Practical implementation of arbitrary nonlocal controlled-unitary gate via indefinite causal order

TL;DR

This ICO-based approach enables full programmability of CU gates by adjusting the inherent single-qubit operations, offering advantages over conventional fixed causal-order methods in terms of reduced circuit complexity and improved experimental flexibility.

Abstract

Quantum gate teleportation enables the implementation of nonlocal quantum operations without direct interactions between distant nodes. We propose an efficient protocol for implementing arbitrary controlled-unitary (CU) gates acting on two spatially separated parties via indefinite causal order (ICO). By establishing a maximally entanglement between two remote nodes and coherently superposing orders of single-qubit gates, our protocol circumvents the drawback of complex local two-qubit operations. This ICO-based approach enables full programmability of CU gates by adjusting the inherent single-qubit operations, offering advantages over conventional fixed causal-order methods in terms of reduced circuit complexity and improved experimental flexibility. Furthermore, we develop an optical construction to implement the polarization CU gate using a stable and reciprocal Sagnac interferometer. Our work establishes a practical framework for scalable distributed quantum computation with flexible operations.

Figures (3)

• Figure 1: (a) Schematic diagram for teleporting an arbitrary CU gate acting on two spatially separated qubits, $A$ and $B$. A maximally entangled state $|\varphi\rangle_{ab} = \frac{1}{\sqrt{2}}(|0\rangle_a|0\rangle_b + \texttt{i}|1\rangle_a|1\rangle_b)$ is shared between Alice and Bob. The ancillary qubits $a$ and $b$ coherently control the order of single-qubit gate operations via quantum switches, shown as purple boxes. If qubit $a$ (or $b$) is in the state $|0\rangle_a$ (or $|0\rangle_b$), the gate $U_{A_1}$ (or $U_{B_1}$) is applied before $U_{A_2}$ (or $U_{B_2}$), i.e., the pink solid circuit. If qubit $a$ (or $b$) is in the state $|1\rangle_a$ (or $|1\rangle_b$), the gate $U_{A_2}$ (or $U_{B_2}$) is applied before $U_{A_1}$ (or $U_{B_1}$), i.e., the blue dashed circuit. By measuring the ancillary qubits in appropriate bases and applying corresponding feed-forward operations, the teleportation of the CU gate is completed. The single-qubit gates $U_{A_1}=R_{z}(\frac{\pi}{2})$, $U_{A_2}=X$, $U_{B_1}=R_\textbf{n}(\frac{\pi}{2})$, and $U_{B_2}=\mathbf{n}^\perp\cdot \bm{\sigma}$. The quantum switches at Alice’s and Bob’s nodes are schematically illustrated in panels (b) and (c), where the symbols "$\circ$" and "$\bullet$" denote control qubits in the states $|0\rangle$ and $|1\rangle$, respectively.
• Figure 2: Optical setup for teleporting an arbitrary two-qubit CU gate on the polarization DOF. A polarization-entangled photon state $|\Psi\rangle_{0}=\frac{1}{\sqrt{2}}(|H\rangle_1|H\rangle_2+\texttt{i}|V\rangle_1|V\rangle_2)$ is generated via a type-I spontaneous parametric down-conversion (SPDC) process. The photon pairs then pass through polarization beam splitters (PBSs), half-wave plates (HWPs), and quarter-wave plates (QWPs) to convert the polarization-entangled state into a general path-entangled state $|\Psi\rangle_{3}=\frac{1}{\sqrt{2}}(|0\rangle_1|0\rangle_2+\texttt{i}|1\rangle_1|1\rangle_2)|\psi\rangle_1'|\psi\rangle_2'$, by exploiting the fact that the PBS transmits $H$-polarized photons while reflecting $V$-polarized photons. The photons are subsequently directed into a reciprocal polarization Sagnac interferometer to implement a superposition of the orders of the single-qubit polarization gates $U_{A_1}$, $U_{A_2}$, $U_{B_1}$, and $U_{B_2}$. Finally, photons from different paths are recombined at either a balanced beam splitter (BS) or a variable beam splitter (VBS). Coincidence measurements on the path states $|0\rangle_1|0\rangle_2$, $|1\rangle_1|1\rangle_2$, $|0\rangle_1|1\rangle_2$, and $|1\rangle_1|0\rangle_2$ are performed by using the detector pairs $\{M_1, M_3\}$, $\{M_2, M_4\}$, $\{M_1, M_4\}$, and $\{M_2, M_3\}$, respectively. Here, each $M_i$ comprises a polarization analyzer (PA) capable of reconstructing the polarization state in any basis.
• Figure 3: Relations between the average gate fidelity $\overline{F}$ of various CU gates and the imperfection parameter $\delta$. When $\delta=0$, all gate fidelities are unity. In addition, $\overline{F}_\text{CY}=\overline{F}_\text{CZ}=1$ for any value of $\delta$.

Theorems & Definitions (1)

• proof