Efficient encoding of the 2D toric code logical state using local Clifford gates

TL;DR

Encoding the logical state of the $L\times L$ toric code is constrained by Lieb–Robinson bounds, but this work introduces a Heisenberg-picture, diagrammatic approach using only local CX and Hadamard gates to achieve a linear-depth circuit of depth $2L+1$. The core technique unitarily transforms the toric code Hamiltonian to a disjoint sum acting on $X$- and $Z$-type qubits via a unitary $U$ built from commuting CX layers, enabling the logical state to be prepared by applying $U^\dagger$ to a simple product state. A concrete construction (including code and visualization tools) shows the encoding depth and locality improvements over prior methods, with a gate count of $3L^2+2L-5$. This approach offers a scalable, locality-respecting method for toric-code state preparation and suggests avenues to further reduce depth toward linear scaling.

Abstract

An algorithm which encodes the $L\times L$ 2D toric code logical state with a circuit of depth $2L+1$, using only local controlled-NOT($CX$) and Hadamard($H$) gates, is presented.

Figures (4)

• Figure 1: Basic operations after conjugation of $CX$ gates.
• Figure 2: Coordinate system used to represent the 2D toric code. Periodic boundary conditions are implied. Qubits are shown without repetition while the triangles on the top/bottom and left/right should be interpreted as one 4-site operator.
• Figure 3: The output of toricCodeVisualization.py when $L=8$. The unitary $U$ is in form of $2L$ layers of commuting $CX$ gates. Each layer is in the same colour. The total gate number is $3L^2+2L-5$.
• Figure 4: The output of toricCodeOperation.py when $L=8$. The trivial sites are at $(0,0),(2L-1, 2L-1)$ which are neighbours. The blue operators are $X$ interactions and the red operators are $Z$ interactions. We choose to plot $2(L^2-1)$ independent stabilizers by not plotting the 2 stabilizers that have initial support on the sites that will become trivial.