Quantum Error Correction beyond <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>S</mml:mi> <mml:mi>U</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:math> :

We develop a framework for constructing quantum error-correcting codes and logical gates for three types of spaces—composite permutation-invariant spaces of many qubits or qudits, composite constant-excitation Fock state spaces of many bosonic modes, and monolithic nuclear state spaces of atoms, ions, and molecules. By identifying all three spaces with discrete simplices and representations of the Lie group <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>S</a:mi> <a:mi>U</a:mi> <a:mo stretchy="false">(</a:mo> <a:mi>q</a:mi> <a:mo stretchy="false">)</a:mo> </a:math> , we prove that many codes and their gates in <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mi>S</e:mi> <e:mi>U</e:mi> <e:mo stretchy="false">(</e:mo> <e:mi>q</e:mi> <e:mo stretchy="false">)</e:mo> </e:math> can be interconverted between the three state spaces. We construct code instances for all three spaces using classical <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:msub> <i:mi>ℓ</i:mi> <i:mn>1</i:mn> </i:msub> </i:math> codes and Tverberg’s theorem, a classic result from convex geometry. We obtain families of quantum codes with distance that scales almost linearly with the code length <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:mi>N</k:mi> </k:math> by constructing <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:msub> <m:mi>ℓ</m:mi> <m:mn>1</m:mn> </m:msub> </m:math> codes based on combinatorial patterns called Sidon sets and utilizing their Tverberg partitions. This compares favorably with the existing designs for all the state spaces. We present explicit constructions of codes with shorter length or lower total spin/excitation than known codes with similar parameters, bosonic codes with exotic Gaussian gates, as well as examples of short codes with distance larger than the known constructions.