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Fault-tolerant architectures aim to reduce the noise of a quantumcomputation. Despite such architectures being well studied a detailedunderstanding of how noise is transformed in a fault-tolerant primitive such asmagic state injection is currently lacking. We use numerical simulations oflogical process tomography on a fault-tolerant gadget that implements a logicalT = $Z({\pi}/8)$ gate using magic state injection, to understand how noisecharacteristics at the physical level are transformed into noisecharacteristics at the logical level. We show how, in this gadget, asignificant phase ($Z$) bias can arise in the logical noise, even with unbiasednoise at the physical level. While the magic state injection gadgetintrinsically induces biased noise, with extant phase bias being furtheramplified at the logical level, we identify noisy error correction circuits asa key limiting factor on the magnitude of this logical noise bias. Our approachprovides a framework for assessing the detailed noise characteristics, as wellas the overall performance, of fault-tolerant logical primitives.

Logical Noise Bias in Magic State Injection