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NTRODUCTION ok eciewy fdsrbtn nageetbe- entanglement Previous distributing [23]. of ways (QKD) describe distribution works be- key (PST) quantum [22] [21], transfer and teleportation [20], state [19], nodes perfect network network tween quantum perform the to in havescenario employed and successfully [16]–[18] been computing are quantum walks Quantum for [15]. universal on walks based quantum computing time discrete quantum distributed for protocol requirements. connectivity universality network maintaining and while switches quantum implemen- of physical tech- tations abstract to network need computing protocols quantum quantum network distributed that of indicates [14]. ecosystem nologies quantities diverse physical information quantum This distinct quantum in exchanging encoded of interconnecting the capable quantum in devices of investigation diverse description a a is name architectural to there [13] addition, [12], In Silicon- diamond few. and in [11] centers [10], color ions vacancy trapped supercon- [9], investigation, qubits ducting for under suited systems computation physical implementa- of hardware quantum plethora to a agnostic is be There to tions. it for need the one with connect directly not network another. are that a communication nodes cases, the between orchestrate both to necessary For is protocol information. exchange quantum continuous the of or of [8] application gates controlled the distinct remote either in demands qubits processors with quantum connectivity. computation network protocol quantum quantum for network Generic distributed accounting quantum desired while a a operation performing for of demand capable a Second, is the [7]. distance there in with rate effect entanglement decrease channel exponential depleting quantum in the e.g known data, physical quantum well of First, exchange a dimensions. have two channels least extends at computing in quantum the distributed considered, of is physical complexity scenario network that [6]. quantum qubits the fact among When talk the cross reduces given directly separation operations, quantum of h olo hsatcei opooeacontrol a propose to is article this of goal The is protocol control a of design the in challenge One tween nodes on a quantum network using the coin The description of the protocol is carried in the space of the walker to propel entanglement generation logical setting under the assumption that quantum error between qubits [20], [21]. In addition, the formalism correction processes ensure unitary operations for both of quantum walks with multiple coins enabled the control and data qubits. description of an entanglement routing protocol, which The remainder of this article is structured as follows. interprets qubits within a network node as vertices In Section 2, we present the mathematical background of an abstract graph used by a quantum walker to needed for the description of the quantum walk pro- generate entanglement [24]. In spite of their relevance, tocol. We describe the quantum walk protocol and the quantum walk approaches defined in the literature demonstrate its universality in Section 3. The descrip- are suited to particular network structures and quantum tion is extended to case of multiple walkers in Section operations. In particular, they consider the case of 4. In section 5, we apply the protocol to recover regular lattices, describing walker dynamics on regular the behavior of entanglement distribution protocols. structures and do not address how the quantum walk Finally, the manuscript is concluded in Section 6. can be used to perform generic distributed quantum op- erators. In this context, this work adds to the literature II. BACKGROUND of both quantum walks and quantum networks with the description of a quantum network control protocol Consider the graph G = (V, E). Let δ(v) denote the that can be applied to arbitrary graphs and perform set of neighbors of vertex v V and d(v) = δ(v) universal quantum computing in a quantum network. denote the degree of v. Let ∆(∈ u, v) denote the| hop-| distance between u and v in G. Throughout this work A. Contributions we refer to the inverse of a binary string x 0, 1 ∗ as x. A quantum network is a set of quantum∈ { hosts} The contributions of this article are three-fold. (quantum processors) interconnected by a set of quan- • We propose a quantum walk protocol for dis- tum channels that allow for the exchange of quantum tributed quantum computing in a quantum net- information [2]. A host is either a quantum repeater, a work. The protocol uses a quantum walker system quantum router or a quantum computer with a fixed as a quantum control signal to perform computa- number of qubits, which performs generic quantum tions among quantum processors that are physi- operations. A quantum