  Fault-tolerant architectures aim to reduce the noise of a quantum
computation. Despite such architectures being well studied a detailed
understanding of how noise is transformed in a fault-tolerant primitive such as
magic state injection is currently lacking. We use numerical simulations of
logical process tomography on a fault-tolerant gadget that implements a logical
T = $Z({\pi}/8)$ gate using magic state injection, to understand how noise
characteristics at the physical level are transformed into noise
characteristics at the logical level. We show how, in this gadget, a
significant phase ($Z$) bias can arise in the logical noise, even with unbiased
noise at the physical level. While the magic state injection gadget
intrinsically induces biased noise, with extant phase bias being further
amplified at the logical level, we identify noisy error correction circuits as
a key limiting factor on the magnitude of this logical noise bias. Our approach
provides a framework for assessing the detailed noise characteristics, as well
as the overall performance, of fault-tolerant logical primitives.
