Electromagnetic duality in the toric code has order two at the level of anyon types and acts as a Hadamard-type logical transformation on the encoded quantum information. Here, we ask whether it can likewise be realized microscopically as an order-two Clifford operation. For the $\mathbb{Z}_2$ toric code, we prove that any locality-preserving Clifford realization of electric-magnetic exchange cannot have order two. Our proof is fully general and requires neither translation symmetry with restricted families of system sizes nor a fixed pairing between vertex and plaquette stabilizers. Geometrically, the proof locally emulates the unavoidable crossing between electric and magnetic strings, similar to a cross-cap in a non-orientable manifold. Extending the problem to the $\mathbb{Z}_N$ toric code, we find that an order-two Clifford realization exists for odd $N$, whereas it is impossible for even $N$. Our results show that the group law of an emergent electromagnetic duality need not lift faithfully to its microscopic Clifford realization.
Electromagnetic duality in the toric code has order two at the level of anyon types and acts as a Hadamard-type logical transformation on the encoded quantum information. Here, we ask whether it can likewise be realized microscopically as an order-two Clifford operation. For the Z2 toric code, we prove that any locality-preserving Clifford realization of electric-magnetic exchange cannot have order two. Our proof is fully general and requires neither translation symmetry with restricted families of system sizes nor a fixed pairing between vertex and plaquette stabilizers. Geometrically, the proof locally emulates the unavoidable crossing between electric and magnetic strings, similar to a cross-cap in a non-orientable manifold. Extending the problem to the ZN toric code, we find that an order-two Clifford realization exists for odd N, whereas it is impossible for even N. Our results show that the group law of an emergent electromagnetic duality need not lift faithfully to its microscopic Clifford realization.
Exact Z2 electromagnetic duality of Z2 toric code is non-Clifford
The 2D Z2 toric code admits a global symmetry exchanging electric and magnetic quasiparticles, known as electromagnetic duality. Known realizations include lattice translation symmetry, an exact Z4 symmetry generated by a Clifford circuit, and an exact Z2 symmetry generated by a non-Clifford circuit. We show that a Clifford electromagnetic duality cannot realize an exact internal Z2 symmetry. This is proved rigorously for symmetries with coarse translation invariance by l lattice units for generic odd l. Therefore an exact internal Z2 electromagnetic duality must be non-Clifford, whereas generic internal Clifford realization necessarily has Z2m algebra with m≥2. Our result suggests an unexpected connection between the algebra of exact electromagnetic duality and Clifford hierarchy of circuits.
Self-dual S3 gauge theory in 2+1d: lattice model and topological phase transitions
Electric-magnetic self-duality of the Z2 gauge theory, realized microscopically as a half-lattice-translation exchanging electric charge and magnetic flux, has been an influential example of a duality symmetry with an exact lattice realization. We construct the first non-Abelian generalization of this construction: a lattice model of the S3 quantum double D(S3) on a tensor product Hilbert space in which the Z2em anyon-permutation symmetry, exchanging the non-Abelian chargeon C and fluxon F, is realized via lattice translation. Consequently we find that the zigzag boundary termination of the model realizes, without fine-tuning, a gapless critical edge state described by the tetracritical Ising CFT. The bulk admits three independent Z2em-preserving bosonic perturbations, driving D(S3) into distinct gapped phases. We analyze these transitions by three independent methods: category-theoretic anyon condensation, microscopic lattice Hamiltonians, and Chern-Simons-Higgs theory, which all agree, yielding a unified picture. These examples motivate a minimal-condensation principle: proliferating a bosonic anyon generically drives condensation of a minimal condensable algebra containing it, with symmetry-related condensates appearing as degenerate vacua that spontaneously break the anyon-permutation symmetry. Our model construction extends to an infinite family of self-dual dihedral quantum doubles D(D2n). Notably, each model is sign-problem-free, opening the door to large-scale numerical exploration of the phases of non-Abelian Chern-Simons-Higgs theories.
Parameterized Families of Toric Code Phase: em-duality family and higher-order anyon pumping
Within the toric-code phase, we study parameterized families of topologically ordered states. We construct 1- and 2-parameter families of local Hamiltonians and confirm their non-triviality via topological pumping. For the 1-parameter family, we show that the em-exchange defect is pumped into the bond Hilbert space of a tensor-network representation. For the 2-parameter case, we construct a ``pump of a pump'' that transports an S1-family of a system in one lower spatial dimension. Using similar methods, we also present a 1-parameter family with a higher-order anyon pump that produces corner-localized anyon modes. These constructions provide explicit lattice realizations and concrete diagnostics of family-level topology. We use recently developed boundary algebra methods to study the non-triviality of these families.
