Topological Preparation of Non-Stabilizer States and Clifford Evolution in $SU(2)_1$ Chern-Simons Theory

TL;DR

The paper develops a topological framework to encode non-stabilizer quantum states in $SU(2)_1$ Chern-Simons theory and to compute their entanglement via Wilson-loop path integrals. It constructs Pauli and Clifford operations from fusion tensors and modular data, linking Clifford evolution to Dehn twists in the mapping class group and providing both algebraic and topological realizations. It explicitly builds topological preparations for $W_n$ and Dicke states and derives their bipartite and multipartite entanglement entropies through replica-like manifold constructions. The work advances a geometric interpretation of quantum resources in TQFT, suggests extensions to higher levels and holographic duals, and points toward potential applications in fault-tolerant topological quantum computing and quantum networks.

Abstract

We develop a topological framework for preparing families of non-stabilizer states, and computing their entanglement entropies, in $SU(2)_1$ Chern-Simons theory. Using the Kac-Moody algebra, we construct Pauli and Clifford operators as path integrals over 3-manifolds with Wilson loop insertions, enabling an explicit topological realization of $W_n$ and Dicke states, as well as their entanglement properties. We further establish a correspondence between Clifford group action and modular transformations generated by Dehn twists on genus-$g$ surfaces, linking the mapping class group to quantum operations. Our results extend existing topological constructions for stabilizer states to include families of non-stabilizer states, improving the geometric interpretation of entanglement and quantum resources in topological quantum field theory.

Figures (12)

• Figure 1: Manifold $\eta$ consisting of a solid torus with two tori removed from its interior Salton:2016qpp. This manifold has three toroid boundaries. The associated fusion tensor $N_{j_1,j_2}^{j_3}$, and thereby $W_n$, can be prepared by path integration over $\eta$.
• Figure 2: Graphical representation of the fusion tensor $N_{j,1}^{j+1}$, which maps two Hilbert spaces into one. The tensor $N_{j,1}^{j+1}$ is prepared by path integration over the manifold $\eta$ in Figure \ref{['Manifold']}. This tensor encodes how maximally-entangled states, e.g. $W_n$, are prepared topologically using path integration.
• Figure 3: Graphical representation for the action $\left(N_{j,1}^{j+1}\right) \cdot\left(N_{j,1}^{j+1}\right)^\dagger$, as in Eq. \ref{['WDiag']}, composed of $N_{j,1}^{j+1}$ with its copy tensor $\left(N_{j,1}^{j+1}\right)^\dagger$ glued together. Both fusion tensors are prepared using path integration along $\eta$, enabling a topological construction for $W_n$.
• Figure 4: The action of $\left(N_{j,1}^{j+1}\right) \cdot \left(N_{j,1}^{j+1}\right)^\dagger$ is topologically prepared as a path integral over the glued manifold $-\eta \cup_{\partial\eta}\eta$. The final manifold is formed by joining two copies of $\eta$, depicted in Figures \ref{['Manifold']} and \ref{['TorusAndGenus']}, along their common boundary with opposite orientation. This construction gives a topological interpretation for the operator used to construct $W_n$.
• Figure 5: Evaluating the sum in Eq. \ref{['SumXj']} corresponds to summing over copies of $-\eta \cup_{\partial_{\eta}} \eta$, each attached at their shared boundaries. Likewise, summing over the results of independent path integrals on each $-\eta \cup_{\partial_{\eta}} \eta$ gives a topological construction for Eq. \ref{['SumXj']}.

Theorems & Definitions (2)

• Conjecture 1
• Corollary 1