Large-scale quantum networks based on graphs

Society relies and depends increasingly on information exchange and communication. In the quantum world, security and privacy is a built-in feature for information processing. The essential ingredient for exploiting these quantum advantages is the resource of entanglement, which can be shared between two or more parties. The distribution of entanglement over large distances constitutes a key challenge for current research and development. Due to losses of the transmitted quantum particles, which typically scale exponentially with the distance, intermediate quantum repeater stations are needed. Here we show how to generalise the quantum repeater concept to the multipartite case, by fully describing large-scale quantum networks, i.e. network nodes and their long-distance links, in the language of graphs and graph states. This unifying approach comprises both the distribution of multipartite entanglement across the network, and the protection against errors via encoding. The correspondence to graph states also provides a tool for optimising the architecture of quantum networks.

We show how general graph states, an important resource state for multipartite quantum protocols, can be distributed over large distances using intermediate repeater stations. To this aim we describe a one way quantum repeater scheme using encoding in the language of graph states. For a general Calderbank-Shor-Steane (CSS) code we do a refined error analysis that allows to correct qubit errors and erasures caused by imperfect preparation, gates, transmission, detection, etc.. We analyze the cost and repeater rate for this general scheme. The concept is exemplified by the 7-qubit Steane code and the quantum Golay code. Several fascinating aspects of quantum theory can be summed up as its "non-classicality", which is particularly striking in quantum entanglement. The benefit of quantum technology increases with the system size, e.g. when doing quantum computations [1,2]. You might expect this from the growth of the state space dimension. But additionally the "non-classicality" of quantum systems increases with the number of parties, i.e. the number of spatially separated subsystems. For example the violation of several families of Bell inequalities is known to increase exponentially with the number of parties [3]. By now already several multipartite protocols with quantum advantages are known, e.g. secret sharing or multipartite cryptography [4][5][6][7][8]. They motivate the investigation of the distribution of the required resource states consumed during the execution of the protocols. In the age of the Internet it is not hard to imagine that the parties will be distributed over distances at the global scale and form a two dimensional network in the future. Distribution of entanglement over long distances of more than ≈ 200 km requires intermediate quantum repeaters to compensate for transmission losses [9][10][11][12]. Here we do the generalization of error correction [13][14][15][16][17] based quantum repeaters [18][19][20][21][22][23] to general networks of such devices that are capable of producing multipartite entangled resource states. Our approach uses the language of graph states [24][25][26][27][28][29][30][31] that gives a graphic description of the network and the corresponding quantum states. Along the way we improve the error analysis by taking many sources of errors into account. The structure of this article is as follows. First we will explain the basic idea of such repeater for two parties using the stabilizer formalism. We then shift the graph state to a logical graph state by encoding the qubits with a quantum error correction code and analyze the performance depending on the amount of imperfections and the distance between the repeater stations. Finally we show how this scheme generalizes to networks. A graph state |G corresponds to a mathematical graph * epping@hhu.de G consisting of a set V of N vertices and a set of edges E ⊂ V × V . It can be defined in two equivalent ways. First, |G is the state that is created from the state |+ ⊗N , |+ = 1 √ 2 (|0 + |1 ) by applying a controlledphase gate C Z on each pair of vertices in E, i.e.
Second, |G is the unique state that is stabilized by the operators (so-called stabilizer generators) i.e. g i |G = |G , for all i ∈ V . Here X i and Z i denote Pauli-operators for the vertex i.
To explain the idea of the graph state repeater we start with a simple line graph that consists of an even number N of vertices (see Figure 1). An odd number of vertices leads to an analogous reasoning. The line graph corresponds to a chain of repeaters connecting two parties, Alice (A) and Bob (B). The repeater scheme is one-way, i.e. signals are sent from left to right and the protocol starts at Alice's site. Each edge of the graph state is created by performing at one station the entangling gate to the two qubits and sending one through the channel to the other station (see Figure 2). The maximal connectedness of a graph implies that it is possible to turn the quantum state into a maximally entangled state shared by Alice and Bob. This can be formulated in the stabilizer formalism. We number the The preparation and the gate of station Ri (a) and the transmission of the qubit produced at Ri to Ri+1 (b) creates the edge (i, i + 1), where the same procedure is repeated to create the next edge (c,d).
vertices from left to right, additionally we denote the first qubit by A and the last by B.
