Maximal success probabilities of linear-optical quantum gates

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 two-qubit controlled-sign [or controlled NOT (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 three-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 to $S={\left(2/27\right)}^2/2\approx 0.00274$, obtainable by combining two CNOT gates and a passive quantum filter [T. C. Ralph, K. J. Resch, and A. Gilchrist, Phys. Rev. A 75, 022313 (2007)]. The general optimization approach can easily be applied to other areas of interest, such as quantum error correction, cryptography, and metrology [M. M. Wilde and D. B. Uskov, Phys. Rev. A 79, 022305 (2009); G. A. Durkin and J. P. Dowling, Phys. Rev. Lett. 99, 070801 (2007)].

Figure 1

(Color online) A general measurement-assisted transformation (e.g., a quantum logic gate or a linear-optical quantum state generator).

Figure 2

The optimized success probability for the CS gate is shown for two, three, and four ancillas and an arbitrary number of vacuum modes. Each point indicates a complete run starting from a randomly chosen starting matrix $\mathbf{U}$. The success rates are arranged in ascending order so that the horizontal axis may be viewed as a cumulative frequency. The $2/27\approx 0.074$ success rate found by Knill [6] is indicated by a horizontal line.

Figure 3

The distribution of success rates for the Toffoli gate.