network can be represented as cally separated. We assume that each processor a symmetric directed graph N = (V, E). Each node dedicates part of its internal quantum register to v V represents a quantum host that has a set v of ∈ M represent the walker control signal and describe qubits that can be processed together at any time and a how the control subsystem interacts with the data set v of qubits that is used to exchange quantum in- N subsystem. The interaction between data and con- formation with nodes in its neighborhood δ(v) through trol is specified by unitary operations that nodes a set of quantum channels. More precisely, each edge need to implement in order to realize the quantum (u, v) E represents a quantum channel connecting ∈ walk control plane. the qubits in u and v which can interact through N N • We show the universality of the protocol by operations mediated by the channel. We will refer describing how a 2-qubit controlled X (CNOT) to v and v as the networking and data registers N M operation between qubits in distinct nodes of the of node v, respectively. The sets = v and M v M network can be performed with the quantum walk. = v are respectively referred to as the network N v N S The protocol is universal in the sense that it allows control plane and the network data plane. S for any quantum operation in the Hilbert space This network model separates the qubit registers formed by all qubits in the data subsystem of the in the nodes into control and data registers. This nodes. Furthermore, it is generic in the sense that choice differs from previous works where the set of it abstracts hardware implementation and channel qubits that can interact with quantum channels transmissions, while being well-defined for any are consideredN alone [25], [26]. Note that considering network topology only networking qubits suffices to define entanglement • We demonstrate how the protocol can be used routing protocols. Nonetheless, it is straightforward for to recover the behavior of entanglement distribu- the goal of describing a quantum network control plane tion protocols previously described in the litera- protocol to consider the separation between control and ture [25], [26]. data registers in the nodes. A. Quantum network protocols information in the logical perspective. In particular, we Consider the system formed by two quantum pro- exploit the representation in (3) considering A and B cessors u and v connected by a channel and their as unitary operators provided by the network instead of respective qubits. A local operation is a quantum generic superoperators and perform the analysis in the transformation represented as a separable operator of state vector formalism. It is worth emphasizing that the the form unitarity assumption for operators A and B translates to the assumption that quantum error correction is O = Ou Ov, (1) ⊗ provided by the network. where Ou and Ov acts on the state space of the qubits B. Quantum walks on graphs at processors u and v, respectively. A local operation There are many ways to define a quantum walk assisted by two-way classical communication (LOCC) on a graph and this article focus on the discrete- is a local operation that depends on classical informa- time coined quantum walk model. Given a symmetric tion exchanged between nodes [7], e.g the unitaries directed graph G = (V, E), a coined quantum walk on of quantum teleportation. The classical information is G is a process of unitary evolution on the Hilbert space used to select what operators are to be applied to the = formed by the edges of the graph, system, which is embedded in the formalism by av- G V C whereH H codifies⊗ H vertices and is the coin space eraging non-unitary measurement operators into trace- V C of theH walker codifying the degreesH of freedom the preserving operators. A quantum network protocol for walker can move on. More precisely, every (v,u) E N is an algorithm that operates on the qubits at the defines a basis vector v,c for through a mapping∈ nodes, transforming their joint state by an LOCC ¯ G Λt between the edges incident| i to vHwith the set of degrees as of freedom v = 0, 1,...,d(v) 1 . As a convention, ¯ ¯ † C { − } ρ(t)= Λtρ(0)Λt , (2) we map the degrees of freedom of the walker at a given vertex v to the order of labels of its neighbors, such where t is the number of rounds in the protocol, ρ that v,c refers to the edge (v,u), where u is the c- is a density operator defined on the Hilbert space | i th smallest label in δ(v). Later, cvu will be used to M formed by all qubits on the network, ρ(0) is a refer to the degree of freedom of v that represents the Hdensity matrix where registers in distinct nodes are edge (v,u) and cv to represent the self-loop (v, v). The in a separable state. We express ¯ t as Λt = k=0 Λk generic state of the walker Ψ(t) = ψ(v,c,t) v,c