Loss tolerant cross-Kerr enhancement via modulated squeezing

Abstract

We develop squeezing protocols to enhance cross-Kerr interactions. We show that through alternating between squeezing along different quadratures of a single photonic mode, the cross-Kerr interaction strength can be generically amplified. As an application of the squeezing protocols we discuss speeding up the deterministic implementation of controlled phase gates in photonic quantum computing architectures. We develop bounds that characterize how fast and strong single-mode squeezing has to be applied to achieve a desired gate error and show that the protocols can overcome photon losses. Finally, we discuss experimental realizations of the squeezing strategies in optical fibers and nanophotonic waveguides.

Figures (5)

• Figure 1: Schematic representation of the squeezing protocols that are developed in this work. A controlled phase gate $\text{CZ}(\phi)$ (green) can be (deterministically) implemented by (a) repeating the evolution $U_{\Delta t}$ generated by the cross-Kerr interaction (yellow) over a time interval $\Delta t$. Since the cross-Kerr interaction is typically weak, this process is unpractical as the time scale required to impart a full $\pi$ phase shift is beyond the coherence time of the system. Instead, in (b) the cross-Kerr evolution $U_{\Delta t}$ is interspersed with single-mode squeezing transformations $S_{\theta}$ (blue) along different quadratures ($\theta=0,\pi$) to enhance the cross-Kerr interaction. The wider blue squeezing box in the middle of the sequence indicates squeezing with twice the strength $r$ as $S^{\dagger}_{\pi}=S_{0}$, which is used to simplify the squeezing sequence given in \ref{['eq:HAsequence']}. In this way only a few repetitions of $U_{\Delta t}$ and a phase shifter $R$ are required to implement $\text{CZ}(\pi)$. The resulting (Trotter) error is determined by the spacings $\Delta t=\frac{t}{2N}$ of $S_{\theta}$.
• Figure 2: Gate error for implementing a controlled phase gate in the presence of photon losses when the cross-Kerr interaction is amplified through single-mode squeezing applied to both modes. The bare phase shift $\chi t = 1.2\times 10^{-3}$ and loss rate $\eta t = 5.76\times10^{-4}\,\text{dB}$ was chosen according to the parameters reported in optical fibers venkataraman2013phaseamrani2021low. The color maps in (a) show the gate error as a function of the squeezing strength $r$ for phase shifts up to $\phi=\pi$ (left panel) and $\phi=\frac{\pi}{100}$ (right panel) based on the quantum channel in Eq. \ref{['eq:lossmodelL']} that is obtained in the limit $N\to \infty$. In (b) the gate error is plotted for a fixed phase shift $\phi=\pi$ (left) and $\phi=\frac{\pi}{100}$ (right) for different Trotter steps $N=1$ (green circles), $N=5$ (purple diamonds) and $N=10$ (orange triangles). In all cases, the gate error was normalized to $1$ by dividing by its largest value.
• Figure 3: Gate error \ref{['eq: upper bound final result']} shown as dashed lines for implementing controlled phase gate $\text{CZ}(\pi)$ as a function of the number of Trotter steps $N$ for different values of the cross-Kerr phase shift $\chi t= \frac{\pi}{2}, \frac{\pi}{7}, \frac{\pi}{10}$ for different amplification factors $\lambda_1 =2,7,10$. The corresponding upper bounds given by \ref{['eq: f(r) in bounds']} are shown as solid lines. The gate errors were normalized to 1 by dividing by its largest value. For comparison, the upper bounds were also normalized by dividing by the corresponding largest gate error.
• Figure 4: Norm difference \ref{['eq:errorChannels']} for different phase shifts as a function of the number of Trotter steps $N$. We used $\chi t = 1.2 \times 10^{-3}$ for the bare phase shift and $\eta t = 5.76\times10^{-4}\,\text{dB}$ for the loss rate venkataraman2013phaseamrani2021low. For each curve, the norm difference was normalized to $1$ by dividing by its largest value.
• Figure 5: Gate error for implementing $\text{CZ}(\frac{\pi}{100})$ gate in the presence of photon losses when the cross-Kerr interaction is amplified through single-mode squeezing transformations that contain errors in the squeezing angle \ref{['eq:perturbedsqueezing']} for the same parameter setting as in Fig. 2 (b), right panel. The inset plot shows the gate error for a smaller range of the squeezing strength. The errors $\delta_a, \delta_b$ are drawn from a Gaussian distribution with mean $\mu = 0$ and standard deviations $\sigma = 0$ (grey line, see Fig. 2 (b), right panel), $1\times10^{-3}$ (green triangles), $1.25\times10^{-3}$ (blue triangles) and $1.5\times10^{-3}$ (red triangles). The data points show the average over $15$ samples.