Adaptive constant-depth circuits for manipulating non-abelian anyons

TL;DR

This work shows that for solvable finite groups $G$, Kitaev's quantum double model supports constant-depth adaptive quantum circuits, interleaving a fixed number of local unitaries with mid-circuit measurements and efficient classical processing, to prepare the ground state, create anyon pairs at arbitrary separations, and perform topological charge measurements. A key no-go result proves that non-adaptive constant-depth ribbons cannot implement certain non-abelian ribbon operators, highlighting a fundamental distinction between abelian and non-abelian anyons. The authors provide explicit adaptive, 1D-local circuits for ground-state preparation (via a recursive factorization of $G$ through normal subgroups), for group-multiplication primitives, and for the implementation and measurement of anyonic ribbon operators, including a probabilistic-to-deterministic construction for ribbon operators. These constructions leverage Clifford-based gate teleportation, stabilizer techniques, and Barrington-style depth reductions, enabling scalable, depth-efficient simulations of non-abelian topological order with near-term hardware. The results pave the way for experimental exploration of non-abelian anyons and topological quantum computation using low-depth adaptive circuits and measurement-based feedback.

Abstract

We consider Kitaev's quantum double model based on a finite group $G$ and describe quantum circuits for (a) preparation of the ground state, (b) creation of anyon pairs separated by an arbitrary distance, and (c) non-destructive topological charge measurement. We show that for any solvable group $G$ all above tasks can be realized by constant-depth adaptive circuits with geometrically local unitary gates and mid-circuit measurements. Each gate may be chosen adaptively depending on previous measurement outcomes. Constant-depth circuits are well suited for implementation on a noisy hardware since it may be possible to execute the entire circuit within the qubit coherence time. Thus our results could facilitate an experimental study of exotic phases of matter with a non-abelian particle statistics. We also show that adaptiveness is essential for our circuit construction. Namely, task (b) cannot be realized by non-adaptive constant-depth local circuits for any non-abelian group $G$. This is in a sharp contrast with abelian anyons which can be created and moved over an arbitrary distance by a depth-$1$ circuit composed of generalized Pauli gates.

Figures (18)

• Figure 1: The action of the site-operators $B^h_s$ and $A^g_s$ on computational basis states (denoted $x$). Here $g,h\in G$ and $s$ is a site of the lattice. Both operators are ribbon operators associated with a closed ribbon (shaded) starting and ending at $s$.
• Figure 2: A primary triangle is specified by an edge $e$ and an adjacent plaquette $p$, and a dual triangle by an edge $e'$ and one of its vertices $e_*$.
• Figure 3: The ribbon operator $F_\xi^{h,g}$ for a ribbon $\xi$ with starting site $s_0$ and ending site $s_1$. The figure illustrates the action of the operator on a computational basis state specified by $(x,y)$. Here $\hat{y}_j=y_1\cdots y_j$.
• Figure 4: Illustration of the proof of identity \ref{['eq:deformationidentity']}
• Figure 5: The base (oriented) lattice. Each edge $e$ carries a qudit $\mathsf{C}_e\cong\mathbb{C}^{|G|}$.

Theorems & Definitions (38)

• Corollary 1.1
• Theorem 1.2
• Theorem 3.1
• Lemma 3.2
• proof
• Corollary 3.3
• Lemma 3.4
• proof
• proof : Proof of Theorem \ref{['thm:nogotheorem']}
• Theorem 4.1