Quantum circuits for toric code and X-cube fracton model

TL;DR

This paper presents an efficient Clifford gate-based quantum circuit for simulating ground states of topological models like the toric code and X-cube fracton model, with extensions to complex 3D phases.

Contribution

It introduces a geometric reformulation of ground state preparation, enabling systematic circuits for 2D and 3D topological models, including arbitrary planar lattices.

Findings

Efficient circuits for 2D toric code ground state in O(L) steps.

Extension to 3D topological phases like the 3D toric code and X-cube model.

A measurement-based gluing method for arbitrary planar lattices.

Abstract

We propose a systematic and efficient quantum circuit composed solely of Clifford gates for simulating the ground state of the surface code model. This approach yields the ground state of the toric code in $\lceil 2L+2+log_{2}(d)+\frac{L}{2d} \rceil$ time steps, where $L$ refers to the system size and $d$ represents the maximum distance to constrain the application of the CNOT gates. Our algorithm reformulates the problem into a purely geometric one, facilitating its extension to attain the ground state of certain 3D topological phases, such as the 3D toric model in $3L+8$ steps and the X-cube fracton model in $12L+11$ steps. Furthermore, we introduce a gluing method involving measurements, enabling our technique to attain the ground state of the 2D toric code on an arbitrary planar lattice and paving the way to more intricate 3D topological phases.