Reference frame agreement in quantum networks

In order to communicate information in a quantum network effectively, all network nodes should share a common reference frame. Here, we propose to study how well m nodes in a quantum network can establish a common reference frame from scratch, even though t of them may be arbitrarily faulty. We present a protocol that allows all correctly functioning nodes to agree on a common reference frame as long as not more than t<m/3 nodes are faulty. Our protocol furthermore has the appealing property that it allows any existing two-party protocol for reference frame agreement to be lifted to a protocol for a quantum network.

Definition 1. For η > 0, a η-reference frame consensus protocol among m network nodes is a protocol such that Termination Each correct node P i terminates the protocol, and outputs a reference frame v i . Consistency For all pairs of correct nodes P i and P j we have η ⩽ ( ) Note that consistency does not require that all the correct nodes share the same reference frame (η = 0), but that each node has an approximation of it (η is small). This is important because already any two-node protocol using only a finite number of rounds of communication cannot allow the two nodes to share a frame exactly.

Results
We introduce the first protocol to solve the reference frame agreement problem in a quantum network of m nodes of which < t m 3 can be arbitrarily faulty. Our protocol has the appealing feature that it can use any two-node protocol as a black box. Such two-node protocols [22] are characterized by the accuracy δ (i.e., the two nodes δ-agree) and the success probability q succ with which such an approximation guarantee is achieved. Theorem 1. Given any two-node protocol to estimate a direction with accuracy δ and success probability q succ , the protocol RF-consensus is a δ ( ) 30 -reference frame consensus protocol tolerant to < t m 3 faulty nodes. It succeeds with probability at least q m succ 2 .
Our protocol is efficient as we need only a linear (in the number of nodes m) number of rounds of quantum communication. As an example, we take the simplest two-node protocol in which the sender encodes the direction in the Bloch vector of a qubit and sends n identical copies of it to the receiver. For accuracy δ > 0, the success probability of this two-node . We also show that this setting is robust to noise on the channel connecting any two nodes. To give some examples of parameters, protocol RFconsensus achieves accuracy δ = 30 0.02 with success probability 99% in a network of m = 10 nodes with noiseless communication, if each node transmits ≈ × n 3.1 10 8 qubits at each round.
Our protocol uses ideas of [24] which solves a simpler problem from classical distributed computing called Byzantine agreement [25]; in particular we use classical consensus as a subroutine. This problem has been extensively studied using synchronous [26,27] and asynchronous [28][29][30][31] classical communication, as well as quantum communication [32], also in a fail-stop model in which the faulty nodes can prevent the protocol from ever terminating [33]. There, the correct nodes should perfectly agree on a single classical bit. Recall that we cannot send a direction classically without a shared reference frame, and hence we cannot use such protocols. In addition, we face two extra challenges: first, we are dealing with a continuous set of outcomes; and second, it is impossible to transmit a direction perfectly using a finite amount of communication, even on an otherwise perfect channel. In quantum networks, furthermore, we also have errors on the communication channel, which are pretty much unavoidable in a regime where we cannot easily perform quantum error correction due to the lack of a common frame. In the Byzantine problem such errors would be attributed to faulty nodes, but in our setting this would mean that all nodes in the network are faulty and no protocol could ever hope to succeed. Here, we thus require a careful treatment of such approximation errors.

Model of communication
We assume that all the communication channels are public (faulty nodes can adapt their strategy depending on the network traffic), authenticated (faulty nodes cannot tamper with the channel connecting correct nodes), and synchronous (correct nodes know when they are supposed to receive a message, and if none is received, e.g. due to communication error, the protocol continues which ensures that our protocol cannot stall indefinitely).
We only use quantum communications to send a direction between a sender and a receiver. As an example we use protocol 2ED, one of the simplest possible protocols as studied in [16]. Here a sender creates many identical qubits with their Bloch vector pointing in the intended direction and the receiver measures them with Pauli measurements. From the statistics of the measurement outcomes, the receiver then estimates the Bloch vectorʼs direction closely with high success probability. We use this protocol since it has some experimental advantages for implementation: it does not require any quantum memory or creation of entangled states, and it succeeds even if the quantum channel has a depolarizing noise. But the downside of this choice is that our protocol is not optimal in the number of qubits sent to achieve a certain accuracy. Optimal protocols [34][35][36] can align frames in the so-called Heisenberg limit [21], they have a quadratic gain over the one we use here.
We prove the following theorem in the appendix.

