Krysta Svore

Abstract Estimating the reducing overhead of existing fault tolerance schemes is a crucial step toward realizing scalable quantum computers. Many of the most promising schemes are based upon two-dimensional (2D) topological codes such as the surface and color codes. In these schemes, universal computation is typically achieved using readily implementable Clifford operations along with a less convenient and more costly implementation of the $T$ gate. In our work, we compare the cost of fault-tolerantly implementing the $T$-gate in 2D color codes using two leading approaches: state distillation and code switching to a 3D color code. We report that state distillation is more resource-efficient than code switching, in terms of both qubit overhead and space-time overhead. In particular, we find a $T$ gate threshold via code switching of $0.07(1)\%$ under circuit noise, almost an order of magnitude below that for distillation with 2D color codes. To arrive at this result, we provide and implement a simplified end-to-end recipe for code switching, detailing each step and providing important optimization considerations. We not only find numerical overhead estimates of this code switching protocol, but also lower bound various conceivable improvements. We also optimize the 2D color code for circuit noise yielding it's largest threshold to date $0.37(1)\%$, and adapt and optimize the restriction decoder and find a threshold of $0.80(5)\%$ for the 3D color code with perfect measurements under $Z$ noise. We foresee that this analysis will influence the choice of which FT schemes and which salable hardware designs should be pursued in future.