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

• URL: http://arxiv.org/abs/2410.02407v1
• Date: Thu, 3 Oct 2024 11:35:57 GMT
• Title: Qudit-based quantum error-correcting codes from irreducible representations of SU(d)
• Authors: Robert Frederik Uy, Dorian A. Gangloff,
• Abstract summary: Qudits naturally correspond to multi-level quantum systems, but their reliability is contingent upon quantum error correction capabilities. We present a general procedure for constructing error-correcting qudit codes through the irreducible representations of $mathrmSU(d)$ for any odd integer $d geq 3.$ We use our procedure to construct an infinite class of error-correcting codes encoding a logical qudit into $(d-1)2$ physical qudits.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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