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arXiv:2512.03131

Generating redundantly encoded resource states for photonic quantum computing

Samuel J. Sheldon, Pieter Kok

quant-phSubmitted Dec 2, 2025• Announced Dec 4, 2025• 0 citations
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Abstract

Measurement-based quantum computing relies on the generation of large entangled cluster states that act as a universal resource on which logical circuits can be imprinted and executed through local measurements. A number of strategies for constructing sufficiently large photonic cluster states propose fusing many smaller resource states generated by a series of quantum emitters. However, the fusion process is inherently probabilistic with a 50% success probability in standard guise. A recent proposal has shown that, in the limit of low loss, the probability of achieving successful fusion may be boosted to near unity by redundantly encoding the vertices of linear graph states using Greenberger-Horne-Zeilinger states [Quantum 7, 992 (2023)]. Here we present a protocol for deterministically generating redundantly encoded photonic resource states using single quantum emitters, and study the impact of protocol errors and photonic losses on the generated resource states and type-II photonic fusion. Our work provides a route for efficiently constructing complex entangled photonic qubit states for photonic quantum computing and quantum repeaters.

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Measurement-based quantum computing relies on the generation of large entangled cluster states that act as a universal resource on which logical circuits can be imprinted and executed through local measurements. A number of strategies for constructing sufficiently large photonic cluster states propose fusing many smaller resource states generated by a series of quantum emitters. However, the fusion process is inherently probabilistic with a 50% success probability in standard guise. A recent proposal has shown that, in the limit of low loss, the probability of achieving successful fusion may be boosted to near unity by redundantly encoding the vertices of linear graph states using Greenberger-Horne-Zeilinger states [Quantum 7, 992 (2023)]. Here we present a protocol for deterministically generating redundantly encoded photonic resource states using single quantum emitters, and study the impact of protocol errors and photonic losses on the generated resource states and type-II photonic fusion. Our work provides a route for efficiently constructing complex entangled photonic qubit states for photonic quantum computing and quantum repeaters.
Entanglement in multiphoton graph states is an important resource with various applications, such as measurement-based quantum computing and memory-less quantum repeaters. While some primitive instances have been successfully prepared, scalable generation of large-scale graph states with programmable connections remains an outstanding challenge. Fusion provides a promising "divide-and-conquer" approach by linking smaller resource states into larger, more complex graphs. However, its practical scalability has been limited by the probabilistic nature of conventional fusion protocols, which necessitates extensive hardware overhead and feed-forward resources to achieve deterministic outcomes. Here, we introduce and experimentally demonstrate a deterministic fusion operation with built-in error mitigation, capable of connecting small time-bin-encoded cluster states--originally generated on-demand from a quantum emitter--into larger, reconfigurable photonic graph states of increased sizes. The fusion operation is realized by nondestructively projecting pairs of flying microwave photonic qubits into Bell states using superconducting circuits with in-situ programmability. Crucially, the protocol features active reset and reuse of matter qubits during graph state generation, thereby circumventing the coherence constraints of the quantum emitter and enabling the generation of genuine multipartite entanglement across 13 photonic qubits. Our results establish a scalable framework for photonic graph-state synthesis, paving the way for fault-tolerant measurement-based quantum computation and distributed quantum networks.
Fusion-based quantum computing (FBQC) relies on a set of small, typically photonic, resource states that are fused together through Bell state measurements. The main bottleneck of FBQC is the low rate of generating the resource states, which stems from the probabilistic nature of photonic fusion gates. Previous work introduced a general algorithm for constructing circuits that deterministically generate photonic resource states from a minimal number of quantum emitters for a specified photon emission ordering. However, finding the minimal number of emitters and CNOT gates across all possible orderings is an NP-hard problem. Here, we exploit the symmetries present in FBQC resource states to dramatically simplify this optimization problem. We find that logically encoded 24-photon FBQC resource states can be produced using as few as 3 emitters and 11 CNOTs.
Fusion-based photonic quantum computing architectures rely on two primitives: i) near-deterministic generation and control of constant-size entangled states and ii) probabilistic entangling measurements (photonic fusion gates) between entangled states. Here, we demonstrate these key functionalities by fusing resource states deterministically generated using a solid-state spin-photon interface. Repetitive operation of the source leads to sequential entanglement generation, whereby curiously entanglement is created between the quantum states of the same spin at two different instances in time. Such temporal multiplexing of photonic entanglement provides a resource-efficient route to scaling many-body entangled systems with photons.
We develop an architecture for measurement-based quantum computing using photonic quantum emitters. The architecture exploits spin-photon entanglement as resource states and standard Bell measurements of photons for fusing them into a large spin-qubit cluster state. The scheme is tailored to emitters with limited memory capabilities since it only uses an initial non-adaptive (ballistic) fusion process to construct a fully percolated graph state of multiple emitters. By exploring various geometrical constructions for fusing entangled photons from deterministic emitters, we improve the photon loss tolerance significantly compared to similar all-photonic schemes.
Fusion measurements are a key primitive for linear-optical quantum computing and quantum networks. Type-I and type-II fusion gates are widely used to combine small entangled resource states into larger photonic states, but without ancillary resources their success probability is limited to 1/21/21/2. Existing 3/43/43/4-efficient type-I schemes rely on entangled Bell-pair ancillary states, whose preparation is itself probabilistic and resource-intensive. Here we propose a boosted type-I fusion gate using only four ancillary single photons and standard linear-optical primitives. For multiqubit GHZ and graph states, the gate succeeds directly with probability 5/85/85/8, while a distillation protocol converts partially entangled outcomes into additional successful events, raising the total success probability to 11/1611/1611/16 after one stage and asymptotically to 3/43/43/4. We quantify the practical advantage of this scheme by estimating the photonic resources required for generating representative large entangled photonic states and show that the proposed gate significantly reduces the required overhead compared with existing schemes. These results expand the set of resource-efficient linear-optical primitives and enable a substantial reduction in the resource requirements for scalable photonic quantum computing and quantum communication.
Multiqubit entangled photonic graph states are an important ingredient for all-photonic quantum computing, repeaters, and networking. Preparing them using probabilistic stitching of single photons using linear optics presents a formidable resource challenge due to multiplexing needs. Quantum emitters provide a viable solution to prepare photonic graph states, as they enable deterministic production of photons entangled with emitter qubits and deterministic two-qubit interactions among emitters. A handful of emitters often suffice to generate useful-size graph states that would otherwise require millions of emitters used as single photon sources, using the linear-optics method. Photon loss, however, impedes the emitter method due to a large circuit depth and hence loss accrual on the photons of the graph state produced, given the typically large number of slow two-qubit controlled-not (cnot) gates between emitters. We propose an algorithm that can trade the number of emitters with the graph-state depth, while minimizing the number of emitter cnot gates. We apply our algorithm to generate a repeater graph state (RGS) for an alternative all-photonic repeater protocol, which achieves a far superior rate-distance trade-off compared to using the least number of emitters needed to generate the RGS. Yet it needs five orders of magnitude fewer emitters than the multiplexed linear-optics method---with each emitter used as a photon source---to achieve a desired rate-distance performance.