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Physical Review A

Optimal success probability of a tunable linear-optical controlled-phase gate

Karel Lemr, Antonín Černoch

Published Sep 5, 2012• 3 citations
Publisher

Abstract

In this Brief Report we provide the physical interpretation of the success probability of an optimal tunable linear-optical controlled-phase gate as a function of the phase shift imposed by the gate. We analyze in detail the theoretical prediction of this success probability as identified by Kieling et al. [New J. Phys. 12, 013003 (2010)] and provide discussion revealing the reasons for its nonmonotonous behavior. Moreover, we show that the success probability can be increased if several requirements imposed on the gate can be lifted.

DOI
10.1103/physreva.86.034304

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In this Brief Report we provide the physical interpretation of the success probability of an optimal tunable linear-optical controlled-phase gate as a function of the phase shift imposed by the gate. We analyze in detail the theoretical prediction of this success probability as identified by Kieling et al. [New J. Phys. 12, 013003 (2010)] and provide discussion revealing the reasons for its nonmonotonous behavior. Moreover, we show that the success probability can be increased if several requirements imposed on the gate can be lifted.
We present a linear-optical scheme for a controlled-phase gate with tunable phase shift programmed by a qubit state. In contrast to all previous tunable controlled-phase gates, the phase shift is not hard-coded into the optical setup, but can be tuned to any value from 0 to π by the state of a so-called program qubit . Our setup is feasible with current level of technology using only linear-optical components. We provide an experimental feasibility study to assess the gate’s implementability. We also discuss options for increasing the success probability up to 1/12; this approaches the success probability of an optimal non-programmable tunable controlled-phase gate.
We show that it is possible to reduce the number of two-qubit gates needed for the construction of an arbitrary controlled-unitary transformation by up to 2 times using a tunable controlled-phase gate. On the platform of linear optics, where two-qubit gates can only be achieved probabilistically, our method significantly reduces the amount of components and increases success probability of a two-qubit gate. The experimental implementation of our technique presented in this Letter for a controlled single-qubit unitary gate demonstrates that only one tunable controlled-phase gate is needed instead of two standard controlled-not gates. Thus, not only do we increase the success probability by about 1 order of magnitude (with the same resources), but also avoid the need for conducting quantum nondemolition measurement otherwise required to join two probabilistic gates. Subsequently, we generalize our method to a higher order, showing that n-times controlled gates can be optimized by replacing blocks of controlled-not gates with tunable controlled-phase gates.
We report on the first experimental realization of optimal linear-optical controlled phase gates for arbitrary phases. The realized scheme is entirely flexible in that the phase shift can be tuned to any given value. All such controlled phase gates are optimal in the sense that they operate at the maximum possible success probabilities that are achievable within the framework of postselected linear-optical implementations with vacuum ancillas. The quantum gate is implemented by using bulk optical elements and polarization encoding of qubit states. We have experimentally explored the remarkable observation that the optimum success probability is not monotone in the phase.
Nonlinear optical quantum gates can be created probabilistically using only single-photon sources, linear optical elements, and photon-number-resolving detectors. These gates are heralded but operate with probabilities much less than 1. There is currently a large gap between the performance of the known circuits and the established upper bounds on their success probabilities. One possibility for increasing the probability of success of such gates is feed-forward, where one attempts to correct certain failure events that occurred in the gate's operation. In this Brief Report we examine the role of feed-forward in improving the success probability. In particular, for the nonlinear sign-shift gate, we find that in a three-mode implementation with a single round of feed-forward the optimal average probability of success is approximately given by psuccess=0.272{p}_{\mathrm{success}}=0.272psuccess​=0.272. This value is only slightly larger than the general optimal success probability without feed-forward, psuccess=0.25{p}_{\mathrm{success}}=0.25psuccess​=0.25.
As primitives for entanglement generation, controlled phase gates have a central role in quantum computing. Especially in ideas realizing instances of quantum computation in linear optical gate arrays, a closer look can be rewarding. In such architectures, all effective nonlinearities are induced by measurements. Hence the probability of success is a crucial parameter of such quantum gates. In this paper, we discuss this question for controlled phase gates that implement an arbitrary phase with one and two control qubits. Within the class of post-selected gates in dual-rail encoding with vacuum ancillas, we identify the optimal success probabilities. We construct networks that allow for implementation using current experimental capabilities in detail. The methods employed here appear specifically useful with the advent of integrated linear optical circuits, providing stable interferometers on monolithic structures.
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)].