  Quantum error mitigation schemes (QEM) have greatly enhanced the efficiency
of quantum computers, specifically focusing on reducing errors caused by
interactions with the environment. Nevertheless, the presence of coherence
errors, typically arising from miscalibration and inter-qubit crosstalk, is a
significant challenge to the scalability of quantum computing. Such errors are
often addressed using a refined Pauli twirling scheme called Randomized
Compiling (RC) that converts the coherent errors into incoherent errors that
can then be mitigated by conventional QEM. Unfortunately for multi-qubit gates,
RC is restricted to Clifford gates such as CNOT and CPHASE. However, it has
been demonstrated experimentally that a direct implementation of multi-qubit
non-Clifford gates, i.e. without using multi-qubit Clifford gates, has reduced
the depth of the circuit by a factor of four and more. Recently, a framework
called pseudo-twirling (PST) for treating coherent error in multi-qubit
non-Clifford gates has been introduced and experimentally demonstrated. We show
analytically that a higher order correction to the existing PST theory yields
an over-rotation coherent error generated by the PST protocol itself. This PST
effect has no analogue in RC. Although the small induced over-rotation can
amount to a significant coherent error in deep circuits, we explain why it does
not degrade the performance of the gate.
