permutation matrix has a certain order r, such that Sr is the identity onC ⊗ W . Define the set of matrices

F := {A, SASr−1, . . . , Sr−1AS}, (8)

where A is the span of matrices of the form∑N−1

j=0 A j ⊗ | j〉〈 j | with A j ∈ u(d).Notice that A is a Lie algebra (a Lie algebra has the structure of a vector space withthe additional Lie bracket operation), since it is closed under the Lie bracket operationas it can be seen by taking

⎡

⎣N−1∑

j=0

A j ⊗ | j〉〈 j |,N−1∑

k=0

Bk ⊗ |k〉〈k|⎤

⎦ =N−1∑

j=0

[A j , B j ] ⊗ | j〉〈 j | ∈ A. (9)

Here, we used the fact that the basis {|0〉, . . . , |N − 1〉} is orthonormal. In fact, A isthe direct sum (a direct sum of Lie subalgebras is a sum of subspaces which commutewith each other) of N subalgebras isomorphic to u(d) as it can be easily seen with achange of coordinates which transforms

∑N−1j=0 A j ⊗ | j〉〈 j | into the block diagonal

form∑N−1

j=0 | j〉〈 j | ⊗ A j . Let L be the Lie algebra generated by F , defined as the

smallest Lie algebra containing F , and let eL be the connected Lie group associatedwith L, that is, the connected component containing the identity.

Proposition 3.1 Recall the definition of S in (4). Let:

1. K be the set defined as:

K := eL ∪ eLS ∪ eLS2 ∪ · · · ∪ eLSr−1 (10)

where eLS j is the set of all matrices X S j with X ∈ eL.2. G be the group generated by eL and {S}.

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3. If p is the smallest integer 1 ≤ p ≤ r such that S p ∈ eL, let C be the set definedas the disjoint union of eL, eLS, . . . , eLS p−1.

Then K = C = G and they are Lie groups.

Before giving the proof, we notice that:

• The set defined in Eq. (10) is the same as the set of all matrices S j Y with Y ∈ eL.That is S j eL = eLS j . To prove eLS j ⊆ S j eL, write X S j as S j Sr− j X S j . SinceSr− j X S j ∈ eL if X ∈ eL the claim follows. The converse inclusion is provedanalogously.

• To see that the sets eL, eLS, . . . , eLS p−1 are disjoint, notice that if there existtwo different indices 0 ≤ k < j ≤ p − 1 and two elements in eL, X and Y suchthat X S j = Y Sk, we would have S j−k = X−1Y ∈ eL which contradicts theminimality of p.

Proof It follows from the definitions that K ⊆ G, and C ⊆ K. The equality K = C =G follows if we show that G ⊆ K and K ⊆ C. An element in G is a product

∏mk=0 Yk,

with Y0 equal to the identity, where Yk ∈ eL or Yk = S, for k ≥ 1. By induction onm, if m = 0, this product is the identity which is in eL and therefore in K. If m > 0,write

∏mk=0 Yk as Y

∏m−1k=0 Yk, with

∏m−1k=0 Yk ∈ K, i.e.,

∏m−1k=0 Yk = X S j for some

0 ≤ j ≤ r − 1 and X ∈ eL. Now, if Y ∈ eL, then Y X S j ∈ eLS j ⊆ K. If Y = Sthen SX S j = SX Sr−1S1S j and since X ∈ eL implies Z := SX Sr−1 ∈ eL, we haveY X S j = Z S j+1 ∈ K.

To see that K ⊆ C, we need to consider only X Sk with kp − 1. Choose n so that0 ≤ k−np mod r < p and define j := k−np mod r . We have X Sk = X Snp Sk−np :=Y S j . The matrix Y := X Snp is in eL, since X and Snp are both in eL. Moreover sincej < p,Y S j is in C, that is, X Sk ∈ C.

To see that C and therefore K and G are Lie groups, we notice that C naturallyhas the structure of a differentiable manifold inherited by eL. In fact, if {Uα, φα} isan atlas for eL, with φα : Uα → RI n, then {UαSk, φα,k}(k = 0, . . . , p − 1), withφα,k(X Sk) := φα(X), is an atlas for C. Thus, if� : eL → eL is C∞, then, for all 1 ≤k, l ≤ p−1, the induced map�k,l : eLSk → eLSl , defined as�k,l(X Sk) := �(X)Sl

is also C∞. (If φα,k and φβ,l are chart maps on eLSk and eLSl , then φβ,l ◦�k,l ◦ φ−1α,k

is by definition the same as φβ ◦�◦φ−1α which is a C∞ map from RI n to RI n since�

is C∞.) In particular, consider the inverse matrix map ϒ : (X Sk) → (X Sk)−1. Writer − k as (r − k − j p)+ j p for some j where r − k − j p < p. We have:

ϒ(X Sk) := (X Sk)−1 = S−k X−1 = Sr−k− j p S jp X−1S−(r−k− j p)Sr−k− j p.

This formula shows that, the inverse matrix map 〉 can be seen as the induced map�k,r−k− j p : eLSk → eLSr−k− j p, where the map � : eL → eL is defined as� := �3 ◦ �2 ◦ �1 with �1 : X → X−1, �2 : Y → S jpY (recall that by definitionof pS jp ∈ eL). and �3 : z → Sr−k− j pzS−(r−k− j p). Since the maps �1, �2 and�3 are C∞ so is � and therefore so is the inverse map. Analogously, one noticesthat a map � : eL × eL → eL which is C∞ induces, for fixed k, l and m, a map�(k,l),m : eLSk ×eLSl → eLSm which is also C∞ defined by�(k,l),m(X Sk,Y Sl) :=

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�(X,Y )Sm . Using this fact one shows that the product operation in C is also C∞.Therefore, since both inverse and product operations on C(= K = G) are C∞, thisgroup is by definition a Lie group. ��

Notice that if S ∈ eL,G has only one connected component which is given by eL.The following theorem characterizes the controllability of quantum walks.

Theorem 1 Let E be the set of possible evolutions of the quantum walk. Then

E = K (11)

Proof E is the set of products of transformations of the form SC with C a coin toss-ing operation and S a conditional shift. Since C ∈ eL ⊆ K and S ∈ K (becauseK = G in Proposition 3.1) then SC ∈ K and therefore E ⊆ K. Vice versa, consideran element X S j ∈ K, for some 0 ≤ j ≤ r − 1. Since X ∈ eL, it can be writtenas the product of matrices of the form SkeA Sr−k with the matrix A ∈ A of the formA = ∑N−1

l=0 Al ⊗ |l〉〈l| and Al ∈ u(d). (This is a consequence of the known fact thatif a set F generates a certain Lie algebra, then every element X in the correspond-ing Lie group can be written as the finite product of exponentials eFt with t ∈ RIand F ∈ F .) The matrix C := eA is a coin operation, and therefore, we can writeSkeA Sr−k as SkC Sr−k . We can obtain SkC Sr−k by performing r − k steps with coinoperations equal to the identity, one step with coin operation equal to C and k − 1steps with coin operation equal to the identity (in the case k = 0, we can use one stepwith coin operation equal to C followed by r −1 operations with coin operation equalto the identity). Therefore every matrix of the form SkeA Sr−k can be obtained as anevolution of the quantum walk. So can every product of such matrices and thereforeevery X ∈ eL. To obtain X S j , just compose the sequence giving X with j steps of thewalk with coin operation equal to the identity. This shows that K ⊆ E and concludesthe proof of the theorem. ��Remark 3.2 An analogous characterization of the set E can be proved with just smallnotational modifications for the ‘centralized’ case where all the matrices Q j in (3) areequal. In this case, the Lie algebra A in (8) has to be replaced by the Lie algebra ofmatrices A ⊗1 with A ∈ u(d) and 1N the N × N identity. This was the case treated in[2,8]. The above discussion goes however further with respect to the results in [2,8]where only the inclusion eL ⊆ E was proved.

