Quantum circuits for toric code and X-cube fracton model

TL;DR

A gluing method involving measurements is introduced, enabling the 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.

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.

Figures (26)

• Figure 1: The black solid net on the left represents the lattice $\Gamma$ and the black dashed net on the right represents the dual of $\Gamma$ induced by the gray net.
• Figure 2: Definitions of $A_{v}$ and $B_{p}$ operators in toric code.
• Figure 3: Initially, a qubit in the state $|0\rangle$ is situated at each gray dot. As quantum gates are applied to these qubits, their color changes to black. A circle positioned on a dot signifies the application of a Hadamard gate to the corresponding qubit, while an arrow indicates a CNOT gate, with the arrowhead pointing from the control qubit to the target qubit.
• Figure 4: The procedure on the basics structure is applying Hadamard gate on any qubit at $|0\rangle$ first and CNOT gates to other qubits in any order.
• Figure 5: Boundaries with the same color are identified to represent a torus.