Single-photon-boosted type-I fusion gates

Abstract

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 $1/2$. Existing $3/4$-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 that achieves a total success probability of $3/4$ using only four ancillary single photons and passive linear optics. The gate succeeds directly with probability $5/8$, while a distillation step converts partially entangled outcomes into additional successful events. 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. 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.

Figures (5)

• Figure 1: Notation for a balanced $50{:}50$ beam splitter acting on two optical modes.
• Figure 2: Type-I fusion gate. (a) Linear-optical circuit realizing a standard type-I gate. (b) A successful event is heralded by detection of exactly one photon in mode $1$ or mode $4$, which implements the map $|0\rangle\langle 00| \pm |1\rangle\langle 11|$ on the input qubits. Detection of zero or two photons corresponds to failure.
• Figure 3: Boosted type-I fusion gate using four ancillary single photons. Part (I) converts the ancillary input $|1111\rrangle$ into two states of the form $(|20\rrangle-|02\rrangle)/\sqrt{2}$. Part (II) interferes the input qubits with these ancillary states and measures modes $1$, $4$, $5$, and $7$. Odd-photon detection events yield the standard type-I fusion output, a subset of four-photon events yields a two-qubit entangled output, and two-photon events can be converted into successful fusion outcomes by the distillation protocol of Fig. \ref{['fig:Distillation']}.
• Figure 4: Distillation protocol for the $n_d=2$ outputs of the boosted type-I fusion gate. The circuit consists of balanced beam splitters and beam splitters with equal transmission coefficient $t$. A successful distillation round is heralded by vacuum detection in mode $3$ and single-photon detection in modes $9$ and $10$. If no photons are detected in modes $3$, $9$, and $10$, a part of the procedure can be repeated, until one or two photons are detected in modes $9$ and $10$.
• Figure 5: Representative fusion-based constructions of photonic resource states. (a) A four-qubit GHZ state generated from one Bell state and one three-qubit GHZ state using a single type-I fusion gate. (b) A four-qubit GHZ state generated from three Bell states using two type-I fusion gates. (c) A six-qubit ring graph state, up to local single-qubit operations, generated from three three-qubit GHZ states followed by single-qubit Hadamard gates on the marked qubits and three type-I fusion gates.