Fusion for High-Dimensional Linear Optical Quantum Computing with Improved Success Probability

TL;DR

This work advances high-dimensional fusion-based quantum computing by delivering the first efficiently scaling Type-II fusion gates for arbitrary qudit dimension. It shows that, for even $d$, a fusion protocol with success probability near $2/d^2$ is achievable using a $(d-2)$-qudit entangled ancilla, and extends this to odd $d$ via embedding into an even dimension to obtain $P_{ ext{succ}} = 2/[d(d+1)]$ (with $D=d+1$). A concrete, experimentally plausible ancilla construction using a silicon spin qudit coupled to a microwave cavity via time-bin multiplexing is proposed, alongside a general framework of extra-dimensional corrections to convert non-maximally-entangled projections into Bell measurements. The paper also analyzes several alternative fusion strategies based on W-state ancillae, Paesani’s state-generation circuits, and qubit boosting, highlighting tradeoffs between ancilla complexity and success probability. Overall, these results establish a strong foundation for high-dimensional FBQC, offering practical paths toward scalable, loss-tolerant linear-optical quantum computing with qudits and guiding future fault-tolerant, high-dimensional architectures.

Abstract

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 which 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 non-deterministically correct non-maximally-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.

Figures (13)

• Figure 1: Examples of qubit graph states. Up to single-qubit Clifford operations, any stabilizer state is expressible as a graph state, so they are commonly used as entangled resource states. On the left is a graph state consisting of six entangled photonic qubits arranged in a hexagonal configuration, often referred to as a $6$-ring Bartolucci2023. On the right is a graph state of four entangled photonic qubits, generally called a linear cluster state.
• Figure 2: Type-II fusion: a) We perform Type-II fusion between chosen qubits of two three-photon graph states. Type-II fusion destroys all the qubits that are involved - measured - in the process. The resulting graph state after the Type-II fusion is a graph state with nontrivial entanglement between the four surviving photons. b) For a simpler example, Type-II fusion can be used to join two linear cluster states into a single joint cluster. Here we obtain a Bell pair; if the initial linear clusters instead had lengths $n$ and $m$, the resulting linear cluster would have length $n+m-2$. c) Standard Type-II fusion circuit: Qubits are represented using dual-rail encoding, where each qubit consists of one photon between two modes. Two 50:50 beamsplitters (Hadamard operations) are applied between modes 0 and 2, and modes 1 and 3, followed by measurements on all modes.
• Figure 3: Boosted Type-II Fusion Ewert_2014. As in Fig. \ref{['fig:fusion-types']}, we apply 50:50 (Hadamard) beamsplitters between corresponding modes of the input qubits. Before measuring, however, we also allow for interference with an ancillary state of the form $\frac{1}{\sqrt{2}}(\ket{20} - \ket{02})$. This protocol has a success probability of $0.625$; if the other qubit undergoes a similar treatment, the success probability is increased to $0.75$.
• Figure 4: Boosted Type-II Fusion bartolucci2021creationentangledphotonicstates. Similarly to Fig. \ref{['fig:boosted_type-II']}, we allow for interference with an ancillary state between the standard fusion circuit and the measurement. This protocol uses only a single ancillary photon and a three-mode Fourier transform. The success probability is $7/12$ as depicted and $2/3$ if the additional interference is implemented on both sides.
• Figure 5: Type-II fusion for $d=4$ using Protocol \ref{['proto:even']}bharos2024efficienthighdimensionalentangledstatebharoshigh. The last two input ports contain the ancilla state $\frac{1}{\sqrt{2}}(\ket{\mathbf{01}} + \ket{\mathbf{23}})$. The second input qudit undergoes the permutation $U_P$ described in the protocol, swapping modes $0$ and $1$ and modes $2$ and $3$. Each jth mode of the ith port undergoes a four-dimensional Fourier transform.

Theorems & Definitions (12)

• Lemma 2.2
• Theorem 3.2
• proof
• Theorem 3.4
• proof
• Theorem 4.1
• Corollary 4.2
• Theorem 4.3
• Lemma A-III.1
• proof