a product of other LOCCs each applied at different | i | i Q is a superposition of the edges of G and the walker time steps [27]. It suffices for ρ(0) to be a separable evolution is defined as P density operator because the preparation of any non- separable state between registers in different network Ψ(t + 1) = S(t)C(t) Ψ(t) , (4) | i | i nodes can be represented by an LOCC. In fact, it is also where C and S are respectively referred to as the coin possible to model a protocol as a sequence of external and shift operators. The coin is a unitary operator of (mediated by channels) and internal (local to vertices) the form time-dependent superoperators At and Bt such that the state of the network is described as C(t)= v v Cv(t), (5) v | ih |⊗ ρ(t +1)= Bt[At[ρ(t)]], (3) X where Cv : C C. The shift can be defined as any where ρ is a density matrix characterizing the joint permutationH operator→ H on the edges of the graph that state of all network memories at discrete time t. As maps an edge (v1,u) to an edge (u, v2). This mapping an example, it is usually the case for entanglement of edges represents a permutation between states v1,c ′ | i distribution protocols that At represents channel en- and u,c , where u is the c-th neighbor of v1. Two shift tanglement protocols performed in all channels of operators| i are used throughout this work: the identity the network and Bt represents either entanglement operator, which is a trivial permutation of the edges, swapping operations or GHZ projections performed and the flip-flop shift operator given by independently in multiple nodes. The LOCC formalism Sf = v,cvu u,cuv , (6) has proven useful since it allowed for the derivation of | ih | v V fundamental bounds for entanglement rate distribution X∈ u∈Xδ(v) [27]. In this article, however, our focus is to address which applies, for every (v,u) E, the permutation ∈ network protocols for the transmission of quantum (v,u) (u, v). Sf reverses edges in the walker → wavefunction and is well defined for every symmetric of the walker. In particular, we consider the case where directed graph. The label of nodes and degrees of only a single data qubit in a given node is used to freedom are numerical and expressed as binary strings. control a unitary operation in the coin space of the In this setting, the degree of freedom c represents walker. Let the bit-wise negation of label c. Note that cvu is not Uvq(t)= Uvqs(t) s s Iq′ , (9) necessarily cuv. ⊗ | ih | q′ q s∈{X0,1} O6= III. QUANTUMWALKNETWORKCONTROLPLANE be a unitary operator acting on the coin space of vertex A direct way of controlling operations in a quantum v controlled by the qubit q v, defined for the joint ∈M network with a quantum walk is to consider a cou- space C M. The complete interaction has the form pled system between a walker and the qubits in the H ⊗H nodes, using supersposition in the walker system to O(t)= v v Uvq(t). (10) v | ih |⊗ implement controlled unitary operations. We describe X this joint system in the logical setting. Let N ′ = The operators correspond to a distributed implemen- (V, E (v, v), v V ) be the graph obtained by tation of a quantum walk system in the network. In ∪{ ∀ ∈ } adding self-loops to a network N. W = V essence, each node v contributes with v to describe H H ⊗ HC N denotes the space of a walker system on N ′ and the space W in N . Thus, (7) and (10) are operations H H = denotes the global Hilbert space controlled by the states of N that represent walker g W M H spannedH H by the⊗H systems. The walker system is assumed position since v is implemented by an entangled state | i to be implemented in the networking control plane, among qubits in . This encapsulates the necessary N such that W is a subspace of N . Since the Hilbert entanglement required for distributed controlled opera- space of theH walker system representsH an edge of N ′ tions between the qubits of the network. As previously as a basis vector, the dimension of the Hilbert space mentioned, we assume quantum error correction and must be at least E for to be a subspace. We describe the walker protocol in terms of logical qubits HN | | HW consider g as a subspace of N M. and operations. In the context of error correcting codes, H H ⊗ H a single logical qubit is implemented by a set of A. Control-data interactions physical qubits [28], [29]. An implementation of the We prescribe two laws of interaction between the walker demands (log( E )) logical qubits in since ′O | | N quantum walk system and the data qubits in the net- each edge of N is a basis vector of W and the di- work. The first generalizes a coin operator to the global mension of the space spanned by qubitsH is exponential Hilbert space g following in the number of qubits. For practical implementations, H network models that consider physical networking C(t)= v v Cv(t) Uv(t), (7) | ih |⊗ ⊗ qubits in the order of ( E ) have exponentially many v O | | X physical qubits as the required number of logical qubits where Cv(t): C C is a unitary operator on the for the walker, which is interesting for the purpose of H → H coin space of the walker and Uv(t): M M is quantum error correction. a unitary operator on the data spaceH of the→ network H written as B. Universal distributed quantum computing The interaction behavior described by (7) and (10) U t K t I , (8) v( )= v( ) Mv suffices for universal quantum computing. Our protocol u v O6= trivially allows for the application of any separable where IMv is the identity operator for the Hilbert space qubit operator in the network nodes, since there is no spanned by all qubits of vertex v. Essentially, C(t) need for the interaction between data qubits and the applies the operator Ku(t) on qubits in v if, and only walker in this case. Thus, we demonstrate universality if, the walker has a non-zero wavefunction component by demonstrating how a CNOT gate between any pair in v. Note that Uv(t) is the extension of Kv(t), which of qubits a and b, respectively at arbitrary nodes A and acts in Mv , to M with the identity operator. Note B, can be performed using the quantum walk control that theH operator describedH in (7) extends the definition plane. In fact, the demonstration works for any two- in (5) to perform unitary operations in the data qubit qubit controlled operation by substituting X with the space controlled by vertex position. desired single-qubit operator. Without loss of gener- The second law of interaction defines controlled ality, assume that a is the control qubit. To simplify operations between the data qubits and the coin space notation, we only define the operators C(t) in (7) and O(t) in (10) for the subspaces spanned by the qubits qubits can be performed with (7) and (10) given states in nodes A and B, considering undefined operators in the superposition (15), although it is not possible to to be identities, and omit the qubits in a,b . evolve the walker to that superposition from A, cA The subscripts following degrees of freedomM\{ inside} a without shift operators. | i ket specifies edges while the ones following qubits are We define coin and shift operators that evolve the considered as indices, e.g A, cA, 0a, 1b W 4 global state from (13) to (15). The quantum walk represents the walker in the| self-loop (A,i ∈ A H), a in⊗H the evolution is restricted to neighbor locality, such that, 0 state and b in the 1 state, where 4 is the Hilbert to have the state given in (15) at time t, all of the space| i spanned by 2 qubits.| i H wavefunction at time t 1 must exclusively be a We consider the initial state of the walker to be superposition of edges incident− to A, B and its neigh- A, cA , which corresponds to the self-loop in A. Thus, bors. There are many ways to define coin and shift the| globali system is described by the state vector operators with this desired behavior and we consider the case where the quantum walk traverses a single Ψ(0) = A, cA (α 0a + β 1a ) Ψb , (11) | i | i⊗ | i | i ⊗ | i path connecting A and B. Some auxiliary definitions where Ψb is the state of b. The interaction operator O and assumptions are required to describe the operators described| i in (10) is applied with the CNOT operation in context. Let p be a minimum path of the network connecting A and B with hop distance ∆(A, B). We UAa = I 0 0 + X 1 1 , (12) CA ⊗ | ih | CA ⊗ | ih | assume that every node knows the network topology which trivially generates the entangled state and that classical information can be transmitted across the nodes. Recall that the edges of N ′ are mapped to O Ψ(0) = (α A, cA, 0 + β A, cA, 1 ) Ψb . walker states following the relation | i | i | i ⊗ | i(13) (v,u) v,cvu , (16) This entangled state can be propagated in the network → | i through the evolution of the walker system such that and that the self-loop of node v is mapped to the degree the state after propagation becomes of freedom cv, for all v. We refer to the edge that connects A to its neighbor in p as A, cp and the | Ai Ψ(t) = (α A, cA, 0 + β B,cB, 1 ) Ψb . (14) reverse edge that connects B to its precedent vertex | i | i | i ⊗ | i in p as B,cp . This notation is depicted in Figure We refer to the propagation of the walker as the B 1 for a 2D-grid| i network. If some quantum operation transmission of the control signal through the network, which will be further explained in detail. The last step is performed between A and B at a given time t, the refers to the application of the extended coin operator propagation process starts at time t ∆(A, B). The extended shift operator used− to route informa- defined in (7), with K = X and all other operators b tion is fixed and given by defined as identity. The final state obtained is

S = Sf IM, (17) Ψ(t) = α A, cA, 0, Ψb + β B,cB, 1, Ψb , (15) ⊗ | i | i where S is defined in (6). The coin operator definition which clearly shows the application of a CNOT con- f is also time-independent, although it depends on the trolled by a with b as a target. minimum path p chosen. All operators C in (7) have Note that the state obtained in (15) is entangled with v the form the walker subsystem. We will later demonstrate how to separate the data qubits from the walker. Cp = c c + c c + c c (18) v | 1ih 2| | 2ih 1| | ih | c∈Cv C. Protocol execution in the network c=6 Xc1,c2

In the context of quantum network protocols, the where c1 and c2 refer to the degrees of freedom that operators defined by (7) and (10) do not capture the represent the edges incident to v in p. Thus, we define p propagation of control information across the network. Cv specifying c1 and c2 for the vertices of interest. p The propagation of quantum control information be- The unitary CA has c1 = cA and c2 = cA. Let u and tween neighbors in the network is embedded in the w be the neighbors of v p A, B on the path. Cv ∈ \{ } formalism through the walker’s shift operator. The has c1 = cvu and c2 = cvw, representing a permutation propagation in the case of non-neighboring nodes between the edges (v,u) and (v, w) in p. Finally, the p needs both coin and shift operators as defined in (5) operator CB has c1 = cB and c2 = cB. It suffices to set and (6). Note that generic computation on the data Kv(t)= IMv in (8) to perform the desired controlled B performed without the walker. In order to overcome this problem, the walker evolution is reversed after |B,cp i |w,c i B wB the controlled operation takes place by applying the B w inverses of the unitary operators used for propagation.

|w,cwv i |u, cvwi Since the coin and shift operators considered are per- mutation operators, they are Hermitian unitaries and, u v w v thus, are their own inverses. It takes ∆(A, B) time |u, cvui |v,cuvi steps to reverse the walker back to A and to transform A u the joint state of the system to the form

p |v,cuAi |A,c i A α A, cA, 0a, Ψb + β A, cA, 1a, Ψb . (19) | i | i A |A,cAi It should be clear that an extended coin operator with (a) Path p connecting A and B. (b) Map interaction of the form given in (7) and a measurement between edges on the walker produces one of the following separable and vectors. states Fig. 1: Notation for edges exemplified in a grid graph. A, cA (α 0a, Ψb + β 1a, Ψb ), Consider that A and B are two nodes connected in a | i⊗ | i | i (20) A, cA (α 0a, Ψb β 1a, Ψb ), 2D grid network. (a) p is a minimum path connecting A ( | i⊗ | i− | i and B with hop-distance 4 traversed by the walker. (b) with equal probability. It is straightforward from the Each edge on the path corresponds to a vector in W , separability between the walker system and the qubits H which appear in the walker wavefunction throughout a and b that the state after a partial trace operation movement. The degrees of freedom are defined such in the walker system is now equivalent to a CNOT that x, cxy represents edge (x, y). As an example, gate, up to a single-qubit Z gate conditioned on the | i the flip-flop operator specified in (6) maps v,cvu measurement outcome. | i → u,cuv , while the operator Cu defined in terms of (18) Note that moving the walker backwards is only one | i maps u,cuA u,cuv . of several ways to achieve a separable state between the | i → | i walker and the qubits a and b. In a nutshell, any walker dynamics that concentrates the walker’s part of the operation between a and b, although it is possible to wavefunction into a single network node suffices for perform operations controlled by a on the qubits in the this purpose, e.g propagating the walker wavefunction intermediate nodes as the walker moves by choosing from both A and B to an intermediate node. Under the Kv(t) accordingly. assumption of quantum error correction, this backward The overall behavior of the walker is straightforward propagation can occur simultaneously with any further and is illustrated in Figure 2. It is assumed that the quantum operation that nodes A and B may perform operator O defined in (13) is applied to the system at on the qubits and does not impact the latency of the the beginning of execution. The first application of the protocol. coin operator in A creates a superposition between the p IV. PROPAGATION OF MULTIPLE CONTROL SIGNALS states A, cA and A, cA that are entangled with a. | i | i p The flip-flop shift allows the wavefunction in A, c We demonstrated in the previous section how the | Ai to propagate along p and ensures that the A, cA quantum walk can propagate through a particular path | i component remains in the superposition throughout of the network connecting nodes A and B. This is protocol execution. Each coin flip routes information easily extended to the traversal of multiple paths as we on nodes internally, ensuring propagation. The net will show next. In this setting, k walkers can be used effect of ∆(A, B) successive applications of SC is the to simultaneously perform operations controlled by a superposition specified in (15). with k target qubits bj located in nodes Bj , for j 0,...,k 1 . This protocol execution translates to the∈ D. Separating data and control {parallel application− } of k +1 1-qubit gates controlled The state prescribed in (15) is an entangled state by a in terms of distributed quantum computation. between control and data. This implies that a partial The analysis carried for a single walker system trace operation in the walker system does not leave the extends to this case by considering a set of paths P state of a and b as it should be if the operation was such that, for every j, the path pj connects A and ∈P B B B B B

