Generalised All-Optical Cat Correction

Abstract

We have generalised an all-optical telecorrection protocol for the higher orders of the cat code, and show that with these higher orders we can achieve target performance at substantially reduced iteration counts at the cost of a higher mean photon-number. We also introduce a probabilistic scheme for correcting deformation of the state, which highlights two interesting abilities of telecorrection: to encode new sets of transformations, and to change the basis of the code. We find that for a target channel fidelity of $99.9\%$ over a channel with $1\text{ dB}$ of loss, a third-order cat code requires $70$ times fewer telecorrection iterations than a first-order one, at a cost of a $3.6$-fold increase in mean photon-number.

Figures (6)

• Figure 1: Fock probabilities for lossless (left) and lossy (right) cat states at different code orders, with cumulative contributions from loss orders beyond the loss capacity of each code ($l>L$) shown in red, taken here to a cutoff in the loss channel of $l=10$. The red therefore corresponds to the magnitude of uncorrectable loss error, where we see that at higher $L$ these errors are much less significant. All states are prepared at amplitudes of $\alpha=3.5$.
• Figure 2: The ratio between the codeword normalisation constants in our choice of basis (a), and the codeword overlap in the basis of Ref. hastrup2022all (b), shown for various code orders. These quantities are related to inherent forms of error in either basis, and are directly related by Eq. (\ref{['eqn:norm_overlap_mismatch']}). At the bottom we show the $L=1$ codewords on a slice of the Bloch sphere in our choice of basis (c) and that of Ref. hastrup2022all or Eq. (\ref{['eqn:Rx_cats']}) (d), where in (d) the codewords $\vert \tilde{0} \rangle$ and $\vert \tilde{1} \rangle$ are shown to be non-orthogonal.
• Figure 3: The cat telecorrection protocol proposed in Ref. hastrup2022all. Here an underbar is used to indicate that the first mode of the ancilla is prepared at a lower amplitude matching that of the input state after loss.
• Figure 4: Maps of the logical $X$ (red) and $Z$ (blue) errors in terms of photon-number measurements, $(n,m)$, on either mode, determined numerically from Eq. (\ref{['eqn:syndrome']}). White indicates that the respective error is not present. $X$ and $Z$ errors may occur simultaneously, and are always correctable. Transmission failure and deformation can be mapped similarly.
• Figure 5: The channel fidelity for different code orders at $1 \text{ dB}$ (left) and $5 \text{ dB}$ (right), shown for increasing numbers of channel iterations. Peak fidelity is indicated with dashed vertical lines, except in cases where the optima are found numerically to be at the edge of the given span of $\alpha$.