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Physical Review Applied

Quantum gate teleportation with the superposition of causal order

Wen‐Qiang Liu, Hai‐Rui Wei

Published Jan 28, 2025• 23 citations
Publisher

Abstract

Quantum gate teleportation provides an approach to introduce nonlinear interactions between long-distance qubits, facilitating distributed quantum computation. We present two alternative protocols for deterministically teleporting controlled-not (cnot) and controlled-phase-flip (cpf) gates, using minimal resources---one entangled qubit (ebit) and two classical bits (cbits). Our protocols eliminate the need for local cnot gates at each party, which are typically required in the standard quantum circuit model [Eisert et al. Phys. Rev. A 62, 052317 (2000)]. Instead, by using a superposition of single-qubit gate orders, our approach makes quantum gate teleportation more experimentally feasible and flexible. Remarkably, the teleportation of the cpf gate and other two-qubit controlled unitary gates can be achieved by adjusting inherent single-qubit operations, reducing the need for single-qubit gates in cnot-based constructions. Additionally, we develop two optical architectures for the teleportation of these gates. The polarization-based and transverse spatial-based setups show that this approach is more feasible and practical, and pave a way for a distributed quantum computing network.

DOI
10.1103/physrevapplied.23.014064

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Quantum gate teleportation provides an approach to introduce nonlinear interactions between long-distance qubits, facilitating distributed quantum computation. We present two alternative protocols for deterministically teleporting controlled-not (cnot) and controlled-phase-flip (cpf) gates, using minimal resources---one entangled qubit (ebit) and two classical bits (cbits). Our protocols eliminate the need for local cnot gates at each party, which are typically required in the standard quantum circuit model [Eisert et al. Phys. Rev. A 62, 052317 (2000)]. Instead, by using a superposition of single-qubit gate orders, our approach makes quantum gate teleportation more experimentally feasible and flexible. Remarkably, the teleportation of the cpf gate and other two-qubit controlled unitary gates can be achieved by adjusting inherent single-qubit operations, reducing the need for single-qubit gates in cnot-based constructions. Additionally, we develop two optical architectures for the teleportation of these gates. The polarization-based and transverse spatial-based setups show that this approach is more feasible and practical, and pave a way for a distributed quantum computing network.
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.
Quantum gate teleportation enables distant parties to implement nonlocal quantum operations without physically transferring the participating qubits, making it a promising primitive for distributed quantum computing. We introduce a general framework for deterministic quantum gate teleportation based on path superposition, in which the target nonlocal operation is specified through the phase of a preshared maximally entangled resource and a suitable family of path-dependent local unitary operators. The framework establishes general design conditions that guarantee deterministic teleportation after measurement of the control qubits and the application of local correction operations. As representative realizations, we construct teleportation protocols for controlled-NOT (CNOT) and controlled-Z (CZ) gates, demonstrating that different nonlocal operations can be implemented within the same protocol architecture through appropriate choices of the design parameters. We further outline a proof-of-concept photonic realization based on spatial-path and polarization degrees of freedom. The proposed framework identifies path superposition as a versatile resource for quantum gate teleportation and distributed quantum information processing.
Abstract Quantum gate teleportation enables spatially separated parties to implement nonlocal quantum operations using preshared entanglement, local operations, measurement, and classical communication. Here, we present a two-branch entanglement-assisted coherent-control architecture for the deterministic remote implementation of nonlocal two-qubit gates. In this architecture, the two parties coherently select between two pairs of local unitary operations acting on their respective input qubits, a structure that can be naturally encoded through spatial-path superposition. The protocol is formulated at the operator level, allowing deterministic operation to be established independently of the input-state parametrization and making its scope explicit. For two-qubit unitary targets, we show that the deterministic branch conditions restrict the nonlocal domain of the balanced two-branch ansatz: apart from local product operations, any implementable nonlocal unitary must be locally unitarily equivalent to the CNOT/CZ class. We construct explicit CNOT and controlled- Z realizations as two representatives of this unique nonlocal equivalence class. We also introduce a photonic path-encoding and destructive branch-readout scheme in which spatial path encodes the coherent-control degree of freedom and polarization carries the computational qubits, illustrating how the proposed architecture can be physically encoded.
Gate teleportation allows us to implement a nonlocal unitary using local operations, classical communication (LOCC), and a shared entangled state. Known deterministic teleportation protocols consume at least one full ebit and achieve optimal entanglement consumption only for Clifford gates. Here, we present a deterministic LOCC protocol for implementing the two-qubit controlled-phase gate Uφ=diag(1,1,1,eiφ)U_φ=\mathrm{diag}(1,1,1,e^{i φ})Uφ​=diag(1,1,1,eiφ) with φ∈[0,π]φ\in [0,π]φ∈[0,π] whose entanglement consumption is close to optimal for every φφφ. In particular, vanishing rotation angles require vanishing entanglement.
Distant quantum control via quantum gates represents an essential step toward realizing distributed quantum networks. An efficient theoretical protocol for the dual non-local implementation of controlled-not (CNOT) gates between two separated partners is presented in this regard. The suggested protocol requires 1~ebit with local operations and classical communication channels. The efficiency of the teleportation scheme is quantified through an infidelity measure. The numerical results show that the infidelity of performing the CNOT gate between legitimate partners depends on the initial qubit settings. It is also shown that the protocol is performed efficiently if the CNOT control qubit and the auxiliary qubit are prepared in the same direction. Furthermore, we provide a noise analysis for the suggested scheme. We find that by maintaining the noise strengths under the threshold 14\frac{1}{4}41​, one can achieve the dual non-local CNOT gate optimally.
Large-scale quantum computers possess the capacity to effectively tackle practical problems that can be insurmountable for classical computers. The main challenge in building these quantum computers is to realize scalable modules for remote qubits and entanglement. By assembling small, specialized parts into a larger architecture, the modular approach mitigates complexity and uncertainty. Such a distributed architecture requires non-local quantum gate operations between remote qubits. An essential method for implementing such operations, known as quantum gate teleportation, requires only local operations, classical communication, and shared entanglement. Till today, the quantum gate teleportation using a photonic chip has remained elusive. Here, we experimentally demonstrate the quantum teleportation of an on-chip controlled-NOT (CNOT) gate, implemented via a silicon polarization directional coupler. Assisted with the scalable silicon chip platform, high-fidelity local quantum logic gates, linear optical components, post-selected entanglement, and coincidence measurements from photonic qubits, we first measure and characterize our teleported chip-scale CNOT gate with an average truth table fidelity of 93.1 ± 0.3%. Second, for different input polarization states, we obtain an average quantum state fidelity of 87.0 ± 2.2% with our teleported on-chip CNOT gate. Third, we use our non-local CNOT gate for remote entanglement creation of four Bell states, with an average quantum state fidelity of 86.2 ± 0.8%. Fourth, we fully characterize our teleported on-chip CNOT gate with a quantum process fidelity of 83.1 ± 2.0%, and an average non-local CNOT gate fidelity of 86.5 ± 2.2%. Our teleported photonic on-chip quantum logic gate could be extended both to multiple qubits and chip-scale modules towards fault-tolerant and large-scale distributed quantum computation.