p |B,cB i w w w w w

u u u u u

|v,cvui

v v v v v

A |A, cAi |A,cp i |v,cvAi

z A A A A A |A,cAi |A,cAi |A,cAi |A,cAi |A,cAi (a) Initial state. (b) Coin in A. (c) Flip-flop shift (d) Coins in path. (e) Final state. Fig. 2: Protocol execution in a 5-by-5 grid with a quantum walk through a single path. Dark edges depict vectors which have non-zero wavefunction component in a given step. (a) The initial state of execution is the state generated by the application of the controlled operation demonstrated in (13). The node z is shown for illustration purposes and is not for propagation. (b) The first coin flip permutes the wavefunction into edges (A, A) and (A, v). (c) The flip-flop shift exchanges edge (A, u) with edge (u, A), moving the walker while mapping the self-loop edge to itself. (d) After the first coin flip, all subsequent coin operators work as shift operators inside a node, mapping degrees of freedom in order to propel the walker towards B. (e) After ∆(A, B) steps, the final wavefunction is a uniform superposition between edges (A, A) and (B, w), which can be used to perform an operation controlled by qubit a located in A with target qubit b located in B.

Bj . The path pj defines extended coin operators for the with S given in (17). j-th walker following (18). By carefully choosing the In addition to simultaneous 2-qubit control opera- qubits controlled by each walker, so that each walker tions, it is trivial to use multiple walkers departing interacts with a unique set of qubits, it is possible from A to perform controlled operations in the nodes to route k = walkers concurrently. This is a B , where the target qubit b in B is controlled by |P| j j j trivial extension to the single walker case because the qubit aj in A. The fundamental difference for this case separability of individual coins for each walker yield is that the O operator described in (13) is controlled by the k-walker coin operator for the set of paths to be qubit a when applied to the j-th walker, generating a P j of the form separable state among the walkers. k−1 P p C (t)= C j (t). (21) V. QUANTUM WALK PROPAGATION AND j=0 O ENTANGLEMENT DISTRIBUTION Since the memories affected by walker j are unique, As described previously, the operators defined in for every j 0,...,T 1 , each walker can be (7) and (10) allow for universal distributed quantum seen as a parallel∈ { control− signal} working on distinct computation among nodes of the network. In this sense, qubits. Note that, even though the walker evolution is the movement of the walker introduces entanglement performed by separable extended coin operators and across nodes. Given this is the case, it is suitable to shifts, the application of the controlled operation O consider entanglement distribution protocols in terms described in (10) before propagation implies that the of the quantum walk control plane. In this section, we walkers are maximally entangled with each other and apply the quantum walk control protocol to recover the with qubit a. Furthermore, the flip-flop shift operator behavior of two entanglement distribution protocols. In for each walker does not depend on a particular path particular, the recovered protocols distribute maximally and the joint operator for all walkers is described by entangled Greenberger-Horne-Zeillinger (GHZ) states k−1 across network nodes. We demonstrate, for both pro- P S = S, (22) tocols, how the entanglement introduced by the walker j=0 O can be used to produce the same entangled states. j Superposition of the walker wavefunction allows for while the interaction operator KA(0) is taken to be the creation of a Bell state between a and b as follows. the identity operator. The state of the system after the Assume that the joint state of the system is at time t application of the extended operator C(0) is described by 1 pj C(0) Ψ(0) = ( A, cA + A, c ). (26) | i √ | i | A i 1 j 2 Ψ(t) = ( A, cA, 0a, 0b + B,cB, 0a, 0b ), (23) | i √ | i | i O 2 Protocol execution is performed by the operators de- which is a separable state between the walker and the fined in (22) and (21) as specified in the previous qubits a and b. By setting UV = Xv I and CV (t)= I section, although the operators Kv are now defined for V A, B in (7), the extended⊗ coin operator C(t) to generate entanglement in the nodes as the walker ∈{ } is passes by. Let qvj0 and qvj1 denote the pair of qubits inside node v pj A, B that need to be entangled 1 ∈ \{ } j C(t) Ψ(t) = ( A, cA, 1a, 0b + B,cB, 0a, 1b ), together by walker j. The operator Uv in equation (8) | i √2 | i | i is defined for walker j by specifying (24) Kv(t)= Xqj Xqj Iq (27) 0 ⊗ 1 which is an entangled state between a and b. The q6=qvj0 ,qvj1 entanglement between a, b and the walker system can O j j be removed by transmitting the walker back to A as for all t, performing an X gate on qubits qv0 and qv1 explained in Section III-D. Note that only operators of controlled by the position of the j-th walker. Since the form prescribed in (7) are needed because there is paths are all edge disjoint, every qubit in the graph no control dependent on the state of either a or b. spanned by interacts with exactly one walker under P A GHZ generalizes Bell states to multiple qubits. the application of the generalized coin operator. For A k-qubit GHZ projection is a von-Neumann mea- walker j, the interaction operator with qubit aj is given surement operation in the GHZ basis and a Bell state by measurement (BSM) is a 2-qubit GHZ projection. Xaj I, when t =1, KA(t)= ⊗ (28) Ia I, when t =0, ( j ⊗ 6 A. Multi-path entanglement distribution with BSMs where the identity operator after the tensor product We start with the multi-path protocol defined in [25]. symbol acts on all qubits in A except aj. Considering Assuming global link state knowledge, it suffices to that uj is the first neighbor of A in the path pj , the choose the same paths that a complete round of the application of the evolution operator S(0)C(0) gives protocol uses to propagate entanglement. Consider that, 1 Ψ(1) = ( A, cA + uj,cu A ), (29) for such round, the set of paths P = p0,...,pk−1 is | i √ | i | j i { } j 2 used. In this protocol, every path pj used to distribute O entanglement yields a set of 2( pj 1) qubits in a GHZ where the data qubits are omitted for simplicity. | |− state, which comes from the net effect of performing S(1)C(1) generates an entangled state between aj and pj 2 BSMs in the intermediate nodes. Moreover, the qubits q and q and moves the walker to | |− uj0 uj1 the protocol assumes a network model where every the neighbors that are 2 hops away from A. Every quantum channel in the network is mapped in a 1-to- application of S(t)C(t) for t> 1 increases by two the 1 fashion with pairs of qubits in the nodes incident number of qubits entangled with walker j such that, to that channel. The logical behavior of this protocol when t = ∆(A, B), the walker j is in an entangled is recovered by sending one quantum walk from A state with the qubits in pj , aj and bj . Figure 3(a) through each path pj such that, once it arrives at B, depicts protocol execution for a 5-by-5 grid network. every qubit incident to the quantum channels in the In this case, P is formed by two shortest-paths in the path are in the same GHZ state. Thus, consider that grid, i.e K =2, and the entire evolution of the walkers all qubits in the network start in the 0 state. Initialize takes 4 steps. | i j k walkers in state A, cA . The operator CA(0) for the j-th walker can be| choseni as any unitary that maps B. Distributing GHZ states across network nodes GHZ projectors are used in [26] to improve the 1 pj multi-path BSM protocol described in [25]. The logical A, cA ( A, cA + A, cA ), (25) | i → √2 | i behavior of the GHZ distribution protocol can be

