  In this work, we migrate the quantum error mitigation technique of Zero-Noise
Extrapolation (ZNE) to fault-tolerant quantum computing. We employ ZNE on
logically encoded qubits rather than physical qubits. This approach will be
useful in a regime where quantum error correction (QEC) is implementable but
the number of qubits available for QEC is limited. Apart from illustrating the
utility of a traditional ZNE approach (circuit-level unitary folding) for the
QEC regime, we propose a novel noise scaling ZNE method specifically tailored
to QEC: distance scaled ZNE (DS-ZNE). DS-ZNE scales the distance of the error
correction code, and thereby the resulting logical error rate, and utilizes
this code distance as the scaling `knob' for ZNE. Logical qubit error rates are
scaled until the maximum achievable code distance for a fixed number of
physical qubits, and lower error rates (i.e., effectively higher code
distances) are achieved via extrapolation techniques migrated from traditional
ZNE. Furthermore, to maximize physical qubit utilization over the ZNE
experiments, logical executions at code distances lower than the maximum
allowed by the physical qubits on the quantum device are parallelized to
optimize device utilization. We validate our proposal with numerical simulation
and confirm that ZNE lowers the logical error rates and increases the effective
code distance beyond the physical capability of the quantum device. For
instance, at a physical code distance of 11, the DS-ZNE effective code distance
is 17, and at a physical code distance of 13, the DS-ZNE effective code
distance is 21. When the proposed technique is compared against unitary folding
ZNE under the constraint of a fixed number of executions of the quantum device,
DS-ZNE outperforms unitary folding by up to 92% in terms of the post-ZNE
logical error rate.
