Repeat-until-success cubic phase gate for universal continuous-variable quantum computation

Kevin Marshall, Raphael Pooser, George Siopsis, and Christian Weedbrook

Phys. Rev. A 91, 032321 – Published 24 March 2015

Abstract

To achieve universal quantum computation using continuous variables, one needs to jump out of the set of Gaussian operations and have a non-Gaussian element, such as the cubic phase gate. However, such a gate is currently very difficult to implement in practice. Here we introduce an experimentally viable “repeat-until-success” approach to generating the cubic phase gate, which is achieved using sequential photon subtractions and Gaussian operations. We find that our scheme offers benefits in terms of the expected time until success, as well as the fact that we do not require any complex off-line resource state, although we require a primitive quantum memory.

Received 3 December 2014

DOI:https://doi.org/10.1103/PhysRevA.91.032321

Authors & Affiliations

Kevin Marshall¹, Raphael Pooser^2,3, George Siopsis³, and Christian Weedbrook⁴

¹Department of Physics, University of Toronto, Toronto, M5S 1A7, Canada
²Quantum Information Science Group, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA
³Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996-1200, USA
⁴QKD Corp., 60 St. George St., Toronto, M5S 1A7, Canada

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Vol. 91, Iss. 3 — March 2015

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Images

Figure 1
Effect of imperfect detectors where the operator $\hat{A} = U_{N} (γ) - e^{i γ {\hat{x}}^{3}}$ is the difference between the approximation and ideal cubic phase operators. The expectation value, both real (solid) and imaginary (dashed) parts, as well as the standard deviation (dotted) of this operator, in a position eigenstate $| x 〉$ , is plotted for $γ = 0.03$ [11, 21] and $N = 1$ . We consider a detector with an efficiency of $90 %$ , a dark count rate of $100 Hz$ and a timing resolution of $100 ps$ ; such a detector is within the reach of current technology [22, 23, 24]. We see that for small values of $x$ the effect of detector imperfections on our approximate cubic phase gate are small and we still approximate the ideal gate closely.
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Figure 2
Variance $σ_{p}^{2}$ plotted for the ideal (solid) cubic phase gate as well as $U_{N} (γ)$ for $N = 1$ (dashed), 3 (dotted), 5 (dash-dot), and 7 (dash-double dot); the strength of the gate is given by $γ = 0.03$ . The input state is chosen to be a coherent state with constant imaginary part $Im (α) = 0.25$ where the real part varies with the horizontal axis. Additional squeezing, omitted in this plot, can be utilized to improve the strength or quality of the gate; here we are interested only in the differences as one varies $N$ and we see that as $N$ increases we obtain a closer approximation. Deviation from the ideal curve will result in extra variance in the momentum quadrature, compared to the ideal curve, after one squeezes the mode.
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Figure 3
Experimental implementation of a single $U_{l}$ operation in the cubic phase gate decomposition. Left: after application of the two-mode unitary a QND measurement is made to project onto the subspace orthogonal to $| 0 〉 〈 0 |$ . This could consist of an inverse V-STIRAP scheme via strong atomic coupling to a photonic cavity mode (cavity QED) [25], or via cross-phase modulation. However, in the case of cross-phase modulation a phase shift can be registered, but not its magnitude, in order to avoid projection onto a specific number state. Middle: After the QND measurement, the resource state is sent through a weak beam splitter, and an avalanche photodiode registers a click to signal a photon-subtraction measurement. The QND summing gate is then run in reverse, where the photon detection triggers inverse operations on the phase modulators and squeezers compared to the original QND summing gate. If the squeezers in the inverse gate are kept phase locked to the first set in the previous measurement, homodyne detectors are unnecessary here because the APD serves as the conditional measurement and the squeezing angles are known, allowing for deterministic feed-forward control. Right: the QND summing gate consisting of feed-forward phase modulation and off-line squeezing resources.
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Figure 4
Repeat-until-success scheme in which a pulsed resource state remains in a cavity until a photon detection is registered.
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