Preparing topological states with finite depth simultaneous commuting gates

TL;DR

This paper introduces efficient protocols using finite depth, simultaneous, long-range commuting gates to prepare various topologically ordered states, including 2D and 3D models, with optimal resource scaling.

Contribution

It develops optimal, scalable circuits for preparing topological states using simultaneous long-range commuting gates, extending to 3D fracton models.

Findings

Protocols efficiently prepare topological states with $O(L^2)$ gates.

Examples include toric code, Kitaev quantum double, and string net models.

Extended to 3D fracton models like Haah's code and X-Cube.

Abstract

We present protocols for preparing two-dimensional abelian and non-abelian topologically ordered states by employing finite depth unitary circuits composed of long-ranged, simultaneous, and mutually commuting two-qubit gates. Our protocols are motivated by recent proposals for circuits in trapped ion systems, which allow each qubit to participate in multiple gates simultaneously. Our circuits are shown to be optimal, in the sense that the number of two-qubit gates and ancilla qubits scales as $O(L^2)$, where $L$ is the linear size of the system. Examples include the ground states of the toric code, certain Kitaev quantum double models, and string net models. Going beyond two dimensions, we extend our scheme to more general Calderbank-Shor-Steane (CSS) codes. As an application, we present protocols for realizing the three-dimensional Haah's code and X-Cube fracton models.