Protocols
In this section, we present a summary of our protocols and an outline of their proof of correctness. For further detail, we refer to the appendix. Our protocol works in two phases: first, a node is elected as the king P k . Second, the king chooses a direction w k and sends it to all the other nodes. We denote w i the direction received by the node P i in its own frame. If the king is not faulty, 2ED ensures that . Then the correct nodes should decide either all to accept this direction (they output ≈ v w i k in their respective own frame), or all to reject it (output ⊥). This second phase is known as kingconsensus. More formally, a king-consensus protocol should satisfy two properties: δ-persistency: if the king is not faulty, all the correct nodes P i , should output v i such that ; and η-consistency: All the correct nodes reach a consensus, that is, they either all output ⊥, or they all output directions that are η-close to each other, i.e., for all correct nodes P i We repeat those two phases with different kings as long as a consensus is not reached. In particular, the protocol will terminate after at most + t 1 rounds since there are at most t faulty nodes.
The rest of this paper is thus devoted to constructing a king-consensus protocol, which is done in three steps.
Step 1: Weak-consensus We first create a weaker protocol than king-consensus by relaxing the condition that the correct nodes either all output a direction, or all output ⊥. In a weak-consensus, some nodes can output ⊥ and the others a direction. However we keep the condition that if two correct nodes P i and P j output directions u i and u j , they should be close to each other. Formally, we define a weak-consensus protocol as a protocol with the following two properties: δ-weak persistency: if there exists a direction w k such that for every correct node where δ is the accuracy achieved with probability q succ by the two-node protocol used to send directions.
Here, with probability at least − q m m succ 2 , for every correct node P i and P j , It is easy to see that this protocol is δ-weak persistent. We sketch the proof of the weak consistency. Consider the sets S i and S j of two correct nodes P i and P j . If ≠ ⊥ u i and ≠ ⊥ u j , then S i and S j contains at least one correct node in common, let us call it α P . Thus,

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Step 2: Graded-consensus. In a king-consensus protocol, the correct nodes should have a 'global' behavior, as they should all either output a direction or ⊥, whereas in the weakconsensus each node has a 'local' strategy. A graded-consensus protocol behaves intermediately. Alongside a direction ≠ ⊥ v i the nodes also output a grade ∈ { } g 0, 1 i which carries a 'global' property, namely, η-graded consistency: If any correct node outputs a grade 1, then the directions between all the correct nodes should be η-close to each other, that is, for every pair (P P , . The main idea of graded-consensus is that the nodes which output ⊥ in the weakconsensus inform the other nodes (by sending the flags f i ). The first consequence is that for all The second consequence is that if a correct node has grade 1, then for all correct nodes P i and P j , the sets T i and T j each contain at least one correct node, let us denote them α P and β P . Thus, Finally, we get, Step 3: King-consensus. We are ready to present the king-consensus protocol that achieves δ-persistency and δ ( ) 30 -consistency. Our protocol uses classical-consensus as a subroutine. It solves a problem which is closely related to Byzantine agreement. Here, every node P i starts with a bit g i and outputs a bit y i . All the correct nodes agree on a bit b, that is if where at least one of the correct nodes, P j has input = g b j . Classical-consensus can be reached if there are < t m 3 faulty nodes; for an example of such protocol, see e.g. [37].
If the king is not faulty, then all the correct nodes will have grade = g 1 i . Hence the classical-consensus will also be reached with value = y 1 i . So, all the correct nodes will accept the direction shared by the king. If the king is faulty and yet the correct nodes reach a consensus with = y 1 i , it means that at least one correct node had grade 1. In this case the δ ( ) 30 -graded i j for all the correct nodes P i and P j . As a consequence, king consensus is ( δ 30 )-consistent, and so is RF-consensus.

