Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2510.16623

Generalized Fusion of Qudit Graph States

Noam Rimock, Yaron Oz

Published Oct 18, 2025• 0 citations
arXivHTMLPDF

Abstract

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 $M$, the total number of measured qudits including ancillae. Consequently, a correct qudit fusion which requires rank $d$ is impossible without ancillae and requires at least $d-2$ 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.

DOI
10.48550/arxiv.2510.16623

Related papers

Fusion for high-dimensional linear-optical quantum computing with improved success probability

Gözde Üstün, Eleanor Gilbert Rieffel, Simon J. Devitt +1 more

Type-II fusion is a probabilistic entangling measurement that is essential to measurement-based linear-optical quantum computing and can be used for quantum teleportation more broadly. However, it remains under-explored for high-dimensional qudits. Our main result gives a Type-II fusion protocol with proven success probability approximately 2 / d 2 for qudits of arbitrary dimension d . This generalizes a previous method that only applied to even-dimensional qudits. We believe this protocol to be the most efficient known protocol for Type-II fusion, with the d = 5 case beating the previous record by a factor of approximately 723 . We discuss the construction of the required ( d − 2 ) -qudit ancillary state using a silicon spin-qudit ancilla coupled to a microwave cavity through time-bin multiplexing. We then introduce a general framework of extra-dimensional corrections, a natural technique in linear optics that can be used to nondeterministically correct nonmaximally entangled projections into Bell measurements. We use this method to analyze and improve several different circuits for high-dimensional Type-II fusion and compare their benefits and drawbacks.

arXiv:2505.16816•Physical Review Applied•Aug 28, 2025•2 citations

Auxiliary Schmidt Rank as a Resource for Photonic Bell Measurements

Pradip Laha, Peter van Loock

In quantum communication and fusion-based quantum computation, photonic Bell measurements are fundamentally limited when only passive linear optics is employed. While for qubits, some Bell states can be unambiguously identified with static beam splitters and no extra photons or entanglement, additional auxiliary photons or at least additional auxiliary degrees of freedom with a certain level of additional entanglement are needed to approach or attain a complete, deterministic Bell measurement. Here, we prove an exact resource threshold when the same two photons carry system qudits of dimension $d$ and a fixed auxiliary entangled state $Φ$, possibly distributed over several additional degrees of freedom, with total Schmidt rank $r_Φ$. We show that a single conclusive Bell-label functional can occur for $r_Φ\geqslant\lceil d/2\rceil$, but deterministic discrimination of all $d^2$ Bell-state labels requires $r_Φ\geqslant d$. A maximally entangled rank-$d$ auxiliary state achieves the bound by local Bell-basis sorting between each photon's system and auxiliary degrees of freedom. Thus, the auxiliary Schmidt rank is a certified resource for ancilla-photon-free, embedded photonic Bell measurements.

quant-pharXiv:2606.24591•Jun 24, 2026•0 citations

Subdimensional linear-optical quantum computation: from qudit resource states to qubit quantum computation

Tomohiro Yamazaki

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 $50\%$ to $1-1/d$ for qudits of dimension $d$ 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.

quant-pharXiv:2610.03408•Oct 5, 2026•0 citations

Type-I and Type-II Fusion Protocols for Weighted Graph States

N. Rimock, Y. Oz

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.

quant-pharXiv:2601.13381•Physical Review A•Jan 21, 2026•0 citations

Minimal ancilla for boosting type-I photonic fusion

A. A. Melkozerov, S. S. Straupe, M. Yu. Saygin

Fusion gates enable the creation of large entangled photonic states for linear-optical quantum computing and communication protocols. However, their success probability is limited to $1/2$ without additional photonic resources. Previous schemes for boosting type-I fusion have used four ancillary single photons or a two-qubit Bell state, whose preparation is probabilistic and resource intensive. Here we propose a boosted type-I fusion gate using only two independent ancillary single photons and no entangled ancillary input. It achieves a direct success probability of $5/8$, matching that of the previous four-single-photon proposal while using half as many ancillary photons. The circuit is predominantly passive, with a single active element that applies a phase shift or swaps two modes according to intermediate photodetection outcomes. Unlike the previous schemes, successful outputs occupy fixed modes and require no further conditional output routing. In particular, we show that the gate is more robust to heralded ancillary-photon loss than previously proposed boosted gates and reduces the average number of single photons required to generate a six-qubit ring graph state by more than $4.6\%$ compared with other type-I schemes. It therefore offers a resource-efficient primitive for scalable fusion-based photonic quantum information processing.

