LinCombEstimatorGradient has incorrect sign for imaginary and complex cases.

Environment

• Qiskit Terra version: 0.23.0
• Python version: 3.9
• Operating system: MacOS

What is happening?

Together with @Cryoris, we noticed that if the IMAG option is set, the resulting gradient has a spurious (undocumented) - sign. Same problem seems to affect the COMPLEX option.

How can we reproduce the issue?

The code

from qiskit import QuantumCircuit
from qiskit.circuit import Parameter
import numpy as np
from qiskit.algorithms.gradients import DerivativeType, LinCombEstimatorGradient
from qiskit.primitives import Estimator
from qiskit.quantum_info import Pauli

estimator = Estimator()
gradient = LinCombEstimatorGradient(estimator, derivative_type=DerivativeType.IMAG)
c = QuantumCircuit(1)
c.rz(Parameter("p"), 0)
grad = gradient.run([c],[Pauli("I")], [[0.0]])
print(grad.result())

prints

EstimatorGradientResult(gradients=[array([-1.])], metadata=[{'parameters': [Parameter(p)], 'derivative_type': <DerivativeType.IMAG: 'imag'>}], options=Options())

What should happen?

For the code example above, the result should be:

EstimatorGradientResult(gradients=[array([1.])], metadata=[{'parameters': [Parameter(p)], 'derivative_type': <DerivativeType.IMAG: 'imag'>}], options=Options())

Any suggestions?

This code fragment

if self._derivative_type == DerivativeType.REAL:
    op1 = SparsePauliOp.from_list([("Z", 1)])
elif self._derivative_type == DerivativeType.IMAG:
    op1 = SparsePauliOp.from_list([("Y", -1)])
elif self._derivative_type == DerivativeType.COMPLEX:
    op1 = SparsePauliOp.from_list([("Z", 1)])
    op2 = SparsePauliOp.from_list([("Y", -1)])

should be replaced with

if self._derivative_type == DerivativeType.REAL:
    op1 = SparsePauliOp.from_list([("Z", 1)])
elif self._derivative_type == DerivativeType.IMAG:
    op1 = SparsePauliOp.from_list([("Y", 1)])
elif self._derivative_type == DerivativeType.COMPLEX:
    op1 = SparsePauliOp.from_list([("Z", 1)])
    op2 = SparsePauliOp.from_list([("Y", 1)])

It also seems that unit tests for IMAG and COMPLEX options are missing.

[Cryoris]

After discussing with @dlasecki @Zoufalc and @a-matsuo it seems that the conclusion is to change the documentation and not the code, because

1. differentiating left or right is equivalent; we just have to fix a definition
2. this way, we don’t have to break the code behavior and just update the docs

[dlasecki]

After discussing with @dlasecki @Zