The solvability of quantum k-pair network in a measurement-based way

Network coding is an effective means to enhance the communication efficiency. The characterization of network solvability is one of the most important topic in this field. However, for general network, the solvability conditions are still a challenge. In this paper, we consider the solvability of general quantum k-pair network in measurement-based framework. For the first time, a detailed account of measurement-based quantum network coding(MB-QNC) is specified systematically. Differing from existing coding schemes, single qubit measurements on a pre-shared graph state are the only allowed coding operations. Since no control operations are concluded, it makes MB-QNC schemes more feasible. Further, the sufficient conditions formulating by eigenvalue equations and stabilizer matrix are presented, which build an unambiguous relation among the solvability and the general network. And this result can also analyze the feasibility of sharing k EPR pairs task in large-scale networks. Finally, in the presence of noise, we analyze the advantage of MB-QNC in contrast to gate-based way. By an instance network \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${G}_{k}$$\end{document}Gk, we show that MB-QNC allows higher error thresholds. Specially, for X error, the error threshold is about 30% higher than 10% in gate-based way. In addition, the specific expressions of fidelity subject to some constraint conditions are given.

is systematically specified. In contrast to the standard GB-QNC, no entanglement is created during the computation, we thus have a clear distinction between the entanglement creation and its use as a resource. Secondly, by reducing k-pair task to a specific clifford group operation, we build an unambiguous functional relationship between the solvability and the network structure with 2k eigenvalue equations. Further, for the question how to construct coding scheme? We present another sufficient condition given by the stabilizer matrix. This conclusion can also analyze the feasibility in the communication task to sharing k EPR pairs over arbitrary networks.
The works above are mainly focused on the noiseless resource states and the perfect quantum operations. In 2016, Satoh 44 find that GB-QNC is more sensitive to noises especially on control operations. Specially, for pauli X and Z error, they prove that final fidelity would drop below 0.5 if the initial resource error is 10%. Motivated by these works, the performance of QNC in presence of noise is another concern in this paper. A second contribution of this paper is that we show the performance advantage of MB-QNC compared with GB-QNC. We will use a heuristic local Pauli-diagonal-noise channels to describe both errors. Since fidelity is usually difficult to calculate for complex network, we will work in stabilizer basis representation of mixed graph states by which the evolution of graph states can be easily tracked and just keeps them in diagonal form. Further, we present a set of constraint conditions to obtain more simpler expressions for fidelity in noisy resource error, noisy measurement error and both of them cases respectively. The property that multiple noises on the same qubit are equal to its the linear superposition makes us summarize all noises effects on resource state noise only. Finally, we apply above results to an instance butterfly network, showing that MB-QNC allows higher error threshold compared with the GB-QNC.

Results
Network Setting. The basic set up of a k-pair network is as follows. Specially, a general k-pair network here refers to a finite, directed and acyclic graph = G V E ( , ) RK , where V is the set of nodes or vertices, and E is the set of edges, whose elements are pairs of nodes that are adjacent. The set of its k source nodes is , , } RK k n k n 1 2 and arbitrary n intermediate nodes is The quantum operations on each node are defined by any trace-preserving completely positive(CP-TP) map. The goal in any given k-pair network is to communicate k independent messages simultaneously from each source to its corresponding sink. Clearly, this task depends on the particular properties of the network and it might or might not be possible to achieve for the given G, S, and T. We shall be concerned with cases where the actual network topology given by G does not allow disjoint paths between the qubits in S and the qubits in T, but we nevertheless want to achieve perfect state transfer. Let = C 2  be a Hilbert space where C is complex field, we say G RK is quantum solvable with fidelity F if there is choice of quantum operations(QNC scheme) over all nodes allowing us to send a quantum state ψ ψ | 〉 ⊗ ⊗ | 〉 ∈ ⊗  k k 1  supported on the source nodes to the sink nodes with fidelity at least F. In particular, when F = 1, it is simply called (perfect) solvable. Here we consider MB-QNC local projective measurements on a highly entangled resource graph state associated to the network are responsible for the transmission of quantum bits.
Among all research hotspots in k-pair network, Butterfly network and G k network keep appearing in a high frequency, as denote in Figs 1 and 2.
Graph state and its properties. Now, we review the concept of graph state, describe some of its properties, and fix the notation. It is sufficient to describe graph states with undirected finite graphs since the interaction of pairs of qubits are not necessarily bound up with the direction of edges. A undirected finite graph = G V G E G { ( ), ( )} is an ordered pair of sets: the set V G ( ) of nodes or vertices, and the set E G ( ) of edges, whose elements are pairs of nodes that are adjacent. In the following, we will mainly consider simple graph which contains neither loops (edges connecting nodes with itself) nor multiple edges between the same vertices. The set of nodes adjacent with v i is denoted by The adjacency relation of node v i can be represent by adjacency vector v i .   (4) and (8) can be verified by direct calculation with λ Λ = ( ) ij . Finally, Eqs (7) and (8) imply that The main task of MB-QNC is to design a suitable measurement pattern for a given network problem. The general steps of the MB-QNC scheme are as follows: (1) Encode information into initial graph state pre-shared among all distant nodes; (2) Apply local single qubit measurements on intermediate nodes and source nodes; (3) Send the measurement results to output nodes and apply local Pauli operations to correct phases.
A specific G k network for k = 2 and k = 4 have been examined in measurement-based way 39 . And for general k-pair network, exact conditions should be developed in measurement-based framework. In following, we will give the sufficient conditions.
The sufficient conditions for solvability of general k-pair networks. For quantum k-pair problem, all source-sink pairs wish to communicate simultaneously in network with common channel. This task would equivalent to a specific unitary operation-k qubits permutation τ U , where τ is a bijective with τ τ is any k qubits state held by source nodes and output by source nodes, we have Raussendorf et al. 43 shows the conditions for simulating any unitary operations successfully in measurement-based way. We generalize this conclusion to quantum k-pair problem.
Since the permutation operation mapped pauli operators onto pauli operators under conjugation, that is, Therefore τ U is Clifford group operations defined as the operations map Pauli group operators to itself under conjugation. Browne et al. 45 shows that all Clifford group operations can be implemented by measurement pattern with Pauli measurements alone which are parallelized in chronological order. This makes the solvability can be examined in relatively simple measurement pattern-Pauli measurement. Let are the function of measurement outcomes. Then k-pair network is solvable up to some local Pauli operators with the measurement pattern on M G ( ) RK and In G ( ) RK described by M  and X-basis measurements respectively, With the correlation between quantum k-pair networks and a highly entangled graph states, by reducing k-pair task to a specific clifford group operation, the sufficient conditions is constructed with 2k eigenvalue Equations (15) and (16), building an unambiguous functional relationship between the solvability and the network structure.
However, how should we design proper measurement pattern meeting these eigenvalue equations and what conditions the measurement pattern should exactly satisfies? From this point, another sufficient condition is given by the stabilizer matrix and adjacency matrix.
Note that by selecting a range of correlation centers denoted by = ∈ , we can construct k 2 stabilizer equations. Thus the product of these stabilizer generators are represented by the stabilizer matrix. Now, we consider the solvability with this stabilizer matrix and adjacency matrix.
Then network G RK is solvable with MB-QNC. Further, the exact parameters a and b in each measurement basis as in Eq. (12) can be obtained from A and B, respectively.
Note that Theorem 1 does not imply anything about how to construct measurement basis, but Corollary 2 does by the conditions stabilizer matrix A meet. It is helpful to propose specific network coding in measurement-based way. Note that Theorem 1 and Corollary 2 can also be used as the analysis of feasibility and the design of schemes for sharing EPR pairs , ( ) simultaneously for some communication tasks only if we do not measure input qubits since it is the unique quantum state (up to a global phase) which is stabilized by An instance network G k . In the following, we apply the results developed above to an instance k-pair network G k as shown in Fig. 2, in which k source-sink pairs sharing a common channel wish to communicate with each other simultaneously. Here we introduce k input nodes for quantum states to be transferred. Now, all + k 3 2 nodes are denoted by = 3 2 as shown in Fig. 3. In maths, all nodes i n n e t w or k g r ap h G k c a n b e d i v i d e d i nt o t w o d i s j oi nt p a r t s = + The solvability of G k network can be easily verified by Corollary 2. The adjacency matrix Λ of graph G k obtained from its partition is We find a matrix A Figure 3. Network G k with k introduced input nodes.
satisfies the conditions (1) and (2). Thus G k is solvable in MB-QNC. In fact, by matrix A, we can construct stabilizer equations , ,  Also with these measurement pattern, we can share k EPR pairs among the k source-sinks pairs simultaneously just leaving the input qubits unmeasured.
The performance benefits of MB-QNC over noisy k-pair network. The transmission of quantum resource states also measurement on them will reduce the performance of long-range quantum information processing schemes. Moreover, encoding on intermediate nodes even makes things worse. Satoh 44 shows that Gb-QNC is more sensitive to noise errors. The performance optimization in presence of noise is another concern of QNC. In this section, we explore the benefits of MB-QNC by k-pair communication instance G k and the merit of it is shown in terms of the output fidelity.
Noise model and assumptions. We mainly consider two kinds of noise sources: noisy resource states and noisy measurement caused by the errors on channel and imperfect projective measurements. Local Pauli-diagonal-noise channels are applied to described these errors. Specially, the effect on the ath qubit according to a probability is That is well justified since in large-scale network the separation among the participants is large enough so that collective effects need not be taken into account. Some special channels of this kind include the bit-flip channel (with random X noise) ε p ( ) Channel transmission errors. Generally speaking, the output fidelity of a general k-pair network is hard to calculate. However, we will work in stabilizer basis representation of mixed graph states by which the evolution of graph states can be easily tracked and just keeps the graph state in diagonal form. Further, we present a set of constraint conditions to obtain some simpler expressions of fidelity.
The noisy evolution of graph state in stabilizer basis: Let the prepared graph state be | 〉 G 0 . Since pauli operators map graph state basis to itself by properties (4) In this form, the fidelity can be described as Thus the final fidelity would be closely related to the change of parameter x. Firstly, we will describe it according the noise evolution of graph state. Since local Pauli-diagonal-noise channels on each subsystem except input qubits, = ∈ x , Under channel noise, the evolution of graph state which embodied in the coefficient of each stabilizer element can be described for a special error parameter as in Eq. (29).
Perfect measurement evolution in stabilizer basis. The action of Pauli measurements can also be easily described in this formalism as a transformation of the graph (up to some local unitaries) and also keep the graph state in stabilizer basis: it just multiplies each stabilizer basis by a coefficient. In terms of the stabilizer operators, the measurement of X u commutes with K u , anticommutes with all K v , ∈ u N v ( ). Thus After tracing out qubit u, the new stabilizer is where the new ′ K x corresponds to a new graph ′ G obtained from G by removing or transforming the neighborhood of vertex u. In fact, as described in Cuquet et al. 's ref. 47 measurement of Z simply disconnects the measured qubit from the rest of the graph, while X and Y transform the neighborhood of the measured qubit and then disconnect it. Here, the phase can be corrected by Z ( ) , each measurement on subsystem will remove some certain K x and leave the remains ′ K x combining like terms. Here, ′ G is closely related the choice of neighbor node v of u. Thus, by Hein et al. 's ref. 48 , w h e r e f o r }. In addition, for u and v, we can divide V as , X-basis measurement on u leads SCIENTIfIC REPORTs | 7: 16775 | DOI:10.1038/s41598-017-16272-x Theorem 2. In network G k , we denote K x as x u is deleted and others remain unchanged. For example, we measure node + v k 1 with X-basis, and ∈ = Thus, this measurement leads a new neighborhood that build a new graph ′ G as shown in Fig. 4. Each K x is transformed as ′ = . Here, all pauli components of the measured subsystem in K x are deleted. Furthermore, a constraint condition can be attained, that is + . The details about the transform of K x and the constraint conditions can be seen in Table 1. Finally, we get the + k 2 constraint conditions:      After all rounds of measurement, K x will be deleted as long as the conditions are not met. From the remaining, the exact fidelity associated with these constraints will be given. Denote x as the subscripts satisfying these constraint conditions, the final fidelity is represented as x is applied after noise acts on all particles that are subjected to the measurement, that is, i . Therefore, in the presence of imperfect operation, the Eq. (35) still holds. For the case of both these two noises, an important property is that multiple noises on the same qubit are equal to the linear superposition of multiple error parameters 49 where ′ p is a linear function of the channel noise p 1 and imperfectively operation noise p 2 . Therefore, one can summarize the effect of all noises by a single noisy channel ε , acting on all the network nodes(except the input nodes) should be understood as representing all kinds of imperfections in noisy resource state and noisy measurements. The final fidelity is represented as  44 , the MB-QNC has significant advantage in the final fidelity which can be shown in the task of sharing EPR pairs in k-pair network. We assume the same error = − p p 1 00 that each single subsystem suffers. Consider network G k instance for = k 2. Firstly, by the Eq. (34), we get the constraint condition + + = + + = + + = + + = . x x f For random X errors, = − p p 1 00 and = p p 10 . The fidelity can be calculated by In fact, for some f and e, P f 0, , P e 0 , will be deleted as the cancellation coefficient and the remaining can be seen in Table 3. Consequently, we get Eqs (38) and (39).
By comparing with the GB-QNC in Satoh et al. 's scheme, as shown in Fig. 5, MB-QNC allows higher error threshold. For X error, the error threshold is about 30% in MB-QNC is significantly better than in GB-QNC 10%. And the threshold for Z error, it is slightly better than it.

Discussion
In this paper, we firstly present sufficient conditions for the solvability of general quantum k-pair network with MB-QNC. It solve the central question in quantum communication whether a given network usually with capacity constraints can handle a specific joint communication task. With the correlation between quantum k-pair networks and a highly entangled graph states, the sufficient conditions is constructed by 2 k eigenvalue Equations (15) and (16). Thus a quantifiable relationship between solvability and network structure is built. Further, for the = x x x x x x x x x ( , , , , , , , )  Table 2. All x meet the constraint conditions in Eq. (37).
question how to construct coding scheme and what conditions the coding operations should exactly satisfies? We present another sufficient condition given by the stabilizer matrix and adjacency matrix. It would helpful in designing specific coding scheme for quantum communication tasks to attain high capacity. We find that it would be also helpful in constructively analyze the feasibility to sharing k EPR pairs over arbitrary networks. Finally, we show that in the same initial error parameter, MB-QNC allows higher error threshold compared with the existing GB-QNC. Thus, we optimize transmission schemes to better defend the effects of noisy resource states and noisy measurement caused by channel errors and imperfect local operations. Take an instance network G k , the analysis shows that for X error, the error threshold about 30% in MB-QNC is significantly better than in GB-QNC 10%. And the threshold for Z error is slightly better. This conclusion can be extended to any quantum k-pair network who has a classical linear network code since it is in effect a measurement-based procedure which performs only X-eigenbasis measurements, on a graph state with similar structure to the coding network.
In experimental implementations, despite the fact that projective measurement involved in measurement-based way can nowadays be implemented, the hardest task from experimental point of view is the distribution of graph states in large scale quantum network. In general, the fidelity drops exponentially with the growth of network size. At present, entanglement purification 50 and entanglement distillation 51 are two main techniques to improve the initial fidelity of the prepared entanglement states. For arbitrary (complex) network, the schemes for graph states distribution in the presence of noise have been implemented efficiently 47 . So, our scheme would be implemented in experiment with the current quantum techniques. This also makes MB-QNC schemes very attractive from an experimental perspective. Table 3. The parameters f and e keep P f 0, , P e 0 , . In Eqs (38) and (39), constant coefficients ∑ − ⋅ ( 1) x f x f , equal to 0 as the cancellation coefficient leading some P f 0, , P e 0 , cancelled. Take = f 11100000 and 01100000 for example which are belong and not belong to Table 3. . Figure 5. Compared with the GB-QNC in Satoh et al. 's scheme. For X error, the error threshold is about 30% in MB-QNC is significantly better than in GB-QNC 10%. And the threshold for Z error, it is slightly better.  (15 and 16). Therefore, k-pair network G RK is solvable by Theorem 1.
Proof of Theorem 2. In order to study the transformation of the subscript after measurement, we will discuss each component of the Eq. (32). First, it is easy to verified that the component ( ), ( ) ( )) 0 / since G k is a bigraph. Thus, Eq. (32) can be reduced to Also by the bigraph property, we have