Table 2 Sampling overhead of error countermeasures under synthesis error of ϵ

Gate	Synthesis	Crafted?	Remnant error		QEM method	Sampling overhead	Comment
			magnitude	type
SU(2)	Unitary		ϵ	coherent	PEC	1 + 4ϵ
SU(2)	Unitary		ϵ	coherent	Rescaling	1 + 2ϵ²	Slow WN convergence
SU(2)	Mixed unitary		ϵ²	not Pauli	PEC	1 + 4ϵ²	Pauli frame unavailable
SU(2)	Mixed unitary	✓	ϵ²	Pauli/depol	PEC	1 + 4ϵ²
SU(2)	Mixed unitary	✓	ϵ²	Pauli/depol	Rescaling	1 + 2ϵ²
R_z	Unitary		ϵ	coherent	PEC	1 + 4ϵ	Pauli frame unavailable
R_z	Unitary		ϵ	coherent	Rescaling	1 + 2ϵ²	Slow WN convergence
R_z	Mixed unitary		ϵ²	not Pauli	PEC	1 + 4ϵ²	Pauli frame unavailable
R_z	Mixed unitary	✓	ϵ²	Pauli/depol/XY	PEC	1 + 4ϵ²
R_z	Mixed unitary	✓	ϵ²	Pauli/depol/XY	ED + PEC	1 + (q_x + q_y + 4q_z)ϵ²	q_z ~ ϵ in numerics
R_z	Mixed unitary	✓	ϵ²	Pauli/depol/XY	ED + Resc.	1 + (q_x + q_y + 2q_z)ϵ²	q_z ~ ϵ in numerics
R_z	Mixed CPTP	✓	ϵ³	Pauli Z	PEC	1 + 4ϵ³	Single ancilla
R_z	Mixed CPTP	✓	ϵ³	Pauli Z	Rescaling	1 + 2ϵ³	Single ancilla

The remnant error indicates the diamond distance \(d({{\mathcal{E}}}_{{\rm{rem}}},{\mathcal{I}})\) where \({\mathcal{I}}\) is the identity channel. We assume that unitary gate synthesis is done with an accuracy of ϵ, and compare the performance when we further perform mixed or error-crafted synthesis. The sampling overhead γ is indicated for a single gate, and the total overhead is multiplicative as ∏_lγ_l when the overhead is γ_l for the l-th gate. The rescaling technique assumes that the underlying quantum circuit is sufficiently deep so that the white-noise approximation holds^55,70,71. We assume that the white-noise approximation holds when the rescaling technique is applied. The cubic suppression in the last two rows is verified numerically up to ϵ³ = 10⁻⁹ as shown in Fig. 5. Note that applying PEC for crafted and uncrafted synthesis differs significantly. We cannot rely on the Pauli frame in the latter, and an actual single-qubit operation is required.

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