A family of permutationally invariant quantum codes

TL;DR

This paper introduces a new family of permutationally invariant quantum codes capable of correcting multiple Pauli errors, deletions, and spontaneous decay errors, with some codes being shorter and including a new optimal single-deletion code.

Contribution

It presents a generalized construction of permutationally invariant quantum codes that correct multiple error types and extends error correction conditions to any number of Pauli errors.

Findings

Codes correct multiple Pauli errors, deletions, and decay errors.

Some codes are shorter than previous best codes.

Includes a new optimal single-deletion-correcting code.

Abstract

We construct a new family of permutationally invariant codes that correct $t$ Pauli errors for any $t\ge 1$. We also show that codes in the new family correct quantum deletion errors as well as spontaneous decay errors. Our construction contains some of the previously known permutationally invariant quantum codes as particular cases, which also admit transversal gates. In many cases, the codes in the new family are shorter than the best previously known explicit permutationally invariant codes for Pauli errors and deletions. Furthermore, our new code family includes a new $((4,2,2))$ optimal single-deletion-correcting code. As a separate result, we generalize the conditions for permutationally invariant codes to correct $t$ Pauli errors from the previously known results for $t=1$ to any number of errors. For small $t$, these conditions can be used to construct new examples of codes by computer.