Quantum

Twisted Fourier Analysis of Quantum Weight Enumerators

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Briefings

A new unified Fourier framework for quantum weight enumerators of prime-power-dimensional qudits was developed, introducing Weyl power spectrum and Weyl-conjugation autocorrelation concepts. This framework establishes a symplectic Fourier transform pair for Weyl-operator quantities, allowing derivation of quantum MacWilliams identities via the Hamming-weight functional. The approach extends to twisted Fourier transforms, including the Rains shadow transform, and identifies identical-copy Bell sampling as a twisted autocorrelation related to the shadow enumerator. All twisted transforms are proven to be Weyl translations of either the ordinary MacWilliams or Rains shadow transform, implying no additional independent positivity constraints. These refined identities improve linear programming bounds for binary and qutrit quantum codes, including nonexistence results for specific parameters.

Key points

• A unified Fourier framework for quantum weight enumerators of prime-power-dimensional qudits was developed, introducing Weyl power spectrum and Weyl-conjugation autocorrelation concepts.
• All twisted Fourier transforms are proven to be Weyl translations of either the ordinary MacWilliams or the Rains shadow transform, implying no additional independent positivity constraints.

Context

This research provides a refined Fourier framework for quantum weight enumerators, extending to twisted transforms including the Rains shadow transform. The framework uses Weyl-operator quantities as formal vectors to derive quantum MacWilliams identities via the Hamming-weight functional, improving linear programming bounds for binary and qutrit quantum codes.

Limitations

The framework relies on formal vector representations of Weyl-operator quantities and does not provide explicit numerical values or experimental data. The proof of Weyl translation relationships is theoretical and does not include computational verification or empirical validation. Results are limited to prime-power-dimensional qudits and specific code parameters requiring further testing.

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