From Theorem 1, it follows that the Lie algebra L plays a crucial role in the charac-terization of the set of available state transformations of the quantum walk. Followingcommon terminology in quantum control, we shall call this Lie algebra the dynamicalLie algebra associated with the quantum walk. The following Theorem holds.

Theorem 2 The quantum walk is completely controllable (every unitary operation ispossible) if and only if L = u(d N ).

Proof If L is u(d N ) then eL = U (d N ). Since eL ⊆ K ⊆ U (d N ), we clearly have,from Theorem 1, E = K = U (d N ). Therefore the system is completely controllable.Conversely if a quantum walk is completely controllable, then U (d N ) = E = K. Thus

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K can only have one connected component since U (d N ) is connected. The numberof connected components is given by p in Proposition 3.1. Therefore p must be equalto 1, S ∈ eL and K = eL. From the fact that eL = U (d N ), using the correspondencebetween connected Lie groups and Lie algebras (see e.g., [15]), it follows that the Liealgebras of eL and U (d N ) must coincide. Therefore L = u(d N ). ��Remark 3.3 Another motivation to study the Lie algebra L is given by the work in [7]where a procedure was described to obtain continuous quantum walks as an appro-priate limit of discrete quantum walks. This procedure generalized a method given in[18] for quantum walks on the line. In particular, the set iL, represents the set of allHamiltonians whose associated continuous dynamics over the full space C ⊗ W canbe obtained with the procedure of [7].

In general, calculating a Lie algebra directly from a set of generators can be cumber-some since we have to compute commutators of possibly very large matrices. There-fore different criteria of controllability are desirable. In the following section, we shallcharacterize the dynamical Lie algebra L for every quantum walk in combinatorialterms, i.e., in terms of the permutations π1, . . . , πd characterizing the walk.

4 Controllability and orbits of permutations

We take a closer look at the generating set F in (8) for the dynamical Lie algebra Land at how it relates to the orbits of the permutations π1, . . . , πd .

Given l and m, with l,m ∈ {1, 2, . . . , d}, define the (l,m)th joint orbit, Ol,m, asthe following subset of V × V,

Ol,m :=⋃

k = 0, 1, . . . , r − 1j = 0, 1, . . . , N − 1

(πkl j, πk

m j). (12)

Notice that ( j, j) is in any joint orbit for every pair (l,m). In the basis given by ei j

(see Eq. 5), we can enumerate the rows and columns of any matrix in F (and L)using an index i to identify a block row (or column) (i = 1, 2, . . . , d) and the indexj ( j = 0, 1, . . . , N −1) to identify a position inside a block. We consider the matricesin F and L in this basis. The following theorem relates the structure of the matricesin F with the orbits defined in (12). In order to state this theorem we introduce F, thevector space of all the skew-Hermitian matrices having the (l, h)− (m, s)th positionl,m = 1, 2, . . . , d, h, s = 0, 1, . . . , N − 1 possibly different from zero and arbitraryif and only if (h, s) ∈ Ol,m . The vector space F is spanned by the set B defined as

B :=⎧⎨

⎩

(|cl〉〈cm | ⊗ |h〉〈s| − |cm〉〈cl | ⊗ |s〉〈h|),

i(|cl〉〈cm | ⊗ |h〉〈s| + |cm〉〈cl | ⊗ |s〉〈h|)

∣∣∣∣∣∣

l,m = 1, . . . , d,h, s = 0, 1, . . . , N − 1,(r, s) ∈ Ol.m .

⎫⎬

⎭ . (13)

Theorem 3 span F = F .

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Proof To show that F ⊆ span F it is enough to show that for any B ∈ B, B ∈ F .Let B := |cl〉〈cm | ⊗ |h〉〈s| − |cm〉〈cl | ⊗ |s〉〈h|, for fixed l and m in {1, . . . , d} andh and s in {0, 1, . . . , N − 1}. Since (h, s) is required to be in Ol,m, there exists aj ∈ {0, 1, . . . , N − 1} and a k ≥ 0 such that πk

l j = h and πkm j = s. For the

given k, consider now the matrix Sk AS−k ∈ F (cf. (8)), with A given by A :=(|cl〉〈cm | − |cm〉〈cl |)⊗ | j〉〈 j | and calculate (we will use the Kronecker delta notationδab = 0 if a �= b, δab = 1 if a = b)

Sk AS−k =(

d∑

a=1

|ca〉〈ca | ⊗ Pka

)((|cl〉〈cm | − |cm〉〈cl |)⊗ | j〉〈 j |)

×(

d∑

b=1

|cb〉〈cb| ⊗ P−kb

)

=d∑

a=1

d∑

b=1

δalδbm |ca〉〈cb| ⊗ Pka | j〉〈 j |P−k

b − δamδbl |ca〉〈cb| ⊗ Pka | j〉〈 j |P−k

b

= |cl〉〈cm | ⊗ Pkl | j〉〈 j |P−k

m − |cm〉〈cl | ⊗ Pkm | j〉〈 j |P−k

l

= |cl〉〈cm | ⊗ |r〉〈s| − |cm〉〈cl | ⊗ |s〉〈h|, (14)

since Pkl | j〉 = |h〉, Pk

m | j〉 = |s〉. Thus Sk AS−k = B ∈ F . A similar calculationchoosing A := i(|cl〉〈cm | + |cm〉〈cl |)⊗ | j〉〈 j | shows that i (|cl〉〈cm | ⊗ |h〉〈s| + |cm〉〈cl | ⊗ |s〉〈h|) is in F , so we conclude F ⊆ span F . To prove that span F ⊆ F, it isenough to show that every element in F of the form Sk AS−k, with A = ∑N−1

j=0 A j ⊗| j〉〈 j | ∈ A, for j = 0, . . . , N − 1, where A j is a general matrix in u(d), can beexpressed as a linear combination of elements in B in (13). We write

Sk AS−k =(

d∑

l=1

|cl〉〈cl | ⊗ Pkl

)⎛

⎝N−1∑

j=0

A j ⊗ | j〉〈 j |⎞

⎠(

d∑

m=1

|cm〉〈cm | ⊗ Pkm

)

=∑

l,m = 1, . . . , dj = 0, . . . , N − 1

|cl〉〈cl |A j |cm〉〈cm | ⊗ Pkl | j〉〈 j |P−k

m .

After defining

x jlm := 〈cl |A j |cm〉, (15)

and noticing that, since A†j = −A j , we have x∗

jlm = −x jml , we can write

Sk AS−k =∑

l,m = 1, . . . , dj = 0, . . . , N − 1

x jlm |cl〉〈cm | ⊗ Pkl | j〉〈 j |P−k

m

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=∑

j=0,1,...,N−1

⎛

⎝∑

l=1,2,...,d

x jll |cl〉〈cl | ⊗ Pkl | j〉〈 j |P−k

l

⎞

⎠

+⎛

⎝∑

l,m=1,2,...,d, l<m

x jlm |cl〉〈cm | ⊗ Pkl | j〉〈 j |P−k

m

+ x jml |cm〉〈cl | ⊗ Pkm | j〉〈 j |P−k

l

⎞

⎠ . (16)

By defining R jlm := Re(x jlm) and I jlm = Im(x jlm) and since we have R jlm = −R jml

and I jlm = I jml , we have

Sk AS−k =∑

j=0,1,...,N−1

∑

l=1,...,d

i I jll |cl〉〈cl | ⊗ Pkl | j〉〈 j |P−k

l

+∑

l<m

R jlm

(|cl〉〈cm | ⊗ Pk

l | j〉〈 j |P−km − |cm〉〈cl | ⊗ Pk

m | j〉〈 j |P−kl

)

+∑

l<m

I jlmi(|cl〉〈cm | ⊗ Pk

l | j〉〈 j |P−km + |cm〉〈cl | ⊗ Pk

m | j〉〈 j |P−kl

).

