Frank Verstraete

We introduce the notion of polynomial-depth duality transformations, which relates two sets of operator algebras through a conjugation by a poly-depth quantum circuit, and make use of this to construct efficient Gibbs samplers for a variety of interesting quantum Hamiltonians as they are poly-depth dual to classical Hamiltonians. This is for example the case for the 2D toric code, which is demonstrated to be poly-depth dual to two decoupled classical Ising spin chains for any system size, and we give evidence that such dualities hold for a wide class of stabilizer Hamiltonians. Additionally, we extend the above notion of duality to Lindbladians in order to show that mixing times and other quantities such as the spectral gap or the modified logarithmic Sobolev inequality are preserved under duality.

Despite recent advances in the lattice representation theory of (generalized) symmetries, many simple quantum spin chains of physical interest are not included in the rigid framework of fusion categories and weak Hopf algebras. We demonstrate that this problem can be overcome by relaxing the requirements on the underlying algebraic structure, and show that general matrix product operator symmetries are described by a pre-bialgebra. As a guiding example, we focus on the anomalous $\mathbb Z_2$ symmetry of the XX model, which manifests the mixed anomaly between its $U(1)$ momentum and winding symmetry. We show how this anomaly is embedded into the non-semisimple corepresentation category, providing a novel mechanism for realizing such anomalous symmetries on the lattice. Additionally, the representation category which describes the renormalization properties is semisimple and semi-monoidal, which provides a new class of mixed state renormalization fixed points. Finally, we show that up to a quantum channel, this anomalous $\mathbb Z_2$ symmetry is equivalent to a more conventional MPO symmetry obtained on the boundary of a double semion model. In this way, our work provides a bridge between well-understood topological defect symmetries and those that arise in more realistic models.

Abstract We consider a two-dimensional quantum memory of qubits on a torus encoding an extended Fibonacci string-net model, and construct error correction strategies when those qubits are subjected to depolarizing noise. In the case of a fixed-rate sampling noise model, we find an error correcting threshold of 4.75% with a clustering decoder. Using the concept of tube algebras, we construct a set of measurements and of quantum gates which map arbitrary qubit errors to the Turaev-Viro subspace. Tensor network techniques then allow to quantitatively study the action of Pauli noise on that subspace. We perform Monte-Carlo simulations of the Fibonacci code, and compare the performance of several decoders. To the best of our knowledge, this is the first time that a threshold has been calculated for a two-dimensional error correcting code in which universal quantum computation can be performed in its code space.