Discussion
We have presented the first protocol for reference frame agreement in a quantum network. Even in the classical setting, the algorithms to solve the Byzantine agreement problem are surprisingly complicated. We would be very keen to know if simpler and more efficient protocols could be designed for our setting, possibly by using entangled states. It is an interesting open question to construct protocols that also work in an asynchronous communication model. The latter is already challenging for the classical case [28][29][30][31], so we expect a similar behavior to hold here. Another interesting question is whether more faulty nodes than < t m 3 can be tolerated. If our protocol were to succeed with probability 1 and η sufficiently small, we can prove that it is optimal in that sense by adapting the classical proof [38] to our setting. However, for equationing reference frames, any protocol can only succeed with probability strictly less than 1. This problem has been partially studied in the classical case [39]. Even in the constant error scenario the optimal number of faulty nodes that can be tolerated is not known for the classical Byzantine agreement problem [40]. This leaves hope to find protocols that can tolerate < t m 2 faulty nodes when allowing constant success probability both for Byzantine and reference frame agreement.
In this section, we analyze the protocol 2ED to exchange a direction between two nodes. Since this cannot be done perfectly, the receiver has to estimate the direction sent by the sender. This task is formally defined by: Definition 2. A δ-estimate direction protocol is a two-node protocol where one node (the sender) sends a direction u to the other node (the receiver). Upon termination the receiver gets a δ-approximation v of u, that is, This simple protocol has several advantages: it does not require any quantum memory or the creation of entangled states, and it succeeds even if the quantum channel has a depolarizing noise. But the downside of this choice is that the protocol is not optimal in the number of qubits sent to achieve a certain accuracy. Any other protocol can be used here [22]. Proof. We will prove this theorem in two steps. First, we consider the case when the communication channel is noise free (ε = 0), and then, we see how depolarizing noise affects the approximation factor. In the noise-free case, let us fix δ > 0 and denote by θ θ , x y , and θ z the angles between u and the x-, y-, and z-axis of the local frame of the receiver. So, We know in the ideal case, when → ∞ n the relative frequency → θ p cos x 2 2 x but in 2ED n is finite. So, using Hoeffdingʼs inequality we get,   Since v is the normalization of ( ) x y z , , , its angle with u is also θ and from a simple trigonometric observation, we have, From equation (A.14) one can see that the effective relative frequency p x is given by where ′ p x is the relative frequency that the receiver would have got if the channel was noise-free, , -weak-consensus protocol is a m-node protocol, in which each node P i has an input direction w i and outputs either a direction u i or ⊥, that satisfies the following two properties:

δ-weak persistency
If there exists a direction s such that for every correct node and node is correct , and node is correct .
We need to prove that ∩ ≠ ∅ C C i j . We do it by contradiction: let us assume that Inequality (B.8) follows because there can be at most t faulty nodes, and inequality (B.10) since < t m 3 . Now, .
(B.20) The factor δ 3 comes from the fact that α P is in S i and the remaining δ since α P is correct. We can do the same reasoning with the node P j , hence we also have: Step 2: Graded-consensus Again, we shall start by giving a formal definition of a graded-consensus protocol.  Step (3): Now using triangular inequality with inequalities (C.10), (C.6), and (C.11) we get, This proves the δ ( ) 30 -graded consistency of the protocol. □ Appendix D.
Step 3: King-consensus , -king-consensus protocol is an m-node protocol in which one node P k , called the king, chooses a direction w k and each of the other nodes P i outputs either a direction v i or each of them outputs ⊥, which satisfies the following two properties: δ-persistency If the king is correct, then all the correct nodes . η-consistency All correct nodes reach a consensus, that is, they either all output ⊥, or they all output directions that are η-close to each other, i.e., for all correct nodes P i and P j , Our protocol to solve the king-consensus problem uses graded-consensus and classicalconsensus as subroutines. The latter is a protocol between m nodes, in which each node starts with an input bit g i and outputs a bit y i , that satisfies the following two properties: Agreement All correct nodes should output the same bit.

Validity
If all correct nodes start with the same input = g b i , they should all output this value, that is = y b i .
Theorem 5. Using a δ-estimate direction protocol that succeeds with probability q succ , kingconsensus is a δ δ ; and from the validity of classical-consensus, we have that = y 1 i for all correct nodes P i . Hence all the correct nodes output a δ-approximation of w k with probability at least q m succ 2 . Consistency. To prove consistency we will show that all the correct nodes output ⊥, or they all output a direction. In this case we also have to show that for every pair ( ) i j for every correct node P i and P j . □