Multimode rotationally symmetric bosonic codes from group-theoretic construction

Abstract

We introduce a new family of multi-mode, rotationally symmetric bosonic codes inspired by the group-theoretic framework of [Phys. Rev. Lett. 133, 240603 (2024)]. Such a construction inverts the traditional paradigm of code design by identifying codes from the requirement that a group of chosen logical gates should be implemented by means of physically simple logical operations, such as linear optics. Leveraging previously unexplored degrees of freedom within this framework, our construction preserves rotational symmetry across multiple modes, enabling linear-optics implementation of the full Pauli group. These codes exhibit improved protection against dephasing noise, outperforming both single-mode analogues and earlier multi-mode constructions. Notably, they allow exact correction of correlated dephasing and support qudit encoding in arbitrary dimensions. We analytically construct and numerically benchmark two-mode binomial codes instances, and demonstrate that, unlike single-mode rotationally symmetric bosonic codes, these exhibit no trade-off between protection against dephasing and photon loss.

Figures (9)

• Figure 1: Application of the logical $X$ gate on a qutrit ($d=3$) encoded in a three mode rotation-symmetric bosonic code of order $N=2$. Each of the lines indicates a single bosonic mode.
• Figure 2: Performance of the simple instances in Eqs.(\ref{['eq:0-N-2N-0logical']}) and (\ref{['eq:0-N-2N-1logical']}) of the two-mode rotation symmetric code constructed in this paper against (a) the dephasing channel with equal rates in both the modes, $\gamma_1 = \gamma_2= \gamma$ , (b) the loss channel with $\kappa_1 = \kappa_2 = \kappa$, and (c) a combination of both of these noisy channels with equal strengths in both the modes and with $\gamma= \kappa$. In all the three cases, we have chosen the optimal values of the beam-splitter angles $(\delta, \phi)$ and we note that under the loss channel, the performance is independent of this choice. We also compare against the $N=2, K=2$ single-mode binomial code, the two-mode CLY code, as well as to the break-even. For dephasing, we also report the performance for $K=2,N=6$ and $K=2,N=8$, corroborating the performance increase at increasing $N$. We note that under the dephasing channel the evolution is restricted to a smaller subspace than the full Hilbert space that is required to describe a general evolution of the codewords. This allows us to go to higher values of $N$ to evaluate the performance of the code under the purely dephasing channel. We note that the optimum value of $N$ under the loss channel displayed in panel (b) is around $N=4$, which is similar to the corresponding case of the single-mode code Grimsmo_2020. Our two-mode codes show comparable performance to the corresponding $N-$order single-mode code under the loss channel, while they enhance the performance quite significantly against the dephasing channel (panel (a)).
• Figure 3: Exact error-correcting circuit for correlated stochastic dephasing noise.
• Figure 4: An encoding and error-correcting procedure for the rotation-symmetric codes on two modes.
• Figure 5: (a) Entanglement infidelity as a function of $\delta$ and $\phi$, showing minima for $\delta=\pi/4,3\pi/4$ for $N=2, K=2$ against dephasing $\gamma_1 t= \gamma_2 t= 10^{-3}$. (b) The variation of entanglement infidelity as a function of $\phi$ is very small (of the order $10^{-3}$). We however still observe that the lowest value is obtained from $\phi= \pi/4$ which is $\pi/(2N)$ for $N=2$ and $m=0$.