Approximating Turaev-Viro 3-manifold invariants is universal for quantum computation

Approximating the Turaev-Viro Invariant of Mapping Tori is Complete for One Clean Qubit

The Turaev-Viro invariants are scalar topological invariants of three-dimensional manifolds. Here we show that the problem of estimating the Fibonacci version of the Turaev-Viro invariant of a mapping torus is a complete problem for the one clean qubit complexity class (DQC1). This complements a previous result showing that estimating the Turaev-Viro invariant for arbitrary manifolds presented as Heegaard splittings is a complete problem for the standard quantum computation model (BQP). We also discuss a beautiful analogy between these results and previously known results on the computational …

Quantum computation with Turaev-Viro codes

The Turaev-Viro invariant for a closed 3-manifold is defined as the contraction of a certain tensor network. The tensors correspond to tetrahedra in a triangulation of the manifold, with values determined by a fixed spherical category. For a manifold with boundary, the tensor network has free indices that can be associated to qudits, and its contraction gives the coefficients of a quantum error-correcting code. The code has local stabilizers determined by Levin and Wen. For example, applied to the genus-one handlebody using the Z_2 category, this construction yields the well-known toric code. …

Quantum error correction thresholds for the universal Fibonacci Turaev-Viro code

We consider a two-dimensional quantum memory of qubits on a torus which encode the extended Fibonaccistring-net code, and devise strategies for error correction when those qubits are subjected to depolarizing noise.Building on the concept of tube algebras, we construct a set of measurements and of quantum gates whichmap arbitrary qubit errors to the string-net subspace and allow for the characterization of the resulting errorsyndrome in terms of doubled Fibonacci anyons. Tensor network techniques then allow to quantitatively studythe action of Pauli noise on the string-net subspace. We perform…

Defects and excitations in the Kitaev model

We construct a Kitaev model with defects using twists or 2-cocycles of semi-simple, finite-dimensional Hopf algebras as defect data. This data is derived by applying Tannaka duality to Turaev-Viro topological quantum field theories with defects. From this we also derive additional conditions for moving, fusing and braiding excitations in the Kitaev model with defects. We give a description of excitations in the Kitaev model and show that they satisfy conditions we derive from Turaev-Viro topological quantum field theories with defects. Assigning trivial defect data one obtains transparent defe…

在拓扑量子神经网络和振幅多面体表示中实现通用量子计算

Amplituhedra for generic quantum processes via the TQNN representation of UQC

本文提出拓扑量子神经网络通过Reshetikhin-Turaev和Turaev-Viro模型实现通用量子计算，并展示其与振幅多面体的对应关系，为通用量子过程的几何表示提供新方法。

Braiding and entanglement in spin networks: a combinatorial approach to topological phases

The spin network quantum simulator relies on the su(2) representation ring (or its q-deformed counterpart at q= root of unity) and its basic features naturally include (multipartite) entanglement and braiding. In particular, q-deformed spin network automata are able to perform efficiently approximate calculations of topological invarians of knots and 3-manifolds. The same algebraic background is shared by 2D lattice models supporting topological phases of matter that have recently gained much interest in condensed matter physics. These developments are motivated by the possibility to store qua…