Improved entanglement-based high-dimensional optical quantum computation with linear optics

TL;DR

This paper tackles the challenge of implementing deterministic high-dimensional CSWAP (Fredkin) gates in linear optics. It introduces a hybrid encoding where the control is carried by polarization and the targets by spatial modes, and shows how to realize 2⊗d⊗d CSWAP gates with a shallow circuit depth of five using only (2+3d) linear-optical elements, without ancillary photons. The authors provide concrete constructions for 2⊗2⊗2, 2⊗3⊗3, and the general 2⊗d⊗d cases, achieving fidelities up to 99.4% under realistic imperfections and outperforming prior work. This approach promises a scalable, deterministic route to high-dimensional photonic quantum computation leveraging entangled photon pairs and hyperentanglement.

Abstract

Quantum gates are the essential block for quantum computer. High-dimensional quantum gates exhibit remarkable advantages over their two-dimensional counterparts for some quantum information processing tasks. Here we present a family of entanglement-based optical controlled-SWAP gates on $\mathbb{C}^{2}\otimes \mathbb{C}^{d}\otimes \mathbb{C}^{d}$. With the hybrid encoding, we encode the control qubits and target qudits in photonic polarization and spatial degrees of freedom, respectively. The circuit is constructed using only $(2+3d)$ ($d\geq 2$) linear optics, beating an earlier result of 14 linear optics with $d=2$. The circuit depth 5 is much lower than an earlier result of 11 with $d=2$. Besides, the fidelity of the presented circuit can reach 99.4\%, and it is higher than the previous counterpart with $d=2$. Our scheme are constructed in a deterministic way without any borrowed ancillary photons or measurement-induced nonlinearities. Moreover, our approach allows $d>2$.

Figures (6)

• Figure 1: a) Schematic diagram of the device for state-preparation of controlled-SWAP gate $U_{\text{CSWAP}}^{2\otimes2\otimes2}$. VBS$_{i}$ ($i=1,2$) represents variable beam splitter, their rules can be given by Equation (\ref{['eq4']}). b) Schematic diagram of the controlled-SWAP gate $U_{\text{CSWAP}}^{2\otimes2\otimes2}$. The left beam displacer (BD) directs horizontally polarized photons to the lower arm and vertically polarized photons to the upper arm, the right beam displacer (BD) recombines photons from two paths with different polarization states. BS$_i$ ($i=1,2,3,4$) denotes the 50:50 beam splitter. P$_\pi$, phase shifter, completes $|H\rangle\rightarrow -|H\rangle$ and $|V\rangle\rightarrow -|V\rangle$ .
• Figure 2: a) Schematic diagram of the device for state-preparation of controlled-SWAP gate $U_{\text{CSWAP}}^{2\otimes3\otimes3}$. b) Schematic diagram of controlled-SWAP gate $U_{\text{CSWAP}}^{2\otimes3\otimes3}$.
• Figure 3: a) Schematic diagram of the device for state-preparation of controlled-SWAP gate $U_{\text{CSWAP}}^{2\otimes d\otimes d}$. b) Schematic diagram of controlled-SWAP gate $U_{\text{CSWAP}}^{2\otimes d\otimes d}$.
• Figure 4: a) The average fidelity $\bar{F}$ of the controlled-SWAP gate $U_{\text{CSWAP}}^{2\otimes 2\otimes 2}$ with $r\in[0,1\times10^{-3}]$ and $\theta\in[0,5\times10^{-3}]$ under realistic conditions $\Delta\phi=\pi/36$ and $\epsilon=0.02$. b) The average fidelity $\bar{F}$ of the controlled-SWAP gate $U_{\text{CSWAP}}^{2\otimes 2\otimes 2}$ with $\epsilon\in[0,0.02]$ and $\Delta\phi\in[0,\pi/36$] under realistic conditions $r=1\times10^{-3}$ and $\theta=5\times10^{-3}$ rad.
• Figure 5: a) Fidelities of the controlled-SWAP gate $U_{\text{CSWAP}}^{2\otimes 2\otimes 2}$ for the input state $|000\rangle$ as a function of the extinction ratio $r$ with $\theta=5 \times 10 ^{-3}$ rad. b) Fidelities of he controlled-SWAP gate $U_{\text{CSWAP}}^{2\otimes 2\otimes 2}$ for the input state $|000\rangle$ as a function of the deviation of mirror mounts $\theta$ with $r=1\times10 ^{-3}$. The solid red curves are for our $\bar{F}_{0ij}$, and the dashed blue curves are for $\mathcal{\bar{F}}_{0ij}$,Fredkin-Meng where $i,j\in\left\lbrace 0,1\right\rbrace$.