Dressed Gates and Pauli Propagation

qc/mitigation qc/math

The algebra underlying all twirling-based mitigation methods. A gate “dressed” with Paulis on either side is the basic operation that both Pauli Twirling and Samplomatic rely on.

Dressed gate definition

For any $n$-qubit unitary $U$, a dressed version is:

$\tilde{U}=V_{\text{out}}U{V}_{\text{in}}$

where $V_{\text{in}}$ and $V_{\text{out}}$ are products of single-qubit unitaries. The dressing is invariant when:

$V_{\text{out}}U{V}_{\text{in}}=U\text{(up to global phase)}$

This means the gate’s logical action is unchanged, but any noise attached to $U$ gets twirled by $V_{\text{in}}$ and $V_{\text{out}}$.

Clifford conjugation rule

For any Pauli $P$ and Clifford $U$:

$UP{U}^{†}=\pm P^{\prime }$

where $P^{\prime }$ is another Pauli. Finding $V_{\text{out}}$ given $V_{\text{in}}$ amounts to propagating $V_{\text{in}}$ through $U$. In Qiskit:

Pauli("IX").evolve(U, frame='s')   # returns U·P·U† with correct sign
Pauli.equiv(other)                  # compare up to global phase

CZ propagation table (key rules)

For the CZ gate (used throughout this section’s Ising circuits):

• Z on either qubit → passes through unchanged
• X or Y on one qubit → picks up a Z on the other qubit
• The pair XY ↔ YX picks up a minus sign

This asymmetry is why CZ is symmetric (both qubits play equal roles) while CX is not.

# Examples
CZ · IX · CZ = ZX    (X on q0 picks up Z on q1)
CZ · XI · CZ = XZ    (X on q1 picks up Z on q0)
CZ · IZ · CZ = IZ    (Z passes through)
CZ · ZI · CZ = ZI    (Z passes through)

Two maps from the propagation table

Map	Contains	Used for
cz_twirl_map	$V_{\text{in}}\to V_{\text{out}}$ (unsigned)	What to physically apply as dressing
cz_commutation_map	$V_{\text{in}}\to (\text{sign},V_{\text{out}})$	Propagation rules including sign

The sign does not affect gate invariance (absorbed as global phase) but does matter when propagating Paulis through a circuit to track observable modifications (as in PNA).

Scale of application

The same Clifford conjugation operation is used at every level, only the scope differs:

Method	Scope
twirling.enable_gates in EstimatorOptions	Whole circuit, uniform
Twirl box in Samplomatic	Per box (layer or gate)
PNA propagation	Per box, forward through entire circuit into observable

Qiskit helpers

from qiskit.quantum_info import Pauli
 
# Propagate P through U: returns ±P'
result = Pauli("IX").evolve(cz_layer, frame='s')
result.to_label()    # e.g. "+ZX" or "-YX"
 
# Compare up to global phase
Pauli("ZX").equiv(Pauli("-ZX"))   # True

Related

• EstimatorOptions and the Five Mitigation Knobs
• Samplomatic — Boxes and Annotations
• PNA — Propagated Noise Absorption

Self-Check

• Could you explain what “dressing” a gate means and why the dressing has to be invariant?
• Why does propagating a Pauli through a CZ sometimes pick up a minus sign, and why does that sign matter for PNA but not for basic twirling?
• What’s the one operation (Clifford conjugation) that shows up at every scale, from whole-circuit twirling to per-box Samplomatic to PNA?