Yingkai Ouyang, Institute for Quantum Computing (IQC)
Abstract
Our goal is to design codes that encode a single qubit into $m$ bosonic modes that protect against $t$ amplitude damping errors on the code. Our designed codes are invariant with respect to permutation of the $m$ bosonic modes. We find that the family of permutation invariant bosonic codes still has a structure rich enough to admit codes with low maximum excitation number $N_T$ for small values of $t$. Minimizing $N_T$ for fixed $t$ is important because the fidelity of recovery from amplitude damping noise of strength $\gamma$ is $\mathcal F \approx 1 - \bi{N_T}{t+1}\gamma^{t+1}$. For $t \ge 2$, we give examples of permutation invariant bosonic codes with low total excitation number $N_T$. We also introduce a family of qubit permutation invariant codes with $t \ge 1$ that is a natural generalization of Leung et. al.'s 4-qubit code. We also study bounds of $N_T$ with respect to $t$, and prove that for $t=1$, $N_T = 3$ is optimal even in the qubit case.