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

Qudits naturally correspond to multilevel quantum systems, which have the potential to enable more efficient 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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mi>SU</a:mi><a:mo>(</a:mo><a:mi>d</a:mi><a:mo>)</a:mo></a:mrow></a:math> for any odd integer <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mrow><b:mi>d</b:mi><b:mo>≥</b:mo><b:mn>3</b:mn></b:mrow></b:math>. 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 <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mrow><c:mi mathvariant="fraktur">su</c:mi><c:mo>(</c:mo><c:mi>d</c:mi><c:mo>)</c:mo></c:mrow></c:math> 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 <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:msup><e:mrow><e:mo>(</e:mo><e:mi>d</e:mi><e:mo>−</e:mo><e:mn>1</e:mn><e:mo>)</e:mo></e:mrow><e:mn>2</e:mn></e:msup></e:math> physical qudits.