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arXiv:2608.18471 · Lecture notes in electrical engineering

Multicast quantum network coding as optimal symmetric universal cloning over a quantum network

Go Kato, Mio Murao, Masaki Owari

quant-phSubmitted Aug 19, 2026• Announced Aug 20, 2026• 0 citations
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Abstract

We study the problem of perfectly multicasting symmetric universal clones of unknown quantum states over quantum networks with free classical communication. We construct a protocol that multicasts symmetric universal clones of input states from multiple source nodes by extending the quantum network coding protocol proposed by Kobayashi et al. We further establish a sufficient condition for perfect multicast in the single-source setting. Specifically, we show that when a single copy of a $q^r$-dimensional input state is available at the source node, where $q$ is a sufficiently large prime power, perfect multicast of the corresponding symmetric universal clone is achievable using a small amount of entanglement shared among the target nodes. This result holds for quantum networks represented by an undirected graph $G$, where each edge corresponds to a noiseless $q$-dimensional quantum channel, provided that there exists an acyclic directed graph $G'$ obtained by assigning directions to the edges of $G$ such that the minimum cut of $G'$ is at least $r$.

Journal reference
Lecture Notes in Electrical Engineering, vol. 1634, pp. 313-335, 2026
DOI
10.1007/978-3-032-25259-3_26
Comments
25 pages, 1 figures. Published in Lecture Notes in Electrical Engineering (LNEE), vol. 1634. Presented at the 6th International Conference on Nano Electronics Research and Education (ICNERE2025) Joint with the 27th Takayanagi Kenjiro Memorial Symposium

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•Jan 1, 2009•57 citations
We study the problem of perfectly multicasting symmetric universal clones of unknown quantum states over quantum networks with free classical communication.
We construct a protocol that multicasts symmetric universal clones of input states from multiple source nodes by extending the quantum network coding protocol proposed by Kobayashi et al. We further establish a sufficient condition for perfect multicast in the single-source setting. Specifically, we show that when a single copy of a qrq^rqr-dimensional input state is available at the source node, where qqq is a sufficiently large prime power, perfect multicast of the corresponding symmetric universal clone is achievable using a small amount of entanglement shared among the target nodes.
This result holds for quantum networks represented by an undirected graph GGG, where each edge corresponds to a noiseless qqq-dimensional quantum channel, provided that there exists an acyclic directed graph G′G'G′ obtained by assigning directions to the edges of GGG such that the minimum cut of G′G'G′ is at least rrr.
Multicasting of quantum states is an essential feature of quantum internet. Since the noncloning theorem prohibits perfect cloning of an unknown quantum state, an appropriate protocol may depend on the purpose of the multicast. In this paper, we treat the multicasting of a single copy of an unknown state over a quantum network with free classical communication. We especially focus on protocols exactly multicasting an asymmetric optimal universal clone. Hence, these protocols are optimal and universal in terms of mean fidelity between input and output states, but the fidelities can depend on target nodes. Among these protocols, a protocol spending smaller communication resources is preferable. Here, we construct such a protocol attaining the min-cut of the network described as follows. Two (three) asymmetric optimal clones of an input state are created at a source node. Then, the state is divided into classical information and a compressed quantum state. The state is sent to two (three) target nodes using the quantum network coding. Finally, the asymmetric clones are reconstructed using LOCC with a small amount of entanglement shared among the target nodes and the classical information sent from the source node.
Multicast of unknown quantum states is an unavoidable issue in the construction of quantum networks. Most existing quantum network coding schemes can only restore an unknown state at one of the target nodes during multicast. In this paper, we propose a quantum network coding scheme to multicast unknown quantum states in networks with linear network coding. Concretely, GHZ states are generated between each sender and all receivers. Unknown states are then sent and stored in the entangled state shared by the receivers. Finally, the receivers use asymmetric optimal universal cloning to obtain the required states. The scheme achieved to multicast within 1 quantum transmission and 2 classical transmissions in networks with constrained classical communication. Compared to other schemes, our scheme allows receivers to autonomously adjust the fidelity without the sender knowing the requirements.
This paper considers a problem of quantum communication between parties that are connected through a network of quantum channels. The model in this paper assumes that there is no prior entanglement shared among any of the parties, but that classical communication is free. The task is to perfectly transfer an unknown quantum state from a source subsystem to a target subsystem, where both source and target are formed by ordered sets of some of the nodes. It is proved that a lower bound of the rate at which this quantum communication task is possible is given by the classical min-cut max-flow theorem of network coding, where the capacities in question are the quantum capacities of the edges of the network.
Network coding is often explained by using a small network model called Butterfly. In this network, there are two flow paths, s_1 to t_1 and s_2 to t_2, which share a single bottleneck channel of capacity one. So, if we consider conventional flow (of liquid, for instance), then the total amount of flow must be at most one in total, say 1/2 for each path. However, if we consider information flow, then we can send two bits (one for each path) at the same time by exploiting two side links, which are of no use for the liquid-type flow, and encoding/decoding operations at each node. This is known as network coding and has been quite popular since its introduction by Ahlswede, Cai, Li and Yeung in 2000. In QIP 2006, Hayashi et al showed that quantum network coding is possible for Butterfly, namely we can send two qubits simultaneously with keeping their fidelity strictly greater than 1/2. In this paper, we show that the result can be extended to a large class of general graphs by using a completely different approach. The underlying technique is a new cloning method called entanglement-free cloning which does not produce any entanglement at all. This seems interesting on its own and to show its possibility is an even more important purpose of this paper. Combining this new cloning with approximation of general quantum states by a small number of fixed ones, we can design a quantum network coding protocol which ``simulates'' its classical counterpart for the same graph.
Classical network coding uses encoding of information over networks to achieve secure and high throughput communication with low latency. Its quantum analog, namely Quantum Network Coding (QNC) is a promising protocol to achieve quantum communication over quantum networks. In this paper, we propose a method for establishing QNC in Measurement-Based Quantum Computing (MBQC), which is a universal quantum computational model. We demonstrate that the problem of multiple unicast or multiple multicast of distinct computational basis states over a quantum network can be solved by simulating existing classical networks, with the CNOT implementation in MBQC as a basis for transmitting and encoding of states. We also show how it can be further extended to simultaneously distribute the Bell pairs and Greenberger–Horne–Zeilinger (GHZ) states required for quantum communication between distant transmitter and corresponding receiver nodes of a network.
This paper considers the problem of efficiently transmitting quantum states through a network. It has been known for some time that without additional assumptions it is impossible to achieve this task perfectly in general — indeed, it is impossible even for the simple butterfly network. As additional resource we allow free classical communication between any pair of network nodes. It is shown that perfect quantum network coding is achievable in this model whenever classical network coding is possible over the same network when replacing all quantum capacities by classical capacities. More precisely, it is proved that perfect quantum network coding using free classical communication is possible over a network with k source-target pairs if there exists a classical linear (or even vector-linear) coding scheme over a finite ring. Our proof is constructive in that we give explicit quantum coding operations for each network node. This paper also gives an upper bound on the number of classical communication required in terms of k, the maximal fan-in of any network node, and the size of the network. x1 x2 s1 ♠ s2 ♠ ❍❍❍❍ ❥ x1 x2 ✟ n1 x1 x1 ⊕ x2 x2 n2 ♠ x1 ⊕ x2✟ ❍❍❍❍ ❥ x1 ⊕ x2