Quantum computation with Turaev-Viro codes

TL;DR

The paper presents a comprehensive framework for quantum computation using Turaev-Viro codes derived from unitary spherical categories, focusing on the doubled Fibonacci theory. It constructs the ribbon-graph Hilbert space on surfaces, identifies fusion-based anyonic bases, and shows how mapping-class group actions (Dehn twists and braids) act as protected unitaries via sequences of local F-moves. It then realizes the Hilbert space as the ground space of Levin-Wen stabilizer Hamiltonians, enabling state preparation, measurement, and computation through triangulation changes and ribbon-graph gluings. The framework generalizes to arbitrary modular categories, linking Turaev-Viro invariants to topological quantum computation and suggesting fault-tolerant fault-tolerance schemes through continuous stabilizer measurements and active error correction. Overall, it provides a concrete, scalable path to universality using local deformations of stabilizer codes derived from topological quantum field theories.

Abstract

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. For other categories, such as the Fibonacci category, the construction realizes a non-abelian anyon model over a discrete lattice. By studying braid group representations acting on equivalence classes of colored ribbon graphs embedded in a punctured sphere, we identify the anyons, and give a simple recipe for mapping fusion basis states of the doubled category to ribbon graphs. We explain how suitable initial states can be prepared efficiently, how to implement braids, by successively changing the triangulation using a fixed five-qudit local unitary gate, and how to measure the topological charge. Combined with known universality results for anyonic systems, this provides a large family of schemes for quantum computation based on local deformations of stabilizer codes. These schemes may serve as a starting point for developing fault-tolerance schemes using continuous stabilizer measurements and active error-correction.

Figures (16)

• Figure 1: In (a) and (b) are shown two different examples of basis choices for $\mathcal{H}_\Sigma$ for the case of $\Sigma$ being the sphere with three punctures (indicated with crosses). Dashed lines mark the point-set triangulation of the punctures, while the dual graph, which is trivalent, is indicated with solid lines. Part (c) gives another example of a basis for the sphere with four punctures. In (a-c), it is assumed that there are no marked points on the boundaries of $\Sigma$. Part (d) shows the more general situation, for the sphere with two shaded holes, each of which has a marked point on its boundary.
• Figure 2: The Levin-Wen model with a boundary, realizing $\mathcal{H}_\Sigma^\ell$: qubits sit on the solid lattice edges. In (a) and (b), dashed edges represent "virtual" qubits fixed to $0$. The thick, blue edge represents a virtual qubit fixed to the label $\ell(p)$ of the marked boundary point $p$. In (c) is illustrated the gluing of anyon fusion basis states described in \ref{['sec:anyonfusionbasislattice']}. The $t$ virtual qubits are replaced by qubits, and prepared in states $|k\rangle \otimes |0\rangle^{\otimes (t-1)}$, for $k \in \{0,1\}$, entangled with states on either side. Subsequently, all plaquette operators touching the boundary (shaded plaquettes) are applied.
• Figure 3: Change of triangulation $\mathcal{T}$ by flipping edge $e$ (left), and the effect on the dual lattice $\widehat{\mathcal{T}}$.
• Figure 4: The procedure $cut(\gamma)$ isolates a region $\Sigma_A$ enclosed by a closed curve $\gamma$, by making $F$-moves in sequence along the edges $e_1, e_2, \ldots, e_n$, colored red, counterclockwise along $\gamma$ on the dual graph $\widehat{\mathcal{T}}$. The result is a tadpole-like structure as in \ref{['lem:tadpolegluing']}, with a single edge $e$ and plaquette $q$ between the two regions. The inverse operation is $glue(\gamma)$.
• Figure 5: The successive figures show the intermediate steps of the $cut(\gamma)$ procedure, after applying $F_{e_i}$. The thick, red edges are $e_1, \ldots, e_n$. The overall effect of the procedure is shown in \ref{['fig:tadpolgenerationsummary']}.

Theorems & Definitions (16)

• Lemma 4.1
• Lemma 4.2
• Lemma 4.3: KoeReiVid09
• Lemma A.1
• proof
• Lemma A.2
• proof
• Lemma A.3
• proof
• Lemma B.1