Why is there a SdgGate before the measurement in the QPD decomposition of CZ gate?

[Jintao YU (yjt98765)]

The source code

qiskit-addon-cutting/qiskit_addon_cutting/qpd/decompositions.py

Lines 447 to 464 in be575fa

	@_register_qpdbasis_from_instruction("cx", "cy", "cz", "ch")
	def _(gate: CXGate | CYGate | CZGate | CHGate):
	# Constructing a virtual two-qubit gate by sampling single-qubit operations - Mitarai et al
	# https://iopscience.iop.org/article/10.1088/1367-2630/abd7bc/pdf
	measurement_0 = [SdgGate(), QPDMeasure()]
	measurement_1 = measurement_0.copy()
	# Specify the operations to be inserted for systems 1 and 2
	maps = [
	([SdgGate()], [SdgGate()]),
	([SGate()], [SGate()]),
	# R-R+ simplifies to I
	(measurement_0, []),
	(measurement_0, [ZGate()]),
	# R-R+ simplifies to I
	([], measurement_1),
	([ZGate()], measurement_1),
	]

corresponds to the following figure in the original paper

I do not understand the SdgGate in Line 451, which is used in four map cases.

I know that $e^{i\pi Z/2}=iZ$ and $(I\pm Z)/2$ is Z-basis measurement. Does the $S^\dagger$ gate relate to the coefficient $i$?

[Jim Garrison (garrison)]

The figure you show is for the RZZGate and similar, code for which is at

qiskit-addon-cutting/qiskit_addon_cutting/qpd/decompositions.py

Lines 336 to 368 in be575fa

	@_register_qpdbasis_from_instruction("rxx", "ryy", "rzz", "crx", "cry", "crz")
	def _(gate: RXXGate | RYYGate | RZZGate | CRXGate | CRYGate | CRZGate):
	# Constructing a virtual two-qubit gate by sampling single-qubit operations - Mitarai et al
	# https://iopscience.iop.org/article/10.1088/1367-2630/abd7bc/pdf
	if gate.name in ("rxx", "crx"):
	pauli = XGate()
	r_plus = SXGate()
	# x basis measurement (and back again)
	measurement_0 = [HGate(), QPDMeasure(), HGate()]
	elif gate.name in ("ryy", "cry"):
	pauli = YGate()
	r_plus = RYGate(0.5 * np.pi)
	# y basis measurement (and back again)
	measurement_0 = [SXGate(), QPDMeasure(), SXdgGate()]
	else:
	assert gate.name in ("rzz", "crz")
	pauli = ZGate()
	r_plus = SGate()
	# z basis measurement
	measurement_0 = [QPDMeasure()]
	r_minus = r_plus.inverse()
	measurement_1 = measurement_0.copy()
	# Specify the operations to be inserted for systems 1 and 2
	maps = [
	([], []), # Identity
	([pauli], [pauli]),
	(measurement_0, [r_plus]),
	(measurement_0, [r_minus]),
	([r_plus], measurement_1),
	([r_minus], measurement_1),
	]

The CRZGate is obtained by sandwiching a RZZGate with certain gate(s) (left as an exercise for the reader), which is why measurement_0, measurement_1, and maps are all different.

[Jintao YU (yjt98765)]

Jim Garrison (@garrison) Thank you for the quick reply! In case someone is still confused, I will show the exercise answer as two SdgGate! This can be confirmed by the following code:

import numpy as np
from qiskit import QuantumCircuit
from qiskit.quantum_info import Operator

circuit = QuantumCircuit(2)
circuit.sdg([0, 1])
circuit.rzz(np.pi / 2, 0, 1)
circuitOp = Operator.from_circuit(circuit)
print(circuitOp)

The printed result is

Operator([[ 0.70710678-0.70710678j,  0.        +0.j        ,
            0.        +0.j        ,  0.        +0.j        ],
          [ 0.        +0.j        ,  0.70710678-0.70710678j,
            0.        +0.j        ,  0.        +0.j        ],
          [ 0.        +0.j        ,  0.        +0.j        ,
            0.70710678-0.70710678j,  0.        +0.j        ],
          [ 0.        +0.j        ,  0.        +0.j        ,
            0.        +0.j        , -0.70710678+0.70710678j]],
         input_dims=(2, 2), output_dims=(2, 2))

which is CZ with a global phase.