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arXiv:2507.15725

Efficiently Generation of Cluster States via Time-Delayed Feedback in Matrix Representation

Jia-Jin Zou, Qin, Jian-Wei, Franco Nori, Xiang, Ze-Liang

Published Jul 21, 2025• 0 citations
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Abstract

Cluster states, as highly entangled multi-qubit states, are widely used as essential resources for quantum communication and quantum computing. However, due to the diverse requirements of applications for cluster states with specific entanglement structures, a universal generation protocol is still lacking. Here we develop a matrix representation according to the characteristics of time-delayed feedback (TDF) and propose a protocol for generating arbitrary cluster states with multiple TDFs. The matrix representation also allows us to optimize the generation process to reduce TDF usage, thus improving efficiency. In particular, we demonstrate a tree-cluster-state generation process that requires only one TDF. Moreover, accounting for the critical loss mechanisms and imperfections in our protocol, we discuss the additional losses caused by multiple TDFs and evaluate the fidelity of the resulting cluster states.

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Cluster states, as highly entangled multi-qubit states, are widely used as essential resources for quantum communication and quantum computing. However, due to the diverse requirements of applications for cluster states with specific entanglement structures, a universal generation protocol is still lacking. Here we develop a matrix representation according to the characteristics of time-delayed feedback (TDF) and propose a protocol for generating arbitrary cluster states with multiple TDFs. The matrix representation also allows us to optimize the generation process to reduce TDF usage, thus improving efficiency. In particular, we demonstrate a tree-cluster-state generation process that requires only one TDF. Moreover, accounting for the critical loss mechanisms and imperfections in our protocol, we discuss the additional losses caused by multiple TDFs and evaluate the fidelity of the resulting cluster states.
Abstract Photonic cluster states with tree-type encoding are an important resource for resisting photon loss. Here we develop an optimization framework for their efficient generation using time-delayed feedback. By analyzing the relation between feedback loops and entanglement structure through the adjacency matrix, we significantly reduce the resource overhead of state preparation by mapping the nonperiodic structure of the tree cluster state onto a periodic lattice. Combined with reinforcement learning, this framework identifies, under fixed resource constraints, the near-optimal tree structure for maximizing loss tolerance. Furthermore, we determine the practical loss-tolerant threshold set by the size of the near-optimal tree, the number of feedback loops, and realistic imperfections. These results provide a practical design strategy for loss-tolerant photonic cluster states in state-of-the-art quantum platforms.
Cluster states are useful in many quantum information processing applications. In particular, universal measurement-based quantum computation (MBQC) utilizes two-dimensional cluster states [R. Raussendorf and H. J. Briegel, Phys. Rev. Lett. 86, 5188 (2001)] and topologically fault-tolerant MBQC requires cluster states of dimension 3 or higher [R. Raussendorf et al., New J. Phys. 9, 199 (2007)]. This work proposes a protocol to deterministically generate multidimensional photonic cluster states using a single atom-cavity system and time-delay feedback. The dimensionality of the cluster state increases linearly with the number of time-delay feedbacks. We first give a diagrammatic derivation of the tensor network states, which is valuable in simulating matrix product states and projected entangled pair states generated from sequential photons. Our method also provides a simple way to bridge and analyze the experimental imperfections and the logical errors of the generated states. In this method, we analyze the generated cluster states under realistic experimental conditions and address both one-qubit and two-qubit errors. Through numerical simulation, we observe an optimal atom-cavity cooperativity for the fidelity of the generated states, which is surprising given the prevailing assumption that higher-cooperativity systems are inherently better for photonic applications.
Large, multi-dimensional clusters of entangled photons are among the most powerful resources for emerging quantum technologies, as they are predicted to enable global quantum networks or universal quantum computation. Here, we propose an entirely new architecture and protocol for their generation based on recurrent quantum photonic neural networks (QPNNs) and focusing on tree-type cluster states. Unlike other approaches, QPNN-based generators are not limited by the the coherence of quantum emitters or by probabilistic multi-photon operations, enabling arbitrary scaling only limited by loss (which, unavoidably, also affects all other methods). We show that a single QPNN can learn to perform all of the many different operations needed to create a cluster state, from photon routing to entanglement generation, all with near-perfect fidelity and at loss-limited rates, even when it is created from imperfect photonic components. Although these losses ultimately place a limit on the size of the cluster states, we show that state-of-the-art photonics should already allow for clusters of 60 photons, which can grow into the 100s with modest improvements to losses. Finally, we present an analysis of a one-way quantum repeater based on these states, determining the requisite platform quality for a global quantum network and highlighting the potential of the QPNN to play a vital role in high-impact quantum technologies.
A photonic cluster state with a tree-type entanglement structure constitutes an efficient resource for quantum error correction of photon loss. But the generation of a tree cluster state with an arbitrary size is notoriously difficult. Here, we propose a protocol to deterministically generate photonic tree states of arbitrary size by using only a single quantum emitter. Photonic entanglement is established through both emission and rescattering from the same emitter, enabling fast and resource-efficient entanglement generation. The same protocol can also be extended to generate more general tree-type entangled states.
We investigate the implementation of photonic cluster state generation protocols using quantum metasurfaces comprising sub-wavelength atomic arrays which enables quantum-controlled reflectivity. These cluster states are generated using fundamental quantum logic gates and enable wide-ranging applications in quantum computation and communication. In the past few years, certain protocols have been developed, but their physical realizations induces natural losses on the system mainly originated from coupling the photonic structures, setting a limit on the efficiency and maximal qubit number. In this paper, we examine a physical implementation of two specific protocols for generating distinct cluster states: a two-dimensional cluster state and a tree cluster state. Our approach leverages the unique properties of a quantum metasurface and its free space settings to implement two-qubit quantum-logic gates, namely CNOT, CZ, and E gates, with practical fidelities exceeding 0.9, and potential speed-up due to parallelism. In addition, we analyze these protocols fidelities for practical conditions of potential implementation experiments, such as thermal fluctuation of trapped atoms.
We propose and analyze deterministic protocols to generate qudit photonic graph states from quantum emitters. We show that our approach can be applied to generate any qudit graph state and we exemplify it by constructing protocols to generate one- and two-dimensional qudit cluster states, absolutely maximally entangled states, and logical states of quantum error-correcting codes. Some of these protocols make use of time-delayed feedback, while others do not. The only additional resource requirement compared to the qubit case is the ability to control multilevel emitters. These results significantly broaden the range of multiphoton entangled states that can be produced deterministically from quantum emitters. Published by the American Physical Society 2024