Abstract We propose a graph method for systematically searching for schemes that can generate multipartite entanglement in linear bosonic systems with heralding. While heralded entanglement generation offers more tolerable schemes for quantum tasks than postselected ones, it is generally more challenging to find appropriate circuits for multipartite systems. We show that our graph mapping from boson subtractions provides handy tactics to overcome the limitations in circuit designs. Within our graph framework, we identify enhanced schemes for qubit N-partite GHZ, W, and the superposition of N = 3 GHZ and W states. Furthermore, we have found a qudit N-partite GHZ state generation scheme, which requires substantially fewer particles than previous proposals. These results demonstrate the power of our approach in discovering optimized solutions for the generation of intricate heralded entangled states. We expect our method to serve as a promising tool in generating diverse entanglement.
Abstract We propose a graph method for systematically searching for schemes that can generate multipartite entanglement in linear bosonic systems with heralding. While heralded entanglement generation offers more tolerable schemes for quantum tasks than postselected ones, it is generally more challenging to find appropriate circuits for multipartite systems. We show that our graph mapping from boson subtractions provides handy tactics to overcome the limitations in circuit designs. Within our graph framework, we identify enhanced schemes for qubit N-partite GHZ, W, and the superposition of N = 3 GHZ and W states. Furthermore, we have found a qudit N-partite GHZ state generation scheme, which requires substantially fewer particles than previous proposals. These results demonstrate the power of our approach in discovering optimized solutions for the generation of intricate heralded entangled states. We expect our method to serve as a promising tool in generating diverse entanglement.
Non-destructive heralded entanglement with photons is a valuable resource for quantum information processing. However, they generally entail ancillary particles and modes that amplify the circuit intricacy. To address this challenge, a recent work \cite{chin2024shortcut} introduced a graph approach for creating multipartite entanglements with boson subtractions. Nonetheless, it remains an essential intermediate step toward practical heralded schemes: the proposition of heralded subtraction operators in bosonic linear quantum networks. This research establishes comprehensive translation rules from subtraction operators to linear optical operators, which provides a seamless path to design heralded schemes with single photons. Our method begets enhanced or previously unreported schemes for the N -partite GHZ state with 2 N photons, N -partite W state with 2 N + 1 photons and superposition of N = 3 GHZ and W states with 9 photons. Our streamlined approach can straightforwardly design heralded schemes for multipartite entangled states by assembling the operators according to the guidence of sculpting bigraphs, hence significantly simplifies the quantum circuit design process.
Heralded multipartite entanglement is a key resource for various quantum information tasks. However, designing linear optical circuits that generate specific target states is generally challenging due to the complexity of the required optical structures. Here we formulate the design of heralded photonic circuits as an algorithmic graph-search problem. Our framework enables the automated construction and optimization of heralded photonic circuits by substantially reducing the search space using the linear quantum graph (LQG) picture. Our strategy reconstructs circuit structures as graphs in the picture and identifies suitable graphs automatically. As a result, we design efficient schemes for a broad range of useful multipartite resource states, including hypergraph magic states, quantum error correcting codes, general three-qubit states and length-1 caterpillar graph states. Our work establishes an algorithmic framework for the systematic discovery of heralded resource states, laying the foundation for the automated design of increasingly complex multipartite entangled resources.
Recent attention has turned to exploring quantum information within larger Hilbert spaces by utilizing qudits, which offer increased information capacity and potential for robust quantum communications. While the efficient generation of multipartite qudit entanglement is crucial for studying quantum correlations in high-dimensional Hilbert spaces, the increased dimension makes the circuit design challanging, especially when the entanglement is generated by heralding detections. In this work, we demonstrate that the graph picture of linear quantum networks (LQG picture) can provide a simplified method to generate qudit multipartite heralded entanglement of high symmetries. The LQG picture enables the reduction of circuit complexity by directly imposing the state symmetry onto the circuit structure. Leveraging this insight, we propose heralded schemes for generating N-partite N-level anti-symmetric (singlet) and symmetric (Dicke) states. Our study shed light on the optimal circuit design of high-dimensional entanglement with a systematic graphical strategy.