recovered in the quantum walk protocol by slightly modifying the description presented for the multi-path BSM protocol. In this protocol, every qubit in a node used to distribute entanglement is part of the final GHZ state achieved. This is in clear contrast with the BSM multi-path protocol, where each path defines an independent GHZ state. The first modification is that, B instead of preparing separable walkers with coins that maps states following the form given in (25), the k walkers need to be entangled in the state

1 p0 pj ( A, cA,...,A,cA + A, cA ,...,A,cA ). (30) A √2 | i

Note that this behavior can be obtained with an inter- action operator of the form (10) and a coin operator in the form specified by (18). Secondly, instead of rout- (a) Multi-path propagation. ing walkers through edge disjoint paths, the walkers traverse a tree of the network. More concretely, let be a spanning tree of the lattice, rooted at A. k is nowT the number of leaves in and is the set of paths T P B in the tree connecting A with leaves. For all vertices v, the interaction operator between the walker and the data qubits is u2 1 Wv Kv = Xqv , (31)

qv A O where Wv is the number of walkers that pass through v, i.e the number of leaves connected to A by v in u1 the tree. Note that Kv is the same for every walker. The exponent 1 is required because more than one (b) Spanning tree propagation. Wv walker may pass through the same vertex, and operate Fig. 3: Recovering protocol behavior with quantum on the same set of qubits. Since paths are taken from walks. Protocols that rely on GHZ projections apply a spanning tree, each walker can take a particular time entanglement swapping to distribute entanglement be- interval to traverse its path. Thus, the time necessary tween A and B. Entanglement swapping is non-unitary, to entangle all of the necessary qubit in the network although the overall effect of applying a GHZ projec- and remove the entanglement between the walkers tion is a maximally entangled state between qubits. and the data qubits is on the order of (maxj pj ), O | | The quantum walk approach recovers this behavior by where pj denotes the hop distance of path pj . Again, generating GHZ states unitarily. (a) A multi-path BSM protocol| execution| for the generic GHZ projector case protocol is recovered by sending separable walkers is exemplified for a 5-by-5 grid in Figure 3(b). In this into each edge-disjoint path used in a round of the case, k =7 and the whole process takes 7 time steps protocol, denoted in the figure by shaded edges. Each to complete, since 7 is the size of the largest path in path forms an independent GHZ state between all the tree. For nodes u1 and u2, the value of W is 1 and qubits in the path. 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http://arxiv.org/ps/2106.09839v1