This is a linear combination (with real coefficient) of elements in B since for every pair(l,m) and every j, (πk

l j, πkm j) ∈ Ol,m . This shows that Sk AS−k ∈ F and concludes

the proof of the theorem. ��To study the nature of the Lie algebra L generated by F ,we shall apply some results

proved in [20]. In order to do that, we associate with the quantum walk a connectivitygraph, which will be denoted by GC , having d N vertices, each corresponding to a pair(l, h), with l ∈ {1, 2, . . . , d} and h ∈ {0, 1, . . . , N − 1}. We connect two pairs (l, h)and (m, s) if and only if (h, s) ∈ Ol,m . It follows from the proof of the above theoremthat (h, s) ∈ Ol,m if and only if there exists a matrix in F with the [(l, h), (m, s)]th ele-ment different from zero. In the connectivity graph GC , we omit all self connections.These correspond to diagonal elements in the matrices in F , which can, in fact, bechosen arbitrarily (but must be purely imaginary). In [20] the authors studied the Liealgebra generated by two skew-Hermitian matrices, i H0 and �0, with H0 Hermitianand diagonal, and �0, purely real, i.e., skew-symmetric. A (connectivity) graph wasassociated with this pair of matrices with edges connecting vertices corresponding tothe row (or column) indices (a, b) if and only if the (a, b)-th entry in�0 was differentfrom zero. These edges were then labeled, with the label corresponding to (a, b), equalto |λa −λb|,where λa(λb) is the diagonal element (eigenvalue) of i H0 correspondingto a(b). The next theorem was proved in [20]; here it is stated in a form suitable forour purposes.

Theorem 4 Assume that the labeled (connectivity) graph associated with the pair{i H0,�0} is connected and it remains connected after eliminating edges having equallabels. Consider the Lie algebra L generated by i H0 and�0. Then the corresponding

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Lie group eL is transitive on the complex sphere, i.e. for every pair of unit vectors

|ψ0〉 and |ψ1〉 in CI n, there exists an element X ∈ eL with |ψ1〉 = X |ψ0〉.

We shall use this theorem along with some results on the controllability of quantumsystems proved in [1], to establish the following result.

Theorem 5 The quantum walk is completely controllable, i.e., L = u(d N ), if andonly if the associated connectivity graph GC is connected.

Proof First assume that the connectivity graph GC is connected. Since F containsarbitrary skew-Hermitian diagonal matrices we can choose a matrix where all thedifferences between two diagonal elements are different from each other. Let i H0denote such a matrix in F . Moreover we can choose a matrix �0 in span F withentries (l, h) − (m, s) different from zero and real if and only if there exists an edgeconnecting the vertices (l, h) and (m, s) in GC . Thus, the connectivity graph GC isthe graph corresponding to the pair i H0 and �0 considered in Theorem 4. Since thisgraph is connected and there are no edges to remove since all edges are labeled witha different label, we have that the Lie algebra L generated by i H0 and �0, which is

a subalgebra of L, is such that the corresponding Lie group eL is transitive on thecomplex sphere. This is also the case for the Lie group eL associated with L since

eL is a subgroup of eL. This fact does not necessarily imply that the quantum walk iscompletely controllable, i.e., L = u(d N ). However, according to general controlla-bility results for quantum systems [1], the only other possibility is that L is conjugateto the symplectic Lie algebra sp( d N

2 ) plus multiples of the identity matrix. (Noticethat here d N must be even since the hand-shaking lemma of graph theory [9] impliesfor regular graphs that d N = 2|E |, where |E | is the number of edges.) This impliesthat there exists a matrix J of the form J := T † J T where

J =(

0 1 d N2−1 d N

20

), (17)

and T is some unitary matrix, such that

AJ + J AT = 0, (18)

for every A ∈ L with T r(A) = 0. We now prove that (18) is not possible. To seethis, partition J into d × d blocks of dimension N × N . Formula (18) has to hold, inparticular, for every A ∈ F , with T r(A) = 0. Let A be an arbitrary skew-Hermitiandiagonal, zero trace matrix. This type of matrices are in F . Denote A by:

A =

⎛

⎜⎜⎝

D1D2

Dd

⎞

⎟⎟⎠ (19)

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with Dl = Diag[iλl1, · · · , iλl

N], λlj ∈ RI and

∑dl=1

∑Nj=1 λ

lj = 0. Fix two block

indices k, l ∈ {1. . . . , d}, then it holds that:

(AJ + J AT)lk = Dl Jlk + Jlk Dk . (20)

Thus, since from Eq. (18), the previous expression must be zero, we get:

(Dl Jlk + Jlk Dk)s j = i(λls + λk

j )( Jlk)s j = 0, ∀ s, j ∈ {0, 1, . . . , N − 1}.

Now, for fixed indices l, k ∈ {1, . . . , d} and s, j ∈ {0, 1, . . . , N − 1}, we can chooseA, in Eq. (19), with λl

s + λkj �= 0; this is clearly possible since the coefficients λ j

l can

be chosen arbitrarily except for the trace condition. Choose for example λls = λk

j =1, λl

s = −2 for one arbitrary s ∈ {0, 1, . . . , N − 1} different from s, and all otherλm

t ,m ∈ {1, 2, . . . , d}, t ∈ {0, 1, . . . , N − 1} equal to zero. So we get ( Jlk)s j = 0. Byvarying the indexes l, k and s, j we conclude J = 0, which contradicts (17). Thisshows that L = u(d N ).

To see that the condition on the connectivity graph GC being connected is alsonecessary, notice that if the graph is not connected then it can be divided in g ≥ 2connected components. Reordering the column and row indices of the matrices inF , according to the various connected components of the graph, we can write all thematrices in F in block diagonal form. The Lie bracket operation preserves this blockdiagonal form. Therefore, not all the matrices in u(d N ) can be generated from theelements of F and L �= u(d N ). ��

Elaborating further on the statement and the proof of Theorem 5, we obtain moreinformation on the controllability of quantum walks on graphs. In particular, noticethat for every j ∈ V, ( j, j) is in the orbit Ol,m for every, l,m = 1, 2, . . . , d. Thismeans that (1, j), (2, j), . . . (d, j) are all connected in the connectivity graph GC .This observation suggests to use a reduced connectivity graph, which will be denotedby G R . The graph G R will have N vertices, each corresponding to a vertex positionin V := {0, 1, . . . , N − 1}, and there is an edge connecting the two vertices h ands if and only if there exist two coin indices l and m so that (l, h) and (m, s) areconnected in the connectivity graph GC . It follows from the fact that for every j ∈{0, 1, . . . , N − 1}, (1, j), (2, j), . . . (d, j) are all connected in GC , that G R is con-nected if and only if GC is connected. Moreover, from the definition of the orbitsOl,m, we get that two vertices h and s in G R are connected by an edge if and onlyif there exists a j ∈ {0, 1, . . . , N − 1} and two coin indices l and m and an integer ksuch that πk

l j = h and πkm j = s, i.e.,

h = πkl π

−km s. (21)

This relation gives a method to construct G R . The algorithm is as follows:

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Algorithm 1

1. Given the permutations π1, . . . , πd characterizing the walk, consider for everypair l < m the permutations πk

l π−km written in the cycle notation, i.e., πk

l π−km =

(· · · )(· · · ) · · · (· · · ).2. Connect in a graph all the vertices that pairwise belong to the same cycle at least

in one instance. This is the reduced connectivity graph G R associated with thequantum walk.

The following theorem describes the structure of the dynamical Lie algebra L, incases where controllability is not verified, i.e., L �= u(d N ), in terms of the graphsG R .