quant-pharXiv:2610.34257•Oct 27, 2026•0 citations

Generalized type-II fusion of cluster states

Noam Rimock, Khen Cohen, Yaron Oz

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.

arXiv:2406.15666•Physical Review A•Dec 5, 2025•2 citations
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.
Type-II fusion is a probabilistic entangling measurement that is essential to measurement-based linear-optical quantum computing and can be used for quantum teleportation more broadly. However, it remains under-explored for high-dimensional qudits. Our main result gives a Type-II fusion protocol with proven success probability approximately 2 / d 2 for qudits of arbitrary dimension d . This generalizes a previous method that only applied to even-dimensional qudits. We believe this protocol to be the most efficient known protocol for Type-II fusion, with the d = 5 case beating the previous record by a factor of approximately 723 . We discuss the construction of the required ( d − 2 ) -qudit ancillary state using a silicon spin-qudit ancilla coupled to a microwave cavity through time-bin multiplexing. We then introduce a general framework of extra-dimensional corrections, a natural technique in linear optics that can be used to nondeterministically correct nonmaximally entangled projections into Bell measurements. We use this method to analyze and improve several different circuits for high-dimensional Type-II fusion and compare their benefits and drawbacks.
In quantum communication and fusion-based quantum computation, photonic Bell measurements are fundamentally limited when only passive linear optics is employed. While for qubits, some Bell states can be unambiguously identified with static beam splitters and no extra photons or entanglement, additional auxiliary photons or at least additional auxiliary degrees of freedom with a certain level of additional entanglement are needed to approach or attain a complete, deterministic Bell measurement. Here, we prove an exact resource threshold when the same two photons carry system qudits of dimension ddd and a fixed auxiliary entangled state ΦΦΦ, possibly distributed over several additional degrees of freedom, with total Schmidt rank rΦr_ΦrΦ​. We show that a single conclusive Bell-label functional can occur for rΦ⩾⌈d/2⌉r_Φ\geqslant\lceil d/2\rceilrΦ​⩾⌈d/2⌉, but deterministic discrimination of all d2d^2d2 Bell-state labels requires rΦ⩾dr_Φ\geqslant drΦ​⩾d. A maximally entangled rank-ddd auxiliary state achieves the bound by local Bell-basis sorting between each photon's system and auxiliary degrees of freedom. Thus, the auxiliary Schmidt rank is a certified resource for ancilla-photon-free, embedded photonic Bell measurements.
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.
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.
Fusion gates enable the creation of large entangled photonic states for linear-optical quantum computing and communication protocols. However, their success probability is limited to 1/21/21/2 without additional photonic resources. Previous schemes for boosting type-I fusion have used four ancillary single photons or a two-qubit Bell state, whose preparation is probabilistic and resource intensive. Here we propose a boosted type-I fusion gate using only two independent ancillary single photons and no entangled ancillary input. It achieves a direct success probability of 5/85/85/8, matching that of the previous four-single-photon proposal while using half as many ancillary photons. The circuit is predominantly passive, with a single active element that applies a phase shift or swaps two modes according to intermediate photodetection outcomes. Unlike the previous schemes, successful outputs occupy fixed modes and require no further conditional output routing. In particular, we show that the gate is more robust to heralded ancillary-photon loss than previously proposed boosted gates and reduces the average number of single photons required to generate a six-qubit ring graph state by more than 4.6%4.6\%4.6% compared with other type-I schemes. It therefore offers a resource-efficient primitive for scalable fusion-based photonic quantum information processing.
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.