Entanglement-efficient bipartite-distributed quantum computing

Jun-Yi Wu1,2, Kosuke Matsui3, Tim Forrer3, Akihito Soeda3,4,5, Pablo Andrés-Martínez6, Daniel Mills6, Luciana Henaut6, and Mio Murao3

1Department of Physics and Center for Advanced Quantum Computing, Tamkang University, 151 Yingzhuan Rd., New Taipei City 25137, Taiwan, ROC
2Physics Division, National Center for Theoretical Sciences, Taipei 10617, Taiwan, ROC
3The University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113-0033, Japan
4Principles of Informatics Research Division, National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan
5Department of Informatics, School of Multidisciplinary Sciences, SOKENDAI (The Graduate University for Advanced Studies), 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan
6Quantinuum, Terrington House, 13-15 Hills Road, Cambridge CB2 1NL, UK

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In noisy intermediate-scale quantum computing, the limited scalability of a single quantum processing unit (QPU) can be extended through distributed quantum computing (DQC), in which one can implement global operations over two QPUs by entanglement-assisted local operations and classical communication. To facilitate this type of DQC in experiments, we need an entanglement-efficient protocol. To this end, we extend the protocol in [Eisert et. al., PRA, 62:052317(2000)] implementing each nonlocal controlled-unitary gate locally with one maximally entangled pair to a packing protocol, which can pack multiple nonlocal controlled-unitary gates locally using one maximally entangled pair. In particular, two types of packing processes are introduced as the building blocks, namely the distributing processes and embedding processes. Each distributing process distributes corresponding gates locally with one entangled pair. The efficiency of entanglement is then enhanced by embedding processes, which merge two non-sequential distributing processes and hence save the entanglement cost. We show that the structure of distributability and embeddability of a quantum circuit can be fully represented by the corresponding packing graphs and conflict graphs. Based on these graphs, we derive heuristic algorithms for finding an entanglement-efficient packing of distributing processes for a given quantum circuit to be implemented by two parties. These algorithms can determine the required number of local auxiliary qubits in the DQC. We apply these algorithms for bipartite DQC of unitary coupled-cluster circuits and find a significant reduction of entanglement cost through embeddings. This method can determine a constructive upper bound on the entanglement cost for the DQC of quantum circuits.

The scalability of quantum computing on a single quantum processing unit is physically limited. To break through this bottleneck, one needs to distribute quantum computing over multiple quantum processing units. It is therefore essential to establish an entanglement-economical architecture to release the requirement on the entanglement generation rate of a quantum network.

In our paper, an entanglement-efficient architecture for bipartite distributed quantum computing is established based on a decomposition of a quantum circuit into a set of distributable blocks. In each block, one maximally-entangled state is consumed to distribute nonlocal gates with local operations and classical communication. In previous protocols, a distributing process ends with single qubit gates. To improve entanglement efficiency, we introduce embedding processes to merge two non-sequential distributing processes into one, saving on the amount of entanglement needed. Such an embedding-enhanced-distributing structure of a quantum circuit can be fully described by a packing graph made up of vertices representing gate nodes that could be merged together.

The vertices of a packing graph are the candidates for distributing processes. However, there may be conflicts between two embeddings that prevent us from implementing them simultaneously. We have to remove some embeddings to solve the conflicts, which leads to a split of vertices in a packing graph. Such a conflict can be fully described by a conflict graph.

The packing graph and conflict graph of a circuit fully describe its distributability, embeddability, and incompatibility. To determine the best way to distribute a circuit, we develop a packing algorithm that takes into account both the minimum vertex covering of the packing graph and its corresponding conflict graph to find a distributing strategy that is both entanglement-efficient and conflict-free.

Overall, our protocol can be summarized as an “embedding-enhanced-distributing'' protocol, which is based on two types of entanglement-assisted quantum operation, namely distributing processes and embedding processes. Compared to previous protocols, it significantly reduces the amount of entanglement required for distributed quantum computing, making it more practical for real-world applications. It can be employed to determine a tighter constructive upper bound on the entanglement cost for a unitary being decomposed into local operations and classical communication. The protocol can be extended to multipartite systems and adapted to network topology for distributed quantum computing over the quantum internet. The “embedding-enhanced-distributing'' protocol can therefore facilitate large-scale quantum computing in a quantum network of QPUs.

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Cited by

[1] Raúl Santos, Lorenzo Buffoni, and Yasser Omar, "Enhancing Quantum Annealing via entanglement distribution", arXiv:2212.02465, (2022).

[2] Pablo Andres-Martinez, Tim Forrer, Daniel Mills, Jun-Yi Wu, Luciana Henaut, Kentaro Yamamoto, Mio Murao, and Ross Duncan, "Distributing circuits over heterogeneous, modular quantum computing network architectures", arXiv:2305.14148, (2023).

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