Theorem 6 The dynamical Lie algebra L is always the direct sum of m > 0 Liealgebras isomorphic to su(dv ) for some positive integers v , = 1, . . . ,m with∑m =1 v = N along with a one dimensional Lie algebra spanned by multiples of the

identity matrix in u(d N ). Each subalgebra isomorphic to su(dv ) corresponds to aconnected component of the reduced connectivity graph G R with v vertices. Completecontrollability is obtained in the case m = 1.

Proof Assume that G R has m connected components, and that the th componenthas v elements (so

∑m =1 v = N ). We can perform a change of coordinates (which

corresponds to a relabeling of the vertices) to regroup together vertices correspondingto the same connected component. More precisely, this change of coordinates on Lhas the form X → U XU †,where U = I ⊗ P,with I the d ×d identity and P the per-mutation on {0, 1, . . . , N − 1} which puts together all vertices in the same connectedcomponent of G R . In these coordinates all matrices in F and L are block diagonal.Since the th connected component has v vertices the th block has dimension dv .For each block, we can repeat the argument of Theorem 5 to conclude that we can getany matrix in u(dv ), and this shows the structure of the Lie algebra L. ��

In the rest of this section, we give two consequences of the results and methodssummarized in Theorems 5 and 6 and Algorithm 1. Appendix A contains some fur-ther analysis which uses the results of the next section to show that the number m inTheorem 6 can only be 1 (controllable case) or 2.

Proposition 4.1 If d > N2 the quantum walk is completely controllable.

Proof We will show that the reduced connectivity graph G R is connected. Fix twodistinct vertices i, j ∈ {0, . . . , N − 1}. Consider the following two subset of vertices:

Ai = {πl i | l ∈ {1, . . . , d}} , A j = {πl j | l ∈ {1, . . . , d}} . (22)

Since, by assumption on the model, all the vertices in Ai and in A j are different, thenumber of different elements in Ai and A j is d, i.e., |Ai | = |A j | = d. Thus we have|Ai | > N

2 and the same inequality holds for |A j |. These two inequalities imply thatAi ∩ A j �= ∅. Thus there exist two coin indices li and l j such that

πli i = πl j j = k.

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Since πl j j =k, in the graph G there is and edge connecting the two vertices j and k.Therefore, there must exist a coin value clk such that πlk k = j ; analogously thereexists a coin value cls such that πls k = i This implies that

i = πlsπ−1lk

j,

so the two indices i and j are connected in G R, since equation (21) holds. By the arbi-trariness of i and j,we get that G R is connected, as desired. Another (graph theoretic)proof of this result can be obtained as a consequence of 4 in Theorem 8 below. ��

The bound in Proposition 4.1 is sharp in the sense that there are quantum walks thatare not controllable with d = N

2 . In fact, we shall see in Sect. 7 that quantum walkson a cycle (therefore of degree 2) with 4 vertices are not controllable. Notice also that,as a special case of Proposition 4.1, quantum walks on complete graphs are alwayscontrollable (we always assume N > 2).

For the last result of this section, we need the concept of product of two quan-tum walks. Consider two quantum walks. The first one, W1, is supported by a graphG1 := {V1, E1} with a set of permutations {π1

1 , . . . , π1d1

} and the second one, W2,

supported by a graph G2 := {V2, E2} with a set of permutations {π21 , . . . , π

2d2

}. Theproduct walk W1 × W2 is the walk whose graph is the Cartesian product of G1 and G2and the associated permutations are {π1

1 , . . . , π1d1, π2

1 , . . . , π2d2

} acting on the vertices

( j, k) ∈ V1 × V2 as π1l ( j, k) := (π1

l j, k) and π2l ( j, k) := ( j, π2

l k). One example is awalk on a 2-dimensional lattice with N1 × N2 vertices connected in a periodic fash-ion horizontally and vertically. Coin results can be labeled R, L ,U, D (Right, Left,(mod N1), Up, Down (mod N2), respectively) and this is the product of two walks oneevolving horizontally on a cycle with N1 nodes and one evolving vertically on a cyclewith N2 nodes.

Proposition 4.2 The product of two controllable walks is controllable.

Proof Let G1,2R denote the reduced connectivity graph of the product walk W1 × W2.

With the above notations, since the walk W1 is controllable, for every j ∈ V2 thevertices (k, j), k = 1, . . . , N1, are all connected in G1,2

R . Analogously, from the con-trollability of W2, it follows that for every k ∈ V1 the vertices (k, j), j = 1, . . . , N2are all connected in G1,2

R . Therefore G1,2R is connected and the walk W1 × W2 is

controllable. ��We remark that the above condition is not necessary and one can find two quantum

walks with one or both of them uncontrollable whose product is controllable.

5 Constructive controllability algorithms

In this section, we discuss constructive controllability. We will focus on finding controlalgorithms to steer the state of the quantum walk between two values. Thus, for anygiven two state vectors |ψ1〉, |ψ2〉 in C ⊗ W we will find a sequence of coin tossingoperations, C1, . . . ,Ck, such that

|ψ2〉 = SCk · · · SC1|ψ1〉.

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Moreover, we will give a bound on the length k of the needed control sequence.Whether such a sequence exists or not can be checked with the methods of the previ-ous two sections.

First, we define, for a given node j, the set of all nodes that one can reach usingthe edges of the graph in a given number of steps. Fix a node j ∈ {0, . . . , N − 1}, let:

N 0( j) := { j},N k+1( j) := {πs(l) | l ∈ N k( j), 1 ≤ s ≤ d}. (23)

With these definitions, l ∈ N k( j) means that there exists a sequence of permutationsπ1, . . . , πk in the set {π1, . . . , πd} such that l = πk πk−1 · · · π1 j . The connectednessassumption on the graph G implies that ∀ i, j ∈ {0, . . . , N − 1} there exists a k ≥ 0such that i ∈ N k( j). The set N k( j) only depends on the graph. It is the set of verticeswhich are connected to j by a path of length k.

From these observations, we can collect two properties of the sets N k( j) in thenext lemma.

Lemma 5.1 Let i, j, l ∈ {0, . . . , N − 1}, k, s ≥ 0, we have:

1. l ∈ N k( j) ⇔ j ∈ N k(l),2. if l ∈ N k( j) and i ∈ N s( j) then i ∈ N k+s(l).

Choose a node j ∈ {0, . . . , N − 1} and consider a state |ψ1〉 with probability 1 tofind the walker in position j . Thus |ψ1〉 is of the form |ψ1〉 = |c0〉 ⊗ | j〉, for somestate |c0〉 ∈ C. If there exists a sequence of coin tossing operations of length k suchthat

SCk · · · SC1|ψ1〉 =∑

s

∑

l

αls |ckl 〉 ⊗ | jks 〉,

then, jks ∈ N k( j) for all ks . This fact, in particular, implies that a necessary conditionto have complete controllability is that ∀ j ∈ {0, . . . , N − 1} there exists a k ≥ 0such that N k( j) = {0, . . . , N − 1} since we have to be able to transfer to an arbitrarystate in C ⊗ W . By using property 2) of Lemma 5.1, we can substitute ∀ with ∃ inthe previous sentence. In fact, if there exists a j such that with a path of length k, wecan reach any l ∈ {0, 1, . . . , N − 1}, with a path of length 2k we can go from anyj ∈ {0, 1, . . . , N − 1} to any l ∈ {0, 1, . . . , N − 1} (just go to j in k steps and thento l in k additional steps).

Thus, we get that:

Claim C1: complete controllability ⇒ ∃ j ∈ {0, . . . , N − 1} and k ≥ 0 suchthat N k( j) = {0, . . . , N − 1}.This necessary condition can be checked indirectly with the methods of the previ-

ous sections. The constructive algorithms we are going to describe will imply that thisnecessary condition is indeed sufficient to get controllability between two arbitrarystates for our model. We shall refer to this type of controllability as state controllabil-ity. Moreover our results will imply an upper bound on the number of steps needed for

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arbitrary state transfer in terms of the maximal (over j)k such N k( j) = {0, . . . , N −1}and of the order r of the conditional shift matrix S.

