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

TL;DR

This work tackles the challenge of realizing deterministic, high-fidelity quantum gates in linear optics by employing a hybrid encoding that uses the control qubit in polarization and the target qudit in spatial DOFs. The authors demonstrate a minimal-resource CNOT gate with a single PBS and scalable generalized CSWAP gates that require exactly $d$ PBSs while maintaining optical depth of 1, extending from $U_{CSWAP}^{2,2,2}$ to $U_{CSWAP}^{2,d,d}$. Fidelity analyses show exceptional performance, with the three-qubit CSWAP gate exceeding $99.7\%$ under realistic imperfections and robust average fidelity formulas across PBS misalignments. The results indicate a practical path toward efficient, high-dimensional photonic quantum processing using current linear-optics technology.

Abstract

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.

Figures (6)

• Figure 1: The polarization-spatial controlled-NOT gate with linear optics. Single photon $(\alpha|H\rangle+\beta|V\rangle)$ emitted from spatial mode $|a\rangle$. The PBS transmits the horizontal component and reflects the vertical component of photons. The $\textrm{BS}$ has variable reflectivity.
• Figure 2: (a) Experimental setup to create the input state of Fredkin gates. (b) The polarization-spatial controlled-SWAP gate with linear optics.
• Figure 3: (a) Experimental setup of the state preparation. (b) The linear-optical controlled-SWAP gate on $\mathbb{C}^2\otimes \mathbb{C}^3\otimes \mathbb{C}^3$.
• Figure 4: (a) Experimental setup of the state-preparation. (b) The controlled-SWAP gate on $\mathbb{C}^2\otimes \mathbb{C}^d\otimes \mathbb{C}^d$.
• Figure 5: Average fidelity $\overline{F}$ of the controlled-SWAP gate $U_{\text{CSWAP}}^{2,2,2}$ with $r\in[0,1\times10^{-3}]$ and $\theta\in[0,5\times10^{-3}]$.