Scaling of magic state distillation with single & two-qubit gate error

Question

How does the resulting error in magic state distillation scale with the fidelity of single qubit gates and two qubit gates? In most experimental systems, two-qubit gate errors are much larger than single qubit gates. For example, one could imagine single qubit gate errors on the order of 1e-6, but two qubit gate errors still around 1e-3. How does the resulting output magic state error scale in this case, and are there ways to fully tap into improved single qubit T gate errors, or do they not matter beyond a certain point?

Answer 1

The physical gate error rates determine the initial magic state fidelity you are starting from, and how big of a code distance you need for the factories as you boost the fidelity.

The initial fidelity is around the same as the two qubit gate fidelity: 2014 Ying Li "A magic state's fidelity can be superior to the operations that created it". This has to be good enough that the state's noise is below the threshold for your distillation method.

You can generally start with a really low code distance for storing the initial state, because it's so noisy that adding a bit more doesn't really hurt. So you might start by storing the injected state with a code distance of, say, 7. Then each stage of distillation needs roughly double or triple the code distance of its input to perform the factory and to store its output. At this point the physical gate error rates only affect the code distances required; technically anything below the threshold of the error correcting code is sufficient.