Stacked codes: universal fault-tolerant quantum computation in a two-dimensional layout

TL;DR

This paper introduces stacked 3D color codes that enable fault-tolerant, universal quantum computation in a 2D layout by implementing a non-Clifford gate through a local measurement-based transformation, avoiding magic state distillation.

Contribution

The authors propose a novel method to realize a non-Clifford gate fault-tolerantly in 2D stabilizer codes by transforming into stacked codes, eliminating the need for magic states.

Findings

Stacked codes enable transversal non-Clifford gates in 2D layouts.

Transformation involves local measurements and a nonlocal boundary operation.

Method reduces overhead compared to magic state distillation.

Abstract

We introduce a class of 3D color codes, which we call stacked codes, together with a fault-tolerant transformation that will map logical qubits encoded in two-dimensional (2D) color codes into stacked codes and back. The stacked code allows for the transversal implementation of a non-Clifford $\pi/8$ logical gate, which when combined with the logical Clifford gates that are transversal in the 2D color code give a gate set which is both fault-tolerant and universal without requiring nonstabilizer magic states. We then show that the layers forming the stacked code can be unfolded and arranged in a 2D layout. As only Clifford gates can be implemented transversally for 2D topological stabilizer codes, a non-local operation must be incorporated in order to allow for this transversal application of a non-Clifford gate. Our code achieves this operation through the transformation from a 2D color code to the unfolded stacked code induced by measuring only geometrically local stabilizers and gauge operators within the bulk of 2D color codes together with a nonlocal operator that has support on a one-dimensional boundary between such 2D codes. We believe that this proposed method to implement the non-local operation is a realistic one for 2D stabilizer layouts and would be beneficial in avoiding the large overheads caused by magic state distillation.