Efficient and deterministic high-dimensional controlled-swap gates on hybrid linear optical systems with high fidelity

Implementation of quantum logic gates with linear optical elements plays a prominent role in quantum computing due to the relatively easier manipulation and realization. We present efficient schemes to implement controlled-NOT (CNOT) gate and controlled-swap (Fredkin) gate by solely using linear optics. We encode the control qubits and target qudits in photonic polarization (two-level) and spatial degrees of freedom ($d$-level), respectively. Based on the hybrid encoding, CNOT and Fredkin gates are constructed in a deterministic way without any borrowed ancillary photons or measurement-induced nonlinearities. Remarkably, the number of linear optics required to implement a CNOT gate has been reduced to one polarization beam splitter (PBS), while only $d$ PBSs are necessary to implement a generalized Fredkin gate. The optical depths of all schemes are reduced to one and dimension-independent. Besides, the fidelity of our three-qubit Fredkin gate is higher than 99.7% under realistic conditions, which is higher than the previous schemes.

💡 Research Summary

The paper presents a resource‑efficient, deterministic implementation of both two‑qubit controlled‑NOT (CNOT) and three‑qubit controlled‑swap (Fredkin) gates using only linear‑optical components. The authors adopt a hybrid encoding scheme: the control qubit is encoded in the photon’s polarization (horizontal = |1⟩, vertical = |0⟩), while the target qudits are encoded in spatial modes that span a d‑dimensional Hilbert space. By converting spatial operations into polarization‑dependent operations, the CNOT gate can be realized with a single polarization beam splitter (PBS). For the Fredkin gate, only d PBSs are required to swap two d‑level target qudits conditioned on the control polarization.

The construction begins with a spontaneous parametric down‑conversion (SPDC) source that produces a polarization‑entangled photon pair. After passing through a set of non‑polarizing beam splitters (BS) that distribute the photons into the required spatial modes, a single PBS (for CNOT) or a cascade of d PBSs (for Fredkin) performs the conditional logic. The PBS transmits horizontal polarization and reflects vertical polarization, thereby effecting the required swap of spatial paths only when the control photon is in the horizontal state. Because the optical depth of each circuit is one, the schemes avoid the cumulative loss and phase instability that plague deeper interferometric networks.

A key contribution is the extension to arbitrary dimension d. The authors show that by using 2(d − 1) BSs to generate d spatial modes for each target photon and then arranging d PBSs, the generalized controlled‑swap U₂, d, d can be implemented deterministically. The number of linear‑optical elements scales linearly with d, while the depth remains constant. Compared with prior work (e.g., Ref.