Cat Codes with Optimal Decoherence Suppression for a Lossy Bosonic Channel

We investigate cat codes that can correct multiple excitation losses and identify two types of logical errors: bit-flip errors due to excessive excitation loss and dephasing errors due to quantum backaction from the environment. We show that selected choices of logical subspace and coherent amplitude significantly reduce dephasing errors. The trade-off between the two major errors enables optimized performance of cat codes in terms of minimized decoherence. With high coupling efficiency, we show that one-way quantum repeaters with cat codes feature a boosted secure communication rate per mode when compared to conventional encoding schemes, showcasing the promising potential of quantum information processing with continuous variable quantum codes.

Figure 1

(a) Wigner functions and excitation number distributions of $|C_3^0⟩$ and $|C_3^3⟩$ with $d=3$. (b) Average excitation number $⟨a^{†}a{⟩}_n$ for cat states with $d=3$. (c) Schematic of alternating LBC ($\mathcal{L}$) and QEC recovery ($\mathcal{R}$). (d) Quantum circuits of QEC recovery for cat codes, consisting of the dispersive coupling gate $U_{\mathrm{DC}}$ followed by ${\mathbb{Z}}_d$ measurement of excitation number, conditional rotation gate $U_k$ compensating the lost excitations, and, finally, amplitude restoration $S$.

Figure 2

Effective error rate (numerical) for logical subspace $s=0$, 1, 2, 3 (blue, red, green, and orange solid curves, respectively) and analytical bounds ${\mathrm{\Gamma }}_{\pm }$ from Eq. (11) (black solid), for $d=4$ and $\gamma =0.5\%$. The two types of errors in ${\mathrm{\Gamma }}_{-}$, logical bit-flip error ${\epsilon }_f$ and dephasing error ${{\epsilon }_d|}_{\cos \theta =1}$, are shown. The purple and black dashed lines mark $\overline{\mathrm{\Gamma }}$ and the minimized decoherence ${\mathrm{\Gamma }}_{-}^{\min }$, respectively. The inset shows the dependence of ${\mathrm{\Gamma }}_{-}^{\min }$ and ${\alpha }_{\mathrm{o}}^2$ on $\gamma$. The approximate ${\alpha }_{\mathrm{o}}^2$ from Eq. (12) (solid line) agrees well with numerics.

Figure 3

Optimized performance of cat codes for QRs with $\eta =99.5\%$ and comparison with selected DV schemes. (a) Minimum error accumulation rate ${\tau }_{-}$ (red) and associated optimum arc length $\pi {\alpha }_{\mathrm{opt}}/d$ (blue). (b) Optimized SKRPM over long distances for one-way QRs with cat codes (red solid), quantum polynomial codes [45] (brown dotted), and quantum parity codes [15, 46] (gray dashed). $t_0$ is the gate operation time taken as the same for three schemes.