The next proposition provides a first k-steps control algorithm to go from a statewith probability 1 to find the walker in a fixed node j, i.e., a state of the type |c0〉⊗| j〉,to one where the probability is arbitrarily distributed on the nodes in N k( j). Even ifthe proof of the next proposition, as well as the proof of Proposition 5.5, will be givenby induction, they are constructive. We present an example where these constructiveprocedures are used in Sect. 7.2.

Proposition 5.2 Let j be any node and Ak = {v1, . . . , vl} be any subset of N k( j).Fix a state of the type |ψ0〉 = |c0〉 ⊗ | j〉 and complex coefficients (α1, . . . , αl) with∑l

a=1 |αa |2 = 1 on the nodes of Ak. Then it is always possible to construct a controlsequence C1, . . . ,Ck of coin operations such that:

SCk · · · SC1|ψ0〉 =l∑

h=1

αh |ch〉 ⊗ |vh〉, (24)

for some values of the coin variables ch (not necessarily distinct).

Proof We will prove the statement by induction on k.If k = 0, then the statement is obvious. Assume that the proposition holds for k.Let Ak+1 = {v1, . . . , vl} ⊆ N k+1( j). By definition of N k+1( j) we have that

Ak+1 = {v1, . . . , vl} = {π1(w1), . . . , πl(wl)}, where for i = 1, . . . , lwi ∈ N k( j),and π1, . . . πl are permutations in the set {π1, . . . , πd}. The nodes wh need not tobe different. Denote by s the number of distinct elements in {w1, . . . , wl}, and letAk = {w1, . . . , wl} = {z1, . . . , zs} where all elements are distinct in the secondset notation. Without loss of generality, we assume that we have ordered the nodesvh ∈ Ak+1 in such a way that the first g1 of wi are equal to z1, the second g2 of wi

are equal to z2 and so on. We have:

z1 = w1 = · · · = wg1 ,

z2 = wg1+1 = · · · = wg1+g2 ,...

zs = wg1+···+gs−1+1 = · · · = wg1+···+gs ,

with g0 := 0. Moreover denote by ch the coin value that corresponds to the transitionfrom wh in N k( j) to vh in N k+1( j), i.e.,

πchwh = vh . (25)

Let α1, . . . , αl be the given coefficients (cf. (24)), satisfying∑l

h=1 |αh |2 = 1. Wecan assume these coefficients all different from zero, without loss of generality, as inthe case where one of them is zero we can eliminate the corresponding vh from thesum (24).

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Define for i = 1, . . . , s,

γi :=

√√√√√g1+···+gi∑

h=g1+···+gi−1+1

|αh |2. (26)

By the inductive assumption, since Ak is a subset of N k( j), it is possible to construct,from |ψ0〉 = |c0〉 ⊗ | j〉, a sequence of k coin operations that steers |ψ0〉 to

|ψ〉 =s∑

i=1

γi |δi 〉 ⊗ |zi 〉,

for some states of the coin |δi 〉. Let Qzi be any unitary matrix such that:

Qzi |δi 〉 := 1

γi

g1+···+gi∑

h=g1+···+gi−1+1

αh |ch〉, (27)

where the |ch〉 are the ones defined in (24) and the γi ’s are all different from zerobecause so are the αh’s.

Define a coin tossing operation Ck+1 as the matrix where, for the nodes zi we usethe previous matrix Qzi , and for the other ones we use an arbitrary Q in U (d), e.g.,the identity. We have:

SCk+1(|ψ〉) = S

(∑

i

γi (Qzi |δi 〉)⊗ |zi 〉)

= S

⎛

⎝∑

i

⎛

⎝g1+···+gi∑

h=g1+···+gi−1+1

αh |ch〉⎞

⎠ ⊗ |zi 〉⎞

⎠

= S

(l∑

h=1

αh |ch〉 ⊗ |wh〉)

=l∑

h=1

αh |ch〉 ⊗ |πihwh〉=l∑

h=1

αh |ch〉 ⊗ |vh〉,

as desired. In the last equality, we used (25).

The next proposition shows how to reach a state of the form given by the right handside of (24), where the |ch〉’s are replaced by an arbitrary superposition of coin states.

Proposition 5.3 Let j be any node, assume that N k( j) = {v1, . . . , vl}, and fix anystate of the type |ψ0〉 := |c0〉 ⊗ | j〉, with |c0〉 an arbitrary state in C. Then in at mostk + r steps (where r is the order of the conditional shift matrix S), we can reach,from |ψ0〉, any state of the type |ψ f 〉 := ∑l

h=1∑d

s=1 αhs |cs〉 ⊗ |vh〉 for arbitrarycoefficients αhs such that

∑lh=1

∑ds=1 |αhs |2 = 1.

Proof Define βh = ∑ds=1 |αhs |2. We can assume, without loss of generality, that

βh �= 0. In fact, if βh = 0, then necessarily αhs = 0 for all s = 1, . . . , d, and so in

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this case we can just eliminate |vh〉 from the sum that defines |ψ f 〉. From Proposition5.2 we have a sequence of k coin operations C1, . . . ,Ck such that

SCk · · · SC1|ψ0〉 =l∑

h=1

βh |ch〉 ⊗ |vh〉,

for some values of the coin variables ch . Let Qvh be any unitary matrix such that

Qvh |ch〉 := 1

βh

d∑

s=1

αhs |cs〉.

Choose a coin tossing operation Ck+1 as the matrix where in the nodes vh we use theprevious matrix Qvh , and in the other nodes we use an arbitrary Q in U (d). LettingCk+2 = · · · = Ck+r = 1, we have:

SCk+r · · · SC1|ψ0〉 = Sr Ck+1

(l∑

h=1

βh |ch〉 ⊗ |vh〉)

=l∑

h=1

βh(Qvh |cih 〉)⊗ |vh〉 =l∑

h=1

d∑

s=1

αhs |cs〉 ⊗ |vh〉,

as desired. ��Remark 5.4 In some cases one can choose values C1 and C2 for the coin transforma-tions so that

C2SC1 = S−1. (28)

In these cases, we can replace Ck+1 above with C1Ck+1 and Ck+2 = 1 with C2 andomit all the following steps to have SCk+2SC1 = 1 in the proof of the above theorem.In these cases, one can replace r with 2 in the statement of the above proposition.

The previous propositions have shown how to go from a state with walker in a singlenode j to a state where the walker is distributed according to an arbitrary superpositionof states v ∈ N k( j). The following proposition shows how to perform the conversetype of state transfer.

Proposition 5.5 Let j be any node, let N k( j) = {v1, . . . , vl}, and fix any state of theform

|ψ0〉 =l∑

h=1

d∑

s=1

αhs |cs〉 ⊗ |vh〉, (29)

for arbitrary coefficients αhs such that∑l

h=1∑d

s=1 |αhs |2 = 1. Then there exists asequence of coin operations of length at most k that steers the initial state |ψ0〉 to astate of the type |ψ f 〉 = ∑d

s=1 γs |cs〉 ⊗ | j〉.

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Proof As in Proposition 5.2, we will prove the statement by induction on k.If k = 0, then the statement is obvious. Assume that the proposition holds for k.Let N k+1( j) = {v1, . . . , vl} = {π1(w1), . . . , πl(wl)},wherewh ∈ N k( j). Notice

that, for all h = 1, . . . , l, since πh(wh) = vh, there exists a coin value c j (h) such thatπ j (h)(vh) = wh . Let

γh :=√√√√

d∑

s=1

|αhs |2,

where αhs are the ones defined in (29). We can assume γh �= 0, otherwise we caneliminate |vh〉 from the sum in equation (29). Let C1 be a coin tossing operation

C1 :=l∑

h=1

Qvh ⊗ |vh〉〈vh | + Q ⊗(

1 −l∑

h=1

|vh〉〈vh |), (30)

where Q is an arbitrary unitary on the coin space C and

Qvh

(1

γh

d∑

s=1

αhs |cs〉)

= |c j (h)〉.

