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Sequential Circuit as Generalized Symmetry on Lattice
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Generalized symmetry extends the usual notion of symmetry to ones that are of higher-form, acting on subsystems, non-invertible, etc. The concept was originally defined in the field theory context using the idea of topological defects. On the lattice, an immediate consequence is that a symmetry twist is moved across the system by a sequential quantum circuit. In this paper, we ask how to obtain the full, potentially non-invertible symmetry action from the unitary sequential circuit and how the connection to sequential circuit constrains the properties of the generalized symmetries. We find that for symmetries that contain the trivial symmetry operator as a fusion outcome, which we call annihilable symmetries, the sequential circuit fully determines the symmetry action and puts various constraints on their fusion. In contrast, for unannihilable symmetries, like that whose corresponding twist is the Cheshire string, a further 1D sequential circuit is needed for the full description. Matrix product operator and tensor network operator representations play an important role in our discussion.
Forward citations
Cited by 4 Pith papers
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Symmetry Spans and Enforced Gaplessness
Symmetry spans enforce gaplessness when a symmetry E embedded into two larger symmetries C and D has no compatible gapped phase that restricts from both.
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Invariants of Sequential Circuits and Generalized Non-Abelian Statistics
Sequential circuit invariants detect non-invertible symmetry anomalies and characterize non-Abelian fermionic loops plus a new mixed topological order in (3+1)D.
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Non-Invertible Symmetries on Tensor-Product Hilbert Spaces and Quantum Cellular Automata
Any weakly integral fusion category admits a QCA-refined realization on tensor-product Hilbert spaces with QCA and symmetry indices fixed by the categorical data under defect assumptions.
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Lattice Gauging Interfaces and Noninvertible Defects in Higher Dimensions
Explicit lattice constructions of gauging interfaces and condensation defects are given for higher-dimensional systems with higher-form symmetries, using movement operators to manage constrained Hilbert spaces.
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