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arXiv:1105.4211 · Physical Review A

Imperfect linear-optical photonic gates with number-resolving photodetection

A. Matthew Smith, Dmitry B. Uskov, L. H. Ying, Lev Kaplan

Published Sep 29, 2011• 3 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.84.032341

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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 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-UUU gate, including controlled-NOT (CNOT) and controlled-sign gates, can be implemented using only two ancilla resources with a success probability S>0.05S>0.05S>0.05, a generic SU(4) operation requires three unentangled ancilla photons, with success S>0.0063S>0.0063S>0.0063. Specifically, we obtain a maximal success probability close to 0.0072 for the BBB 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-BBB gate decompositions.
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=2/27S=2/27S=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 40S=0.003\text{ }40S=0.003 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)].
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.
We propose an experiment in Linear Optical Quantum Computing (LOQC) in the style rst suggested by Knill, La amme, and Milburn. This experiment is intended to test the theories proposed in the authors' previous work on imperfect LOQC gates using number-resolving photon-detectors. We develop a simple physical apparatus capable of producing controllable delity less than 1 and success rates higher than the current theoretical maximum (S=2/27) for perfect delity. This experimental setup is within the reach of many experimental groups and would provide an interesting experiment in photon-based quantum computing.
In this Letter, the problem of finding optimal success probabilities of linear optics quantum gates is linked to the theory of convex optimization. It is shown that by exploiting this link, upper bounds for the success probability of networks realizing single-mode gates can be derived, which hold in generality for postselected networks of arbitrary size, any number of auxiliary modes, and arbitrary photon numbers. As a corollary, the previously formulated conjecture is proven that the optimal success probability of a nonlinear sign shift without feedforward is 1/4, a gate playing the central role in the scheme of Knill-Laflamme-Milburn for quantum computation. The concept of Lagrange duality is shown to be applicable to provide rigorous proofs for such bounds, although the original problem is a difficult nonconvex problem in infinitely many objective variables. The versatility of this approach is demonstrated.