Qudit-based quantum error-correcting codes from irreducible representations of SU(d)

TL;DR

This paper introduces a general method for constructing quantum error-correcting codes for qudits using irreducible representations of SU(d), enabling reliable quantum information processing with multi-level systems.

Contribution

The authors develop a universal procedure for qudit error correction codes based on SU(d) representations, applicable for any odd integer d ≥ 3, expanding quantum error correction techniques.

Findings

Constructed an infinite class of error-correcting codes encoding one logical qudit into (d-1)^2 physical qudits.

Utilized Weyl character formula and branching rules to identify valid code spaces.

Exploited permutation invariance and Heisenberg-Weyl symmetry to simplify code construction.

Abstract

Qudits naturally correspond to multi-level quantum systems, which offer an efficient route towards quantum information processing, but their reliability is contingent upon quantum error correction capabilities. In this paper, we present a general procedure for constructing error-correcting qudit codes through the irreducible representations of $\mathrm{SU}(d)$ for any odd integer $d \geq 3.$ Using the Weyl character formula and inner product of characters, we deduce the relevant branching rules, through which we identify the physical Hilbert spaces that contain valid code spaces. We then discuss how two forms of permutation invariance and the Heisenberg-Weyl symmetry of $\mathfrak{su}(d)$ can be exploited to simplify the construction of error-correcting codes. Finally, we use our procedure to construct an infinite class of error-correcting codes encoding a logical qudit into $(d-1)^2$ physical qudits.