Then we have:

SC1

(l∑

h=1

d∑

s=1

αhs |cs〉 ⊗ |vh〉)

= S

(l∑

h=1

γh |c j (h)〉 ⊗ |vh〉)

=l∑

h=1

γh |c j (h)〉 ⊗ |wh〉.

This concludes the inductive step, since the nodes w1, . . . , wh are in N k( j). ��The following theorem combines the previous propositions to give an algorithm to

transfer between arbitrary states. It also gives an upper bound on the number of stepsrequired for an arbitrary state transfer.

Theorem 7 If a quantum walk is completely controllable then there exists a node jsuch that N k( j) = {0, 1, . . . , N − 1}, for some finite k. Denote by k j the minimumk such that this is possible. In that case the property is true for every j . Vice versa ifsuch a j exists, we can transfer between two arbitrary states (state controllability).Assume such a j exists and let k be defined as

k := minj∈V

{k j | N k j ( j) = {0, 1, . . . , N − 1}

}. (31)

Let r be the order of the conditional shift matrix S. Then any state transfer can beperformed in at most 2k + r steps.

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Proof The fact that complete controllability implies that there exists a j and a k withN k( j) = {0, 1, . . . , N − 1} was already proven in Claim C1. We now prove that theexistence of such a j implies state controllability. The previous results show how it ispossible to go from a state of the form |ψ0〉 := |c0〉 ⊗ | j〉 to any state of the form (29)where the vh’s are in N k( j) (Proposition 5.3) and vice versa (Proposition 5.5). If thereexists a j such that N k( j) = {0, 1, . . . , N −1}, then the state in (29) is just an arbitrarystate and we can go from an arbitrary state to a state of the form |ψ0〉 = |c0〉 ⊗ | j〉 ink steps and from this state to an arbitrary state in k + r steps. Therefore every statetransfer is possible and it takes at most 2k + r steps. This shows state controllabilitywith at most 2k + r number of steps. Now k depends on j, and we can choose theminimum value k = k j . By choosing k as k in (31), i.e., minimizing over j ∈ V, wehave that an arbitrary state transfer can be obtained with at most 2k + r steps. ��

6 Graph theoretic characterization of controllability

Theorem 7 deals with complete controllability and state controllability of a quantumwalk. These two controllability notions, in general, are not equivalent [1] as completecontrollability implies state controllability but not vice versa. The conditions provedin Theorems 5 and 6, (such as G R connected) which are equivalent to complete con-trollability, imply the existence of a j with N k( j) = {0, 1, . . . , N − 1}. However,from Theorem 7 only follows (constructively) that the condition on N k( j) implies theweaker notion of state controllability. The next result fills this gap and gives a per-fect if and only if condition. The theorem also summarizes the controllability criteriaobtained so far and adds one more criterion of graph theoretic type (point 4 below).

Theorem 8 For a quantum walk, with dynamics described by Eqs. (3)–(7), the fol-lowing conditions are equivalent:

1. The quantum walk is completely controllable.2. The reduced connectivity graph G R is connected.3. The graph G is such that for every couple of vertices w and s in V there exists a

path of even length connecting w and s.4. The connected graph G is not a bipartite graph.5. For every j ∈ V, there exists a k j such that N k j ( j) = {0, 1, . . . , N − 1}.6. The quantum walk is state controllable.

Proof The equivalence between conditions 1 and 2 was proved in Theorem 6, whilethe equivalence between conditions 5 and 6 was proved in Theorem 7.

We next prove the equivalence of conditions 2 and 3. Two vertices r and s areconnected by an edge in G R if and only if (21) is verified. Therefore two verticesw and s are in the same connected component of G R if and only if there exists asequence of permutations of the form πk

l π−km , with l,m ∈ {1, 2, . . . , d} and some

k = 0, 1, 2, . . . transferring w to s. This is equivalent to the fact that there exists asequence of permutations of even length transferring s to w. To see this, first assumethat

w =∏

t

πktltπ−kt

mts, (32)

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for some lt ,mt ∈ {1, 2, . . . , d} and positive integers kt . For any y ∈ V, and anyπm,m ∈ {0, 1, . . . , d}, y and π−1

m y are connected in the graph G. This means thatthere exists a πl such that π−1

m y = πl y. Therefore we can replace every permutationwith a negative power with a (possibly different) permutation with positive power in(32) and obtain our claim. Vice versa if

w =∏

t

πltπmt s, (33)

we can replace all the permutations πmt with negative powers of permutationsand obtain an expression of the form (32). Notice that this also shows that wecan restrict ourselves to considering kt = 1 in (32). Since to every permutationπl , l = {1, 2, . . . , d}, there corresponds an edge in G, this shows the equivalencebetween 2 and 3.

The equivalence between 3 and 4 is a standard fact in graph theory [9]. We givea proof for completeness. If the graph G is bipartite, divide accordingly the set ofvertices V in two disjoint sets V = V1

⋃V2 such that only edges between elements

in V1 and elements in V2 exist. Therefore if w is in V1 and s is in V2 the only pathsconnecting w and s have an odd number of edges and property 3 is not verified. Thisproves by contradiction 3 ⇒ 4. To prove (again by contradiction) that 4 ⇒ 3, assumethat 3 is not verified. Introducing an equivalence relation saying that two vertices areequivalent if and only if there exists an even path between them, this partitions the setV in two subsets V1 and V2 which are both non-empty if 3 is not verified. Every edge,being a path of odd length can only connect a vertex in V1 and a vertex in V2 and thegraph is bipartite.

We conclude by proving the equivalence of 3 and 5. Assume 3 is verified and fix aj ∈ V . Then for any w ∈ V there exists a path of even length from j tow. Let 2kw bethe length of the path, depending on w, and let 2kmax be the maximum length, maxi-mized over the w’s. We can go from j to any w ∈ V in exactly 2kmax steps. We justfollow the previously mentioned path for 2kw steps and then ‘oscillate’ back and forthwith any neighbor kmax − kw times. Therefore condition 3 implies that, given j, thereexists a k j = 2kmax (even) such that we can reach any vertex in V in exactly k j steps onthe graph. Vice versa, if, given j, there exists such a k j with N k j = {0, 1, . . . , N −1},taken any pair w and s in V, they are both in N k j . Therefore, we can connect w ands with a path which connects first w to j in k j steps and then j to s in k j steps andobtain an even path from w to s. This shows that 5 implies 3.

An important consequence of the controllability criterion given in Theorem 8 is thatalthough the quantum walk and the concept of controllability were studied in connec-tion with the defining permutations {π1, . . . , πd}, the controllability of the model doesnot depend on the particular set of permutations π j . In fact, we have the following:

Theorem 9 Controllability of a quantum walk on a graph only depends on the topol-ogy of the graph and not on the particular permutations {π1, . . . , πd}.

In view of this result, one may in principle use the criteria of Theorems 6 and 8 toinvestigate purely graph theoretic properties of a graph G by constructing a quantumwalk on it.

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7 Examples

7.1 Graphs of degree 2

The simplest non-trivial example are quantum walks on cycles, i.e., graph of degree2. For these examples, the controllability properties for the fully centralized case, i.e.,with the coin operation identical for every vertex, were studied in [8] and generalizedto lattices in [2]. Let us denote by |+〉 and |−〉 an orthonormal basis of the two-dimen-sional coin space C. Thus the coin tossing operation will be of the form (3) withQ j ∈ U (2), and the conditional shift will be of the type:

S =(

P+ 00 P−

). (34)

Here P+ and P− are two matrices representing the permutations associated with thetwo coin values + and −, respectively. The possible quantum walks on the cycle aredescribed in the following proposition.