The product of all stabilizer generators (see Eq. 2) centered on odd qubits and the analogous product for the even ones leads to two stabilizers connecting A and B (see also [32]). We denote the first by S A and the second by S B . Due to their importance for our analysis, we call these two stabilizers the main stabilizers. All qubits except A and B are measured in the X-basis. This projects the state onto one stabilized by g A = X A ⊗ Z B and g B = Z A ⊗ X B , up to byproduct operators which depend on the measurement outcomes. After the measurement one of four orthogonal Bell states is shared by A and B, where the measurement outcomes determine which one. The local measurements commute with all operations on other repeater stations. Therefore any qubit can be measured directly after it has been processed by both gates. This is equivalent to the creation of the whole graph state, but easier to implement experimentally. If parties A and B want to perform X or Z measurements on the final Bell state, then the byproduct operators can be applied on the classical data of the measurement outcomes: X flips a Z outcome, Z flips an X outcome and H = 1 √ 2 (X + Z) flips the basis label. The previous considerations have to be expanded by an error model. We distinguish between two main types of errors: noticed (losses) and unnoticed ones (noise). Measurement outcomes are denoted by 0, 1, and ? for no outcome. In our error model a corrupted qubit is set to a completely mixed state, i.e. all errors are modeled by the depolarizing channel. It is convenient to take the equivalent viewpoint [33] that these imperfections are caused by discrete X and Z errors randomly occurring on the qubits after a perfect distribution of the graph state. In this case X and Z errors each occur on this qubit independently with probability 1 2 . One also has to account for spreading of errors by gates. C Z -gates propagate X-errors on the control qubit to Z-errors on the target qubit while they do not propagate Z-errors. Thus in the present circuit (see Figure 3), errors can only propagate to neighboring qubits. X-errors before a Z-measurement and Z-errors before an X-measurement lead to flipped measurement outcomes. The probability for wrong outcomes on "inner" qubits (i.e. all qubits except A and B) is denoted by f q . In the used error model any process acting on a physical qubit can lead to an error. We include the failure rates f P (preparation), f G (gates), f T (transmission), and f M (measurements). Because the losses will usually be dominated by the transmission, we mark the outcome of repeater i as ? whenever the qubit i − 1 got lost. One can adopt the analysis in case that the detection efficiency is the main cause of losses. There are nine sources of an unnoticed phase-flip error on an inner physical qubit i: three preparations, three gates, two transmissions and one measurement (see black circles in Figure 3). All of them lie in between repeater stations i − 2 and i, because errors only propagate to neighboring qubits in this circuit. Thus, for inner physical qubits the probability of phase-flip errors reads where P odd (p, n) denotes the probability that in n tosses of a coin the side appearing with probability p occurs an odd number of times and P odd ( p) is used when the probabilities pooled in p may be different for each toss (see Eqs. (A2) and (A4) in the appendix). The distinction between noticed and unnoticed errors is done by using the index n and u, respectively. Notice how a loss of a qubit can lead to an error on an adjacent qubit. We assume that the single qubit errors are independent and identically distributed. Inner physical qubits may get lost in the preparation, in one of the two gates processing them, during the transmission or in the measurement. Thus the overall probability for a ?-outcome is (see white circles in Fig 3) Typical losses for optical fibers are where f C,n is a coupling failure probability, L 0 is the repeater distance and L att ≈ 20 km defines the attenuation of the fiber. The last parameter strongly depends on the wavelength of the transmitted light.
The errors on the inner qubits propagate to A or B via the application of the byproduct operators. Additionally imperfections in the processing of A and B can directly lead to errors. We denote these error probabilities f A and f B . In a prepare-and-measure scenario of a quantum key distribution protocol, which is equivalent to producing the Bell state and A is measured in X or Z basis, qubit 2 is prepared in one of the states |0 , |1 , |+ , and |− and qubit N − 1 is measured. Thus f A = f B = 0 in this case. An even number of errors on the same main stabilizer will cancel each other. We denote the resulting error rate of the main stabilizer S A by e A and for S B by e B : With probability e A (e B ) the produced state is stabi- We theoretically estimate these error rates by where we have introduced the logical error probabilitȳ f q , which at the moment equals the physical error rate f q . We come back to this point later.
