Deterministic nonlinear phase gates induced by a single qubit

Abstract

We propose deterministic realizations of nonlinear phase gates by repeating a finite sequence of noncommuting Rabi interactions between a harmonic oscillator and only a single two-level ancillary qubit. We show explicitly that the key nonclassical features of the ideal cubic phase gate and the quartic phase gate are generated in the harmonic oscillator faithfully by our method. We numerically analyzed the performance of our scheme under realistic imperfections of the oscillator and the two-level system. The methodology is extended further to higher-order nonlinear phase gates. This theoretical proposal completes the set of operations required for continuous-variable quantum computation. Quantum linear harmonic oscillators described by an infinite-dimensional Fock space are at the center of continuous-variable (CV) quantum information processing [1]. Physical realizations of quantum harmonic oscillators employ traveling light [2, 3], collective modes of atomic spins [4], motional modes of trapped ions [5], and cavity [6] or circuit quantum electrodynamics [7]. In order to fully utilize these systems for use as a quantum computational platform we need the ability to implement an arbitrary nonlinear operation on them. This principal requirement can be reduced to an access to nonlinearity of the third order-the cubic nonlinearitywhich can be then employed for implementation of nonlinearities of higher-order [8,9]. However, obtaining the cubic nonlinearity is not straightforward. The evolution in cubic potential is inherently unstable even in the over-damped regime [10] and the naturally existing nonlinear media either do not exhibit cubic nonlinearity with sufficient strength or do not provide it with sufficient purity [11]. Alternative approaches rely on crafting the nonlinearity by direct manipulation of individual quanta of the oscillators. This can be done by nonlinear conditional measurements [12-14], deterministic feed-forward aided by nonlinear ancillary states [15-21], or deterministic control over a two-level system coupled to the oscillator [22,23]. Recently proposed deterministic schemes for achieving cubic phase nonlinearity obtained by nonlinear feed-forward control and nonlinear squeezed states [24] can be feasibly realized in a loop architecture [25]. An alternative deterministic scheme based on challenging multi-path atomic interferometry has been proposed as well [21].By far the most effective nonlinear approaches towards constructing the nonlinearity exploit the basic coupling between the harmonic oscillator and an associated two-level system, which is the standard scenario in trapped ions and circuit QED. This efficiency originates from the inherent nonlinearity of the two-level system, which can impart nonclassical behavior to the oscillator simply by 'being there' [26]. However, this nonlinearity is of a specific type caused by the discrete nature of two-level system, and it has been not clear for a long time if it allows for the conversion into totally different CV nonlinearities of the quantum ...

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