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arXiv:2406.15666 · Physical Review A

Generalized type-II fusion of cluster states

Noam Rimock, Khen Cohen, Yaron Oz

Published Dec 5, 2025• 2 citations
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Abstract

Measurement based quantum computation is a quantum computing paradigm that employs single-qubit measurements performed on an entangled resource state in the form of a cluster state. A basic ingredient in the construction of the resource state is the type-II fusion procedure, which probabilistically merges two separate photonic cluster states by a quantum measurement. We generalize the type-II fusion procedure by generalizing the measurement setup, and classify the resulting final states, which also include cluster states up to single-qubit rotations. We prove that the probability for the success of the generalized type-II fusion is bounded by fifty percent, and classify all the possibilities to saturate the bound. We analyze the enhancement of the fusion success probability above the fifty percent bound, by the reduction of the entanglement entropy of the resulting state. We prove that the only states that can be obtained with a hundred percent probability of success, are product states.

DOI
10.1103/w9w5-2h7r

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Measurement based quantum computation is a quantum computing paradigm that employs single-qubit measurements performed on an entangled resource state in the form of a cluster state. A basic ingredient in the construction of the resource state is the type-II fusion procedure, which probabilistically merges two separate photonic cluster states by a quantum measurement. We generalize the type-II fusion procedure by generalizing the measurement setup, and classify the resulting final states, which also include cluster states up to single-qubit rotations. We prove that the probability for the success of the generalized type-II fusion is bounded by fifty percent, and classify all the possibilities to saturate the bound. We analyze the enhancement of the fusion success probability above the fifty percent bound, by the reduction of the entanglement entropy of the resulting state. We prove that the only states that can be obtained with a hundred percent probability of success, are product states.
Weighted graph states extend standard graph states by associating phases with entangling edges, and may serve as resources for measurement-based quantum computation. We analyze how the two main fusion operations, Type-I and type-II, act on weighted graph states. Type-I fusion operates identically to the unweighted case, merging two one-dimensional weighted graphs, while preserving edge weights and success probabilities. In addition, the pool of two-qubit weighted graph states can be generated easily from Greenberger-Horne-Zeilinger states or Bell pairs. In contrast, type-II fusion requires a logical qubit, which can be formed only for specific weight configurations, and with a success probability below one-half. When successful, it fuses the states correctly, but its failure outcomes destroy the structure of the graphs, removing the good-failure feature known from ordinary graph states. These two obstacles can be overcome by using extra qubits and π weights in the one-dimensional weighted graph states that are being fused, which enables scalability. We compute the change in the entanglement entropy of the resulting link due to the fused states being weighted graph states (for generalized fusion), and classify the resulting states of a general non-Bell projection. These results define the practical limits of the fusion-based construction of weighted graph states. To our knowledge, this is the first time that fusion processes have been used to construct weighted graph states.
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.
We formalize a generalized type-II fusion operation for qudit cluster states within linear optics. Two designated qudits, one from each input cluster, interfere with optional ancilla qudits via a passive linear-optical network, followed by number-resolving detection; conditioned on measurement outcome, the remaining qudits form the post-selected fused state. We prove a general rank bound: for any such interferometer and outcome, the reduced density matrix across the two parent clusters has Schmidt rank at most MMM, the total number of measured qudits including ancillae. Consequently, a correct qudit fusion which requires rank ddd is impossible without ancillae and requires at least d−2d-2d−2 ancilla qudits. Our analysis extends previous no-go results for Bell-type qubit fusion to the qudit setting and to generalized, non-Bell projections. We analyze the probabilities and entanglement of the relevant measurement outcomes, and discuss how our lower bound aligns with existing constructive schemes. These results set a clear resource threshold for high-dimensional, fusion-based photonic MBQC.
The quantum fusion technique enables the bonding of two cluster states together, and therefore provides a promising method to grow large-scale cluster states. Quantum fusion was first proposed in a discrete-variable regime. In terms of the resource usage, quantum fusion can be divided into two categories, which are type-I and type-II fusions. Type-I fusion consumes one qubit in the fusion procedure, while type-II consumes two qubits. However, for continuous-variable (CV) analogs, concrete schemes for realizing quantum fusions of cluster states with arbitrary topologies have not been shown yet. In this paper, we present the CV analog of quantum fusion schemes. We show that CV type-I fusion can be realized by an ancilla and two quantum nondemolition interactions, and CV type-II fusion can be realized by an entanglement swapping scheme. With finite squeezing considered, we analyze the extra errors added by the fusion procedures in CV cluster-state quantum computation. Our schemes are universal and can be generalized to fuse cluster states with arbitrary topologies.
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.
In this paper, we propose a linear-optical quantum computation scheme that starts from high-dimensionally entangled qudits but performs quantum computation on qubits defined in their subspaces. This approach enables us to increase the success probability of linear-optical fusions from 50%50\%50% to 1−1/d1-1/d1−1/d for qudits of dimension ddd without relying on existing approaches using ancilla photons and quantum codes. In particular, we show that a 2D cluster state of qubits can be generated from 5-qudit 1D cluster states using pairwise fusion gates while preserving their boosted success probability. We numerically demonstrate that this approach can improve the loss tolerance of a percolation-based scheme while keeping the photon number per resource state constant.