Maximal Success Probabilities of Linear-Optical Quantum Gates

TL;DR

This paper uses numerical optimization to determine the maximum success probabilities of linear-optical quantum gates like CNOT and Toffoli, revealing fundamental limits and efficiency improvements in their implementation.

Contribution

It provides the first numerical evidence for the maximum success rate of the 2-qubit CNOT gate with unentangled ancilla and demonstrates a more resource-efficient implementation of the Toffoli gate.

Findings

Maximum success rate for 2-qubit CNOT is 2/27 with two unentangled ancilla.

Toffoli gate achieved with only three unentangled ancilla photons, success rate 0.00340.

Additional ancilla resources do not increase success probability for CNOT.

Abstract

Numerical optimization is used to design linear-optical devices that implement a desired quantum gate with perfect fidelity, while maximizing the success rate. For the 2-qubit CS (or CNOT) gate, we provide numerical evidence that the maximum success rate is $S=2/27$ using two unentangled ancilla resources; interestingly, additional ancilla resources do not increase the success rate. For the 3-qubit Toffoli gate, we show that perfect fidelity is obtained with only three unentangled ancilla photons -- less than in any existing scheme -- with a maximum $S=0.00340$. This compares well with $S=(2/27)^2/2 \approx 0.00274$, obtainable by combining two CNOT gates and a passive quantum filter [PRA 68, 064303 (2003)]. The general optimization approach can easily be applied to other areas of interest, such as quantum error correction, cryptography, and metrology [arXiv:0807.4906, PRL 99 070801 (2007)].