Permutation-invariant qudit codes from polynomials

TL;DR

This paper introduces a new algebraic method to construct permutation-invariant quantum codes for qudits, capable of correcting multiple errors, using polynomials with specific root properties.

Contribution

The paper presents a novel polynomial-based algebraic construction of permutation-invariant qudit codes that can correct multiple errors, expanding the known family of such codes.

Findings

Constructed new families of permutation-invariant codes for qudits.

Codes can correct up to t errors with at least (2t+1)^2(d-1) qudits.

Existence of uncountably many such codes when the number of qudits exceeds a threshold.

Abstract

A permutation-invariant quantum code on $N$ qudits is any subspace stabilized by the matrix representation of the symmetric group $S_N$ as permutation matrices that permute the underlying $N$ subsystems. When each subsystem is a complex Euclidean space of dimension $q \ge 2$, any permutation-invariant code is a subspace of the symmetric subspace of $(\mathbb C^q)^N.$ We give an algebraic construction of new families of of $d$-dimensional permutation-invariant codes on at least $(2t+1)^2(d-1)$ qudits that can also correct $t$ errors for $d \ge 2$. The construction of our codes relies on a real polynomial with multiple roots at the roots of unity, and a sequence of $q-1$ real polynomials that satisfy some combinatorial constraints. When $N > (2t+1)^2(d-1)$, we prove constructively that an uncountable number of such codes exist.