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Quantum error correcting codes from quiver moduli: Grassmannians, cyclic monodromy, and argyle collision divisors
arXiv:2611.05526v1 Announce Type: new Abstract: Building on aspects of a recent diagrammatic field-theory programme for quantum error correction put forward by the first-named author (arXiv:2607.08911), we construct exact finite-dimensional codes from twisted cyclic and argyle quiver moduli on the projective line. A regular fibre of the node-labelled cyclic Hitchin map records the ways to distribute simple spectral zeroes among the arrows with prescribed degrees. The Hilbert space spanned by these allocations is a Young permutation module. Root monodromy acts on its Specht factors and leaves their multiplicity spaces noiseless. Resetting the Specht factor therefore recovers the encoded state for every channel in the monodromy noise model. We prove that the largest Kostka multiplicity is the optimal dimension of a subspace code correcting the full monodromy algebra, even among codes spanning several isotypic blocks. Quantizing an argyle master space recovers the cyclic sheet space as a weight sector of a tensor power, with the identification preserving monodromy. For the balanced three-cycle with $t$ zeroes per arrow, the multiplicity of a partition $(a,b,c)$ of $3t$ is $1+\min(a-b,b-c)$. The optimal logical dimension is thus $t+1$, attained uniquely by the block indexed by $(2t,t)$. At $t=1$ we give explicit codewords for a qubit on six sheets. The commuting right action generates its full logical algebra, yielding errors outside the monodromy model that leave the measured footprint statistics unchanged yet require different recoveries. We also compute collision-parity ranks, give a higher-rank Grassmannian example, and derive a conditional criterion for recovery across splitting types.
Abstract:Building on aspects of a recent diagrammatic field-theory programme for quantum error correction put forward by the first-named author (arXiv:2607.08911), we construct exact finite-dimensional codes from twisted cyclic and argyle quiver moduli on the projective line. A regular fibre of the node-labelled cyclic Hitchin map records the ways to distribute simple spectral zeroes among the arrows with prescribed degrees. The Hilbert space spanned by these allocations is a Young permutation module. Root monodromy acts on its Specht factors and leaves their multiplicity spaces noiseless. Resetting the Specht factor therefore recovers the encoded state for every channel in the monodromy noise model. We prove that the largest Kostka multiplicity is the optimal dimension of a subspace code correcting the full monodromy algebra, even among codes spanning several isotypic blocks. Quantizing an argyle master space recovers the cyclic sheet space as a weight sector of a tensor power, with the identification preserving monodromy. For the balanced three-cycle with $t$ zeroes per arrow, the multiplicity of a partition $(a,b,c)$ of $3t$ is $1+\min(a-b,b-c)$. The optimal logical dimension is thus $t+1$, attained uniquely by the block indexed by $(2t,t)$. At $t=1$ we give explicit codewords for a qubit on six sheets. The commuting right action generates its full logical algebra, yielding errors outside the monodromy model that leave the measured footprint statistics unchanged yet require different recoveries. We also compute collision-parity ranks, give a higher-rank Grassmannian example, and derive a conditional criterion for recovery across splitting types.
Quantum Physics (quant-ph); High Energy Physics - Theory (hep-th); Algebraic Geometry (math.AG); Combinatorics (math.CO); Representation Theory (math.RT)
arXiv:2611.05526 [quant-ph]
(or arXiv:2611.05526v1 [quant-ph] for this version)
https://doi.org/10.48550/arXiv.2611.05526 arXiv-issued DOI via DataCite (pending registration)
Full text under the publisher's open licence · 275 words