We consider the Z 2 toric code, surface code, and Floquet code defined on a nonorientable surface, which can be considered as families of codes extending Shor’s nine-qubit code. We investigate the fault-tolerant logical gates of the Z 2 toric code in this setup, which corresponds to e ↔ m exchanging symmetry of the underlying Z 2 gauge theory. We find that nonorientable geometry provides a new way for the emergent symmetry to act on the code space, and discover the new realization of the fault-tolerant Hadamard gate of the two-dimensional surface code with a single cross cap connecting the vertices nonlocally along a slit, dubbed a nonorientable surface code. This Hadamard gate can be realized by a constant-depth local unitary circuit modulo nonlocality caused by a cross cap. Via folding, the nonorientable surface code can be turned into a bilayer local quantum code, where the folded cross cap is equivalent to a bilayer twist terminated on a gapped boundary and the logical Hadamard only contains local gates with intralayer couplings when being away from the cross cap, as opposed to the interlayer couplings on each site needed in the case of the folded surface code. We further obtain the complete logical Clifford gate set for a stack of nonorientable surface codes and similarly for codes defined on Klein-bottle geometries. We then construct the honeycomb Floquet code in the presence of a single cross cap, and find that the period of the sequential Pauli measurements acts as a H Z logical gate on the single logical qubit, where the cross cap enriches the dynamics compared with the orientable case. We find that the dynamics of the honeycomb Floquet code is precisely described by a condensation operator of the Z 2 gauge theory, and illustrate the exotic dynamics of our code in terms of a condensation operator supported at a nonorientable surface. Published by the American Physical Society 2024
We present an entirely 2D constant-depth realization of topologically protected phase gates at any level of the Clifford hierarchy, and beyond, using non-Abelian surface codes. Our construction encodes a logical qubit in the quantum double D(G) of a non-Abelian group G on a triangular spatial patch. The logical gate is implemented by a constant-depth circuit constructed from stacking on the spatial region a symmetry-protected topological (SPT) phase specified by a group 2-cocycle and boundary counter-terms. The Bravyi-König theorem limits the unitary gates implementable by constant-depth quantum circuits on Pauli stabilizer codes in D dimensions to the D-th level of the Clifford hierarchy. We bypass this limitation, by constructing constant-depth unitary gates at arbitrary levels of the Clifford hierarchy purely in 2D, without sacrificing locality or fault tolerance, at the cost of using the quantum double of a non-Abelian group G. Specifically, for G=D4N, the dihedral group of order 8N, we realize the phase gate T1/N=diag(1,eiπ/(4N)) in the logical Z basis. In this context, we propose a non-abelian stabilizer group formalism, which we work out for dihedral groups. For 8N=2n, the logical gate lies at the n-th level of the Clifford hierarchy and, importantly, has a qubit-only realization: we show that it can be constructed in terms of Clifford-hierarchy stabilizers for a code with n physical qubits on each edge of the lattice. We also discuss code-switching to the double surface-code D(Z2×Z2), to complete a universal gate-set in this setup.
Recent work has shown that quantum double models can be used to circumvent the Bravyi-Koenig bound. In particular, this is achieved through code-switching protocols in which non-Pauli stabilizer phases appear as intermediate codes. We introduce the XCd(k) stabilizer formalism, which generalizes the Pauli stabilizer formalism by combining Pauli-X operators with diagonal operators drawn from a fixed level k of the Clifford hierarchy on prime-dimensional qudits. This yields a natural class of beyond-Pauli stabilizer models, including many twisted quantum double phases. We classify a large class of these XCd(k) stabilizer states, showing that they are uniform superpositions over affine-linear subspaces with diagonal phases. In particular, when the stabilizers of a state require operators from level k of the Clifford hierarchy, we show that the diagonal gates required to produce the phases in the wavefunction sit one level higher. We further show that every 2D twisted quantum double Dω(Zpn) with ω a product of type-III cocycles and p an odd prime admits a realization using only Clifford (k=2) stabilizers, and we conjecture that phases where ω contains a type-I or type-II cocycle require level-p Clifford-hierarchy stabilizers for p-dimensional qudits. Finally, we demonstrate that a pair of qudit toric codes may be deterministically prepared in a logical magic state by measuring XCd(k) stabilizers. Together, these results provide a concrete stabilizer framework for topological codes beyond Pauli stabilizer models and their logical gates.