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Algorithmic Graph-Search Design of Heralded Linear Optical Circuits for Multipartite Entanglement: A Reconciled Quantitative Assessment

Rowan Brad Quni-Gudzinas

Published Oct 3, 2026• 0 citations
Publisher

Abstract

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/

DOI
10.5281/zenodo.23120229

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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/
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.
Graph states are central resources for quantum information processing, supporting applications in computation, communication, and error correction. In photonic systems, they are typically assembled from smaller entangled states using probabilistic fusion gates, which demand many photons and suffer from low success rates. We present an optimized scheme for directly generating caterpillar graph states (CGSs)—essential resource states for constructing high-dimensional lattice graph states—using only single-photon sources, linear optics, and heralded measurements. Based on the linear quantum graph (LQG) picture, our method produces CGSs efficiently. For CGSs of length l ≥ 3 , it requires l − 2 fewer photons and achieves a success rate 2 l − 2 times higher than fusion-based approaches. These results demonstrate that the LQG picture provides a powerful and flexible route to generating complex photonic graph states for efficient quantum information processing.
Designing photonic circuits that prepare graph states with high fidelity and success probability is a central challenge in linear optical quantum computing. Existing approaches rely on hand-crafted designs or fusion-based assemblies. In the absence of multiplexing/boosting, both post-selected ballistic circuits and sequential fusion exhibit exponentially decreasing single-shot yields - a fundamental limitation that makes optimizing individual resource state generators particularly important, as these serve as building blocks in larger FBQC architectures. We present a general-purpose optimization framework for automated photonic circuit discovery using a novel polynomial-based simulation approach, enabling efficient strong simulation and gradient-based optimization. Our framework employs a two-pass optimization procedure: the first pass identifies a unitary transformation that prepares the desired state with perfect fidelity and maximal success probability, and the second pass implements a novel sparsification algorithm that reduces this solution to a compact photonic circuit with minimal beamsplitter count while preserving performance. This sparsification procedure often reveals underlying mathematical structure, producing highly simplified circuits with rational reflection coefficients. We demonstrate our approach by discovering optimized circuits for 333-, 444-, and 555-qubit graph states across multiple equivalence classes. For 4-qubit states, our circuits achieve success probabilities of 2.053×10−32.053 \times 10^{-3}2.053×10−3 to 7.813×10−37.813 \times 10^{-3}7.813×10−3, outperforming the fusion baseline by up to 4.7×4.7 \times4.7×. For 5-qubit states, we achieve 5.926×10−55.926 \times 10^{-5}5.926×10−5 to 1.157×10−31.157 \times 10^{-3}1.157×10−3, demonstrating up to 7.5×7.5 \times7.5× improvement. These results include the first known state preparation circuits for certain 5-qubit graph states.
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.
Entanglement is a fundamental feature of quantum mechanics and a key resource for quantum information processing. Among multipartite entangled states, Dicke states ∣Dnk⟩|D_n^k\rangle∣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⟩|D_n^k\rangle∣Dnk​⟩ with 3n+k3n+k3n+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.
Photonic graph states with advanced topologies can enable measurement-based quantum computing, distributed quantum sensing, and quantum interconnects. However, the efficient generation of photonic graph states is limited by the probabilistic nature of photonic entangling operations and the exponential dependence of generation rate on resource cost. In this work, we study photonic graph state synthesis as a cost-aware decomposition problem, exploiting local Clifford (LC) equivalence to identify more synthesis-friendly representations of the target graph state before decomposition. Specifically, we propose Cost-aware Fusion-based Decomposition (CFD), a three-stage heuristic framework that decomposes a target graph state into ring, star, and linear motifs, and assembles them via Type-I fusion operations to minimize fusion overhead and physical-qubit consumption. We further show that selecting the LC-equivalent graph state with the minimum number of edges provides a highly effective proxy for near-optimal synthesis: in many cases it matches the best generation rate observed within the LC equivalence class under CFD, and in most remaining cases it remains close to it. Numerical evaluations on graph state orbit data and 2D and 3D lattice graph states demonstrate that CFD achieves up to 84.6\% reduction in resource overhead compared to baseline constructions, and yields improvements in photonic generation rate spanning multiple orders of magnitude. These results suggest that combining structure-aware motif decomposition with LC equivalence is a practical and scalable strategy for photonic graph state synthesis.