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arXiv:0808.1926 · Physical Review A

Maximal success probabilities of linear-optical quantum gates

Dmitry B. Uskov, Lev Kaplan, A. Matthew Smith, Sean D. Huver, Jonathan P. Dowling

Published Apr 20, 2009• 49 citations

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 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 $S=0.003\text{ }40$. This compares well to $S={(2/27)}^{2}/2\ensuremath{\approx}0.002\text{ }74$, 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)].

DOI: 10.1103/physreva.79.042326

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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=2/27

. This compares well to

S={(2/27)}^{2}/2\ensuremath{\approx}0.002\text{ }74

, 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)].

We combine numerical optimization techniques [Uskov et al., Phys. Rev. A 79, 042326 (2009)] with symmetries of the Weyl chamber to obtain optimal implementations of generic linear-optical Knill-Laflamme-Milburn-type two-qubit entangling gates. We find that while any two-qubit controlled-

U

gate, including controlled-NOT (CNOT) and controlled-sign gates, can be implemented using only two ancilla resources with a success probability

S>0.05

, a generic SU(4) operation requires three unentangled ancilla photons, with success

S>0.0063

. Specifically, we obtain a maximal success probability close to 0.0072 for the

B

gate. We show that single-shot implementation of a generic SU(4) gate offers more than an order of magnitude increase in the success probability and a two-fold reduction in overhead ancilla resources compared to standard triple-CNOT and double-

B

gate decompositions.

We use the numerical optimization techniques of Uskov et al. [Phys. Rev. A 81, 012303 (2010)] to investigate the behavior of the success rates for Knill-Laflamme-Milburn-style [Knill et al., Nature (London) 409, 46 (2001)] two- and three-qubit entangling gates. The methods are first demonstrated at perfect fidelity and then extended to imperfect gates. We find that as the perfect fidelity condition is relaxed, the maximum attainable success rates increase in a predictable fashion depending on the size of the system, and we compare that rate of increase for several gates.

We introduce the method of using an annealing genetic algorithm to the numerically complex problem of looking for quantum logic gates which simultaneously have highest fidelity and highest success probability. We first use the linear optical quantum nonlinear sign (NS) gate as an example to illustrate the efficiency of this method. We show that by appropriately choosing the annealing parameters, we can reach the theoretical maximum success probability (1/4 for NS) for each attempt. We then examine the controlled-z (CZ) gate as the first new problem to be solved. We show results that agree with the highest known maximum success probability for a CZ gate (2/27) while maintaining a fidelity of 0.9997. Since the purpose of our algorithm is to optimize a unitary matrix for quantum transformations, it could easily be applied to other areas of interest such as quantum optics and quantum sensors.

Abstract Universal quantum gates lie at the heart of designing quantum computer. Here two compact quantum circuits for implementing post‐selected controlled‐phase‐flip (CPF) gate and Toffoli gate with linear optics are constructed assisted by one and two single photons, respectively. The current existing maximum success probability of 1/4 for linear optical CPF gate is achieved by resorting to an ancillary single photon rather than an entangled photon pair or two single photons. Remarkably, the presented Toffoli gate is accomplished with current maximum success probability of 1/30 and unity fidelity in principle, without using additional entangled photon pairs and the standard decomposition‐based approach. Linear optical implementations of the presented two universal gates are feasible and economical under current technology, and provide a potential application in large‐scale optical quantum computing.

Here we propose an experiment in Linear Optical Quantum Computing (LOQC) using the framework first developed by Knill, Laflamme, and Milburn. This experiment will test the ideas of the authors' previous work on imperfect LOQC gates using number-resolving photon detectors. We suggest a relatively simple physical apparatus capable of producing CZ gates with controllable fidelity less than 1 and success rates higher than the current theoretical maximum (S=2/27) for perfect fidelity. These experimental setups are within the reach of many experimental groups and would provide an interesting experiment in photonic quantum computing.