We focus on the application of the repeater scheme for quantum key distribution, where the crucial quantity is the secret fraction r ∞ , i.e. the number of secret bits per Bell pair that result after the data post-processing. For the BB84 protocol it solely depends on the error rates e A and e B and is given by [34] where is the binary entropy. We have already discussed how a Bell pair is gained from the graph state. In the following we describe how our scheme can be used to decrease the error rates.
To tackle imperfections we make use of an error correction code, i.e. we now use several physical qubits to encode the state of the logical qubit corresponding to a vertex of the graph. This leads to a logical graph state. The operators in the graph state stabilizers are replaced by logical operators, which we denote (like all logical quantities) with a bar. The idea of creating a Bell state from the main stabilizers transfers to the logical level. We focus on Calderbank-Shor-Steane (CSS) codes. Transversal (i.e. qubitwise) implementations of controlled-NOT gates are valid in all CSS codes [17]. We use two codes, the code C and the code C , which arises from C by exchanging the role of theX andZ. Even numbered logical qubits are encoded using C, the odd ones using C . In this way the transversal application of the controlled-NOT gates acts like a logical controlled-Phase gate and we can stick to the usual notation of graph states. The error analysis is analogous for both codes. Here we elaborate on the analysis for C only. We assume, that theX i -operator consists exclusively of single qubit X-operators. Hence the measurement outcome ofX i is affected by phase flip errors (Z). Due to the transversal implementation of theC Z gate the physical error rate remains as described in the previous paragraphs. An [[n, k, d]] quantum code encodes k logical qubits into n physical ones, such that t = d−1 2 single qubit errors can be corrected. In general the graph state repeater simultaneously creates k Bell pairs. The code space stabilizers containing X and 1 operators correspond to the rows of a parity check matrix H X . Thus in absence of any errors the X measurement outcomes are valid codewords of the corresponding linear code. The failing pattern e can be seen as caused by a binary erasure channel [35]. Decoder for common codes and their error rates can be found in the literature [13]. AnX-error remains, if an odd number of bit errors occurred on the involved measurement outcomes. This leads to the logical error ratē f q . The error correction is performed with the classical measurement data only. If specific loss patterns occurred, e.g. more than d losses in one block, we may choose to abort the protocol. The effective secret fraction then decreases by a factor P succ . In our analysisf q and P succ are the only quantities that explicitly depend on the employed code (in practice f P will also strongly depend on the code). Let us become more specific. A popular example of a CSS code is the 7-qubit Steane code, which is a [ [7,1,3]] code [15]. It is symmetric in X and Z, thus we can simply use one code and transversal C Z -gates. The stabilizers consisting of X and 1 can be read from the parity check matrix of the (7,4)-Hamming code [13]. The code is able to correct a single error. We choose to abort if we notice three losses. The logical error ratef q and success rate P succ for this code and choice of fatal errors is given in Table III in the appendix. From these quantities the effective secret fraction can be calculated using Eq. (7) and Eq. (9). Some example values can be found in Table I.
To be able to tolerate more errors, e.g. for an increased repeater distance or to relax the requirements for the gates, larger codes are required. We discuss an application of the [ [23,1,7]] quantum Golay code using a decoding described in [36]. We take the word error rate p w , i.e. the probability of recovering to a wrong codeword, from that reference. Half of the 4096 codewords correspond to a +1 and −1 outcome ofX, respectively. We thus assumef q ≈ p w /2. Fault tolerant preparation schemes have been investigated in [37]. Table I also    repeater stations w divided by the total distance L and the effective secret fraction R, i.e.
