Quantum circuits for toric code and X-cube fracton model

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 ⌈2L+2+log2(d)+L2d⌉ 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.