The self-duality defects under discrete gauging in a categorical symmetry
C can be classified by inequivalent ways of enriching the bulk SymTFT of
C with
Z2 0-form symmetry. The resulting Symmetry Enriched Topological (SET) orders will be referred to as
SymSETs and are parameterized by choices of
Z2 symmetries, as well as symmetry fractionalization classes and discrete torsions. In this work, we consider self-dualities under gauging
non-invertible 0-form symmetries in
2-dim QFTs and explore their SymSETs. Unlike the simpler case of self-dualities under gauging finite Abelian groups, the SymSETs here generally admit multiple choices of fractionalization classes. We provide a direct construction of the SymSET from a given duality defect using its
relative center. Using the SymSET, we show explicitly that changing fractionalization classes can change fusion rules of the duality defect besides its
F-symbols. We consider three concrete examples: the maximal gauging of
RepH8, the non-maximal gauging of the duality defect
N in
RepH8 and
RepD8 respectively. The latter two cases each result in 6 fusion categories with two types of fusion rules related by changing fractionalization class. In particular, two self-dualities of
RepD8 related by changing the fractionalization class lead to
RepD16 and
RepSD16 respectively. Finally, we study the physical implications such as the spin selection rules and the SPT phases for the aforementioned categories.
Long-range entanglement-the backbone of topologically ordered states-cannot be created in finite time using local unitary circuits, or, equivalently, adiabatic state preparation.Recently, it has come to light that single-site measurements provide a loophole, allowing for finite-time state preparation in certain cases.Here we show how this observation imposes a complexity hierarchy on long-range entangled states based on the minimal number of measurement layers required to create the state, which we call "shots."First, similar to Abelian stabilizer states, we construct single-shot protocols for creating any non-Abelian quantum double of a group with nilpotency class 2 (such as D 4 or Q 8 ).We show that after the measurement, the wave function always collapses into the desired non-Abelian topological order, conditional on recording the measurement outcome.Moreover, the clean quantum double ground state can be deterministically prepared via feedforward-gates that depend on the measurement outcomes.Second, we provide the first constructive proof that a finite number of shots can implement the Kramers-Wannier duality transformation (i.e., the gauging map) for any solvable symmetry group.As a special case, this gives an explicit protocol to prepare twisted quantum doubles for all solvable groups.Third, we argue that certain topological orders, such as nonsolvable quantum doubles or Fibonacci anyons, define nontrivial phases of matter under the equivalence class of finite-depth unitaries and measurement, which cannot be prepared by any finite number of shots.Moreover, we explore the consequences of allowing gates to have exponentially small tails, which enables, for example, the preparation of any Abelian anyon theory, including chiral ones.This hierarchy paints a new picture of the landscape of long-range entangled states, with practical implications for quantum simulators.
We give efficient quantum algorithms for the problems of Hidden Translation and Hidden Subgroup in a large class of non-abelian groups including solvable groups of constant exponent and of constant length derived series. Our algorithms are recursive. For the base case, we solve efficiently Hidden Translation in Z pn, whenever p is a fixed prime. For the induction step, we introduce the problem Orbit Coset generalizing both Hidden Translation and Hidden Subgroup, and prove a powerful self-reducibility result: Orbit Coset in a finite group G is reducible to Orbit Coset in G/N and subgroups of N, for any solvable normal subgroup N of G.