We optimized C over w for distances L up to 10000 km for several codes, see Figure 4. The optimal separation distance of the repeater stations decreases with increasing total distance. This quantity also allows the comparison with other repeater schemes, e.g. the analysis in [23] of the quantum parity code (QPC, a generalization of the Shor code with special abortion strategy [38]), when one pays attention to the fact that the authors considered less error sources than Eq. (3). The Golay code performs better than the QPC for f G = 2 3 × 10 −4 and f G = 2 3 × 10 −3 , while large QPCs outperform it for higher error rates, like f G = 1 3 × 10 −2 (the gate error rates correspond to the examples in [23]). In the previous paragraphs we considered a repeater setup corresponding to the generation of a line graph which can be used to distribute a Bell pair. For more than two parties it is possible to distribute other more complex and highly entangled states based on other graphs. We generalize the idea of main stabilizers to general graphs. In order to create the graph state |G shared by the parties 1, 2, ..., |V | the edges of G are replaced by line graphs with repeater stations as discussed above (see Figure 5). We assume that the number of repeater stations w ij on all these transmission lines (i, j) ∈ E is even for simplicity. The main stabilizers S i are the stabilizer generators g i of |G connected by chains of X-operators on every second repeater station (see also [32]). Measuring these intermediate qubits in the X-basis projects the state onto one stabilized by the generators of |G up to byproduct operators, i.e. this procedure can be used to create |G . Cycles in a graph may increase the storage time of some qubits, since their measurement can be performed only after all incoming neighbors gates acted. Note that the distribution of a state |G which is local-unitary-equivalent to |G can be less demanding, e.g. the star graph should be less demanding then the complete graph [39,40]. The performance analysis of the previous paragraphs can be transferred to networks. Analogous to Eq. (6) one can calculate the error probability for all main stabilizers S i . Table II lists these values for examples of two small networks, a line and a star graph. The in-degree of a vertex corresponding to a party influences the noticed and unnoticed error rates at this position (compare to Eqs. (3) and (4)) and the out-degree will in practice influence the preparation error in a prepareand-measure-scenario. The last is due to the fact that an (N n o )-qubit state is sent, where N is the number of qubits of one block and n o is the out-degree. However, the performance mainly depends on the error rates at the repeater stations (assuming that there are few parties and many repeater stations). Thus in conclusion we described how any multipartite graph state can be distributed via repeater stations. The procedure can be understood as the creation of a large graph state followed by a projection onto the desired state up to byproduct operators by local measurements on the repeater stations. Operational errors and channel erasures are treated equally. CSS Quantum error correction codes have been employed to tackle these imperfections. In particular the 23-qubit Golay code turns out to be remarkably efficient in the considered scenario of operational losses and noise. General graph states can be distributed with effort, rate and quality comparable to the distribution of a single Bell pair by one-way quantum error correction based repeaters.

ACKNOWLEDGMENTS
M.E. acknowledges helpful discussions with S. Muralidharan and financial support by BMBF. of heads is and P odd (P, N ) = 1 respectively. Suppose we use a different coin for each toss, and the probability for head in the toss i is p i . Then the probability to count an even and odd number of heads is and P odd ( p) = respectively. Here |n| H is the Hamming weight of n in binary representation and n (k) is the k-th binary digit of n, which denotes whether a head or tail was seen on the k-th coin. The probability of a majority of N identically constructed coins to show the side that each coin shows with probability P u is where we assumed that the outcome is chosen randomly when both sides are shown by an equal number of coins. If each coin gets lost with probability P n , then the majority vote on the remaining qubits has a probability of (A6) to decide in favor of the side corresponding to P u .

Appendix B: Logical error rate of some CSS codes
In order to calculate the secret key rate for a specific encoding, we need the error rates on odd and even logical qubits. The first is used in the calculation of e Z , the second for e X . The decoder assigns a codeword to any word given by the measurement, i.e. to the true outcomes altered by the error pattern. This recovered codeword is used to calculate the value of the logical observable. If this recovered value is different from the "true outcome" this word of measurement outcomes contributes to the logical error rate. The decoder may trigger an abort on any error from the set F. The naive approach to calculating this rate thus is where P e (e) is the probability of the error e and f (e) is the number of logical errors after decoding. The success probability depends on F and reads Let us consider a decoder that returns the most likely codeword c in the sense that an error pattern e that changes c to the observed data has maximal probability P e (e). If this is not unique the decoder chooses any such c with equal probability. We use it for the 7-qubit Steane code and use a fatal error set F of the form F = {e|e contains more than n max losses}, where n max ∈ N, i.e. the protocol is aborted if more than n max losses occurred. The error rates of the 7-qubit Steane code listed in Table III where obtained by implementing Eq. (B1).