Proposition 7.1 If d = 2 then the matrices P+ and P− of equations (34) are neces-sarily of the following form:

(a) P+ is the matrix representing a complete cycle,π+ i.e., (after possibly relabelingthe vertices) π+ := (0 1 2 · · · N − 1) and P− = P−1+ .

(b) P+ and P− are the matrices representing permutations π+ and π−, respec-tively, that are sequences of exchanges of two adjacent symbols, i.e., (afterpossibly relabeling the vertices) π+ := (0 1)(2 3) · · · (N − 2 N − 1), π− :=(1 2)(3 4) · · · (N − 3 N − 2)(N − 1 0). This case is possible only when N iseven.

Proof Let π+ be the permutation on the nodes given by the matrix P+.Write π+ as a sequence of cycles, (01 · · · r1)(r1 + 1 · · · r1 + r2) · · · (r1 + r2 +

· · · rk · · · N − 1), for k ≥ 1. Since from the assumption H2) (cf. Sect. 2) we do nothave self-loops, all cycles must have length ≥ 2. If all cycles are of length 2, thenwe have a sequence of N

2 exchanges, and we must necessarily have that N is even.Assume now that there exists a cycle of order p +1 > 2, therefore, modulo a possiblerelabeling of the vertices, we have

π+ = (01 · · · p)π ′.

We need to show that p = N −1. Assume, by the way of contradiction, that p < N −1.Since the permutation π+ corresponds to the edges of the graph G, all the nodes{0, 1, . . . , p} must have two edges, one connecting i to i + 1 and the other connectingi to i − 1(mod N ). If p < N − 1, since G is regular and of degree 2, there cannotbe any edge connecting one of the first p nodes with the remaining nodes. This factcontradicts the connectedness assumption on G, thus the only possibility is p = N −1.

Now if we are in the case where π+ = (0 1 · · · N − 1), then, π+ corresponds tomotion along every edge in one direction. Necessarily π− will correspond to motionalong the edges in the opposite direction, i.e., π− = π−1+ .

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On the other hand, assume that π+ is a sequence of exchanges, and let π− be thepermutation corresponding to P−. By repeating the same argument as before, we con-clude that π− is either a sequence of exchanges or a complete cycle. However thelast choice is not possible otherwise the permutation given by π+ would have to be itsinverse, which is again a complete cycle. By examining the graph, it also follows that ifπ+ := (0 1)(2 3) · · · (N −2 N −1), then π− := (1 2)(3 4) · · · (N −3 N −2)(N −1 0).

��As we have seen in Theorem 9, the controllability of the quantum walk does not

depend on the particular walk considered but only on the graph. According to theprevious proposition, in the case N odd we have only one possible type of quantumwalk, while in the case N even, for the same N there may be two non-equivalent walks.However their controllability properties must coincide according to Theorem 9.

Let us treat the case N odd first. Apply Algorithm 1, and computeπ+π−1− . We obtain

π+π−1− = π2+ = (0 2 4 · (N − 1) 1 3 · · · N − 2), (35)

which is a full cycle. Therefore the reduced connectivity graph G R, in this case, isconnected and the system is controllable.

Alternatively, we can apply the test of Theorem 7, that is 5 of Theorem 8. In thiscase, we also get an upper bound on the number of steps needed for controllabil-ity. Consider the node 0 and the associated sets N k(0). We have that N N−1(0) ={0, 1, 2, . . . , N − 1}. In order to see this order the nodes of the cycle in clockwiseorder from 0 to N −1. To see that from 0 it is possible to reach in N −1 steps any node0, 2, . . . , N − 1, notice that for j = 0, . . . , N−1

2 , we can reach the node N − 1 − 2 jby moving j times between 0 and 1 (so having 2 j steps) plus performing N − 1 − 2 jadditional steps clockwise. Analogously, one can see that {1, 3, . . . , N − 2} are inN N−1(0). To reach 1 + 2 j, for j = 0, 1, . . . , N−3

2 in N − 1 steps, one can movej times between 0 and 1 (and this gives 2 j steps) and then move counterclockwisewith N − 1 − 2 j additional steps. It is also easy to see that N − 1 is the minimumk so that N k(0) = {0, 1, . . . , N − 1} and this minimum value would be the sameif we considered another node instead of 0. Therefore k in (31) is N − 1 and sincer = N in this case the upper bound on the number of steps given by Theorem (7) is2(N − 1) + N . One can in fact get a better bound since in this case the conditionsdescribed in Remark 5.4 are verified with

C1 :=(

0 1−1 0

)⊗ 1 and C2 = C−1

1 :=(

0 −11 0

)⊗ 1. (36)

Extensions of these controllability results can be obtained. For instance, applyingProposition 4.2 one has that q-dimensional lattices (with q ≥ 1) with on odd numberof vertices in every dimension necessarily give rise to controllable quantum walks.

For the case N even, consider first the case where the two permutations π+ andπ− are full cycles. Applying the criterion of Algorithm 1 we study the permutationsπk+π−k− = π2k+ . We see that for every k, π2k+ is given by two cycles of length N

2 eachcontaining only even or odd numbered vertices. Therefore the reduced connectivity

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Fig. 2 Graph with N = 6 andd = 3

graph has two connected components each with N2 vertices and the system is not con-

trollable. The dynamical Lie algebra is the direct sum of two su(N ), plus multiplesof the identity, according to Theorem 6. If we apply the criterion of Theorem 7 wefind that N k(0) contains only even (odd) numbered nodes for k even (odd) and thisimplies that the system is not controllable. In the remaining case, an application ofAlgorithm 1 gives the same dynamical Lie algebra and using the criterion of Theorem7 gives the same sets N k(0) (the criterion is independent of the defining permutationsand the graph is the same) (Fig. 2).

7.2 Example of the controllability algorithm of Sect. 5

Consider the quantum walk whose graph is given in Fig. 1. The graph has 6 nodesand degree d = 3. Thus any associate quantum walk has state space of dimension18 = 6 · 3.

For this graph, it is easy to see that we have:

N 1(0) = {1, 3, 5},N 2(0) = {0, 1, 2, 4, 5},N 3(0) = {0, 1, 2, 3, 4, 5}.

This fact implies that any quantum walk on this graph will be completely controllable.Let us consider the problem to steer the initial state

|ψ0〉 = |+〉 ⊗ |0〉, (37)

i.e., a state where the probability is concentrated in the 0 node, to a final state |ψ f 〉with the probability uniformly distributed among all the nodes, i.e., |ψ f 〉 of the form

|ψ f 〉 = 1√6

5∑

j=0

|c j 〉 ⊗ | j〉 (38)

where |c j 〉 are general (not necessarily basis) states in C.

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We assume, as described in the picture, that the two coin values |+〉 and |−〉 cor-respond to permutations π+ = (0 1 2 3 4 5) and π− = (0 5 4 3 2 1) while with thethird coin value, which will be denoted by |c〉, we associate the permutation πc =(0 3)(1 5)(2 4). We proceed by using the procedure described in Proposition 5.2. Firstconsider N 3(0).

N 3(0) = {0, 1, 4, 2, 3, 5}= {π+(5), πc(5), π−(5), πc(4), π−(4), π+(4)}. (39)

Thus, using the notations of Proposition 5.2, here we have z1 = 5 and z2 = 4. Noticethat this choice is not unique, in fact for example 1 = πc(5) = π+(0). Any possiblechoice will lead to different sequence of coin tossing operations.

