Invertible subalgebras for chiral semion topological order

arXiv:2610.18953v1 Announce Type: new Abstract: Chiral topological order is incompatible with strictly two-dimensional local commuting-projector Hamiltonians on finite-dimensional tensor-product Hilbert spaces. Invertible subalgebras (ISAs) provide an alternative framework. We construct an ISA supporting chiral semions, together with its anti-semion commutant, on a square lattice with one qubit per edge. We give explicit non-Pauli generators for both algebras and prove that together they generate all local operators on the Hilbert space. Starting from the $\mathbb{Z}_4$ toric code, we condense the boson $e^2m^2$ and identify the semion and anti-semion algebras. We then decompose each $\mathbb{Z}_4$ qudit into two qubits and use a finite-depth circuit to disentangle one qubit per edge fixed by the condensation constraints. Pumping the semion ISA along the $z$ direction defines an explicit three-dimensional non-Clifford quantum cellular automaton (QCA). We show that pumping eight times is equivalent to a translation followed by a finite-depth quantum circuit with ancillary qubits. We also give an explicit finite-depth circuit relating this construction to the semion QCA previously obtained from topological quantum field theory. A similar condensation construction yields a $U(1)_4$ ISA on a square lattice. Together with the semion and earlier odd-prime ISAs, finite stacking realizes every pointed Witt class of bosonic Abelian anyon theories. Finally, we extend both constructions to all spatial dimensions $d=4k{-}2$, with one qubit or $\mathbb{Z}_4$ qudit on each $(2k{-}1)$-cell for the semion or $U(1)_4$ ISA, respectively.

Abstract:Chiral topological order is incompatible with strictly two-dimensional local commuting-projector Hamiltonians on finite-dimensional tensor-product Hilbert spaces. Invertible subalgebras (ISAs) provide an alternative framework. We construct an ISA supporting chiral semions, together with its anti-semion commutant, on a square lattice with one qubit per edge. We give explicit non-Pauli generators for both algebras and prove that together they generate all local operators on the Hilbert space. Starting from the $\mathbb{Z}_4$ toric code, we condense the boson $e^2m^2$ and identify the semion and anti-semion algebras. We then decompose each $\mathbb{Z}_4$ qudit into two qubits and use a finite-depth circuit to disentangle one qubit per edge fixed by the condensation constraints. Pumping the semion ISA along the $z$ direction defines an explicit three-dimensional non-Clifford quantum cellular automaton (QCA). We show that pumping eight times is equivalent to a translation followed by a finite-depth quantum circuit with ancillary qubits. We also give an explicit finite-depth circuit relating this construction to the semion QCA previously obtained from topological quantum field theory. A similar condensation construction yields a $U(1)_4$ ISA on a square lattice. Together with the semion and earlier odd-prime ISAs, finite stacking realizes every pointed Witt class of bosonic Abelian anyon theories. Finally, we extend both constructions to all spatial dimensions $d=4k{-}2$, with one qubit or $\mathbb{Z}_4$ qudit on each $(2k{-}1)$-cell for the semion or $U(1)_4$ ISA, respectively.

Quantum Physics (quant-ph); Strongly Correlated Electrons (cond-mat.str-el); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph); Quantum Algebra (math.QA)

arXiv:2610.18953 [quant-ph]

(or arXiv:2610.18953v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2610.18953 arXiv-issued DOI via DataCite (pending registration)

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Invertible subalgebras for chiral semion topological order

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