Entanglement is a fundamental feature of quantum mechanics and a key resource for quantum information processing. Among multipartite entangled states, Dicke states ∣Dnk⟩ are distinguished by their permutation symmetry, which provides robustness against particle loss and enables applications for quantum communication and computation. Although Dicke states have been realized in various platforms, most optical implementations rely on postselection, which destroys the state upon detection and prevents its further use. A heralded optical scheme is therefore highly desirable. Here, we present a linear-optical heralded scheme for generating arbitrary Dicke states ∣Dnk⟩ with 3n+k photons through the framework of the linear quantum graph (LQG) picture. By mapping the scheme design into the graph-finding problem, and exploiting the permutation symmetry of Dicke states, we overcome the structural complexity that has hindered previous approaches. Our results provide a resource-efficient pathway toward practical heralded preparation of Dicke states for quantum technologies.
Heralded multipartite photonic entanglement underpins quantum communication, distributed sensing, and fault-tolerant computation, yet the manual synthesis of linear-optical circuits that generate a prescribed target state remains a combinatorial bottleneck. A recent preprint [1,2] recasts this design task as an algorithmic graph-search problem in the linear quantum graph (LQG) picture, reporting automated discovery of heralded circuits for hypergraph magic states, quantum error-correcting code states, general three-qubit states, and length-1 caterpillar graph states. This paper provides a single reconciled synthesis and quantitative appraisal. We formalize the search-space reduction claimed by the LQG representation with explicit combinatorial counts: for a representative six-mode, three-beam-splitter budget, raw circuit enumeration (3,375 ordered-slot instances) reduces to 455 graph instances, a factor of ≈7.4 (up to ≈44.5 under a reordering-inclusive convention), growing to ≈151 at ten modes and five beam splitters. We derive heralding probabilities for canonical fusion-based constructions: a three-qubit GHZ state via type-II fusion succeeds with probability p²/2 per trial (5×10⁻³ at source probability p = 0.1), and a four-qubit caterpillar projection spans [3.2×10⁻⁷, 7.5×10⁻³] across plausible hardware ranges — showing that hardware parameters, not circuit topology, dominate success probability. We further derive multiplexing arithmetic (a p = 0.01 circuit needs ≈458 attempts for 99% cumulative success; 64-fold multiplexing converts this to 8 rounds of 64 executions (512 total, ≈12% more than the unmultiplexed 458) and loss-limited fidelity bounds F ≈ η^{2q}. We identify falsification criteria and the decisive open problems: graph-isomorphism overhead, ancilla-mode blowup, loss-aware search objectives, and scalability beyond the demonstrated q ≤ 4 target classes. ## 1. Introduction Photons are attractive carriers of quantum information: they decohere negligibly Full text and updates: papers.qnfo.org/papers/algorithmic-graph-search-design-of-heralded-linear-optical-circuits-for-multipar/
Indistinguishability of identical particles is a resource for quantum information processing and has been utilized to generate entanglement from independent particles that spatially overlap only at the detection stage. Here, we introduce a controllable scheme capable of generating via three steps, i.e., initialization, deformation, and postselection, different classes of multipartite entangled states starting from a product state of N spatially distinguishable identical qubits. While our scheme is generalizable to any class of entangled bosonic and fermionic systems, we provide an explicit recipe for the generation of W , Dicke, GHZ, and cluster states, which are resource states for quantum information processing. Using graph-based representations within the framework of spatially localized operations and classical communication (sLOCC), we mathematically demonstrate a direct translation of the generation schemes of specific entangled states into colored, complex, and weighted digraphs, each corresponding to a given experimental setup. We also show that this graph-theoretical approach allows for the optimization of the generation efficiency of specific multipartite entangled states by exploring a variety of generation schemes. The presented theoretical approach, while already implementable with current linear optics architectures, has the potential to bring clear advantages over existing technologies, such as in quantum computing search algorithms and in the design of new experiments in quantum optics or other platforms. Published by the American Physical Society 2024