The expression (39) suggests that if we were in a state

|ψ2〉 = 1√2|c4〉 ⊗ |4〉 + 1√

2|c5〉 ⊗ |5〉, (40)

and applied a coin operation (we denote by In the identity of dimension n)

Q5 ⊗ |5〉〈5| + Q4 ⊗ |4〉〈4| + I3 ⊗ (I6 − |5〉〈5| − |4〉〈4|) , (41)

with Q5(Q4) a unitary transformation mapping |c5〉 (|c4〉) to 1√3(|+〉 + |−〉 + |c〉) we

would obtain state of the form (38). Therefore the problem is reduced to obtaining astate of the form |ψ2〉 in (40). To do that we examine 4 and 5 in N 2(0) and we have4 = π−(5) and 5 = πc(1). This suggests that if we have a state

|ψ1〉 := 1√2|d5〉 ⊗ |5〉 + 1√

2|d1〉 ⊗ |1〉, (42)

we could transfer to a state of the form (40) by applying a coin transformation depend-ing on the walker which maps |d5〉 into |+〉 and |d1〉 into |c〉 followed by a conditionalshift. Finally, examining 5 and 1 which are in N 1(0), we have that 5 = π−(0) and1 = π+(0). Starting from a state |ψ0〉 in (37) and applying a coin transformationmapping |+〉 into 1√

2|−〉 + 1√

2|+〉 followed by a conditional shift S, we obtain the

state in (42). The procedure to go from |ψ0〉 to |ψ f 〉 applies the above procedure inreverse.

7.3 Controllability for density matrices

If a quantum walk is controllable and in particular state controllable we can apply thealgorithm described in Sect. 5 to transfer the state between two values |ψ1〉 and |ψ2〉 inC ⊗ W . In some applications, the state of the quantum walk is described by a densitymatrix ρ. Density matrices are Hermitian, trace 1, positive semi-definite matrices ofdimension equal to the dimension of the underlying Hilbert space (C⊗W in this case).They represent the state of an ensemble of quantum systems (cf., e.g., [16]). In this

123


case, the algorithm of Sect. 5 cannot be applied, unless ρ represents a pure state, i.e.,ρ := |ψ〉〈ψ | for some |ψ〉 ∈ C⊗W . However since we have complete controllability,for every two density matrices ρ1 and ρ2 with the same spectrum, we know that it ispossible to find a sequence of transformations X = ∏

j SC j to transfer, according to

ρ → XρX†, the density matrix ρ1 to ρ2, i.e., ρ2 = Xρ1 X†. We did not give in thispaper a general constructive algorithm for this. We remark however that the situationis a very familiar one in quantum control for which there exist many tools and tech-niques. The situation can be described as follows. We have a set of matrices (F in(8)) which generate all of u(d N ) and we are able to perform the exponential of eachone of these matrices (in this case via a sequence

∏j SC j ; cf. the proof of Theorem

1 and Proposition 3.1). Then, given X in the corresponding Lie group (U (d N ) in thiscase), we want to find a way to express X as a product of these exponentials. In ourcase, this means that we obtain X by concatenating the various sequences giving theexponentials.

Let us illustrate this using a simple example on a cycle with 3 vertices which, asdiscussed in Sect. 7.1, is controllable. Assume we want to transfer the density matrixfrom a value ρ1 = diag( 1

2 , 0, 0, 12 , 0, 0) to a value ρ2 = diag( 1

2 , 0, 0, 0, 0, 12 ). ρ1

represents an ensemble where half of the systems are in |+〉 ⊗ |0〉 and half are in|−〉⊗ |0〉, while ρ2 represents an ensemble where half of the systems are in |+〉⊗ |0〉and half are in |−〉 ⊗ |2〉. A matrix X ∈ U (6) which performs such a transfer is apermutation matrix corresponding to the permutation π := (1)(2)(3 6 4)(5) of rowsand columns. We need to express such a matrix X in U (6) as a sequence

∏j SC j .

In order to do that, we consider the matrices in F . A general matrix A1 in A and ageneral matrix S A2S−1, with A2 ∈ A for this example have the form

A1 =

⎛

⎜⎜⎜⎜⎜⎜⎝

ia1 0 0 α1 0 00 ia2 0 0 α2 00 0 ia3 0 0 α3

−α†1 0 0 ia4 0 0

0 −α†2 0 0 ia5 0

0 0 −α†3 0 0 ia6

⎞

⎟⎟⎟⎟⎟⎟⎠,

S A2S−1 =

⎛

⎜⎜⎜⎜⎜⎜⎝

ib1 0 0 0 β1 00 ib2 0 0 0 β20 0 ib3 β3 0 00 0 −β†

3 ib4 0 0−β†

1 0 0 0 ib5 00 −β†

2 0 0 0 ib6

⎞

⎟⎟⎟⎟⎟⎟⎠, (43)

for arbitrary real numbers a1, . . . , a6, b1, . . . , b6 and complex numbersα1, α2, α3, β1, β2, β3. Choosing all elements equal to zero but α3 = π

2 in A1 we obtainthat C1 = eA1 corresponds to a permutation π1 := (1)(2)(4)(3 6)(5). Choosing allentries equal to zero in S A2S−1 except β3 = π

2 we obtain that eS A2 S−1 = SeA2 S−1 =SC2S−1 corresponds to a permutation π2 := (1)(2)(3 4)(5)(6). Therefore C1SC2S−1

corresponds to the desired permutation π1π2 = (1)(2)(3 6 4)(5) = π . Notice now that

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C1 and C2 are available coin tossing transformations. Using C1 and C2 as in Remark5.4 (cf. (28)) and in (36), we obtain

C1SC2S−1 = SC2SC1C1SC2C2SC1, (44)

which is an admissible sequence (of length 4) performing the desired transfer fromthe density matrix ρ1 to the density matrix ρ2.

The symplectic Lie group Sp(3) would be sufficient to obtain arbitrary transfersbetween two pure states because of its transitivity on the complex sphere (see, e.g.,[1]) but it would not be sufficient to perform the transfer between the two densitymatrices ρ1 and ρ2 in this example. In order to see this write ρ1 and ρ2 as blockdiagonal matrices

ρ1 := 1

2

(E1 00 E1

), ρ2 := 1

2

(E1 00 E3

),

with E j the 3 × 3 matrix which is all zeros except for the ( j, j)−th position whichis occupied by 1. Using the parametrization of a general element X of the symplectic

group as, X := (A −BB A

),with A and B3×3 matrices, a straightforward computation

shows that Xρ1 X† has the form

Xρ1 X† :=(

K1 K2

K †2 K1

),

which is incompatible with the form of ρ2.

Acknowledgments D. D’Alessandro research was supported by NSF under Grant No. ECCS0824085and by the ARO MURI grant W911NF-11-1-0268. D. D’Alessandro also acknowledges the kind hospitalityby the Institute for Mathematics and its Applications (IMA) in Minneapolis where most of this work wasperformed. The authors also would like to thank the referees for a careful reading and pointing out a mistakein the previous version of the proof of Theorem 5.

Appendix A: Further remarks on the structure of the dynamical Lie algebra L

In this short appendix, we remark that the the number m of connected componentsof the reduced controllability graph in Theorem 6 can only be 1 or 2. In order tosee this, define an equivalence relation ∼ on the set of vertices V saying that a ∼ bif there exists a path of even length connecting a and b. The partition of the set Vconsidered in Theorem 6 corresponds to partition in equivalence classes with respectto this equivalence relation according to the discussion in the proof of Theorem 8.Now, fix a j ∈ V and consider a set Vo( j) as the set of vertices that can be reachedby j in an odd number of steps and a set Ve( j) of vertices that can be reached inan even number of steps. Clearly V = Vo( j)

⋃Ve( j). Moreover if a and b are in

Vo( j) (or Ve( j)), a ∼ b. Therefore either V = Vo( j) = Ve( j) or Vo( j) and Ve( j) are

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disjoint and they give two connected components in the reduced connectivity graph.This discussion shows that the example of the cycle discussed in Sect. 7 is somehowprototypical. It also shows that another equivalent condition of controllability is thatgiven a j ∈ V we are able to find a vertex which we can reach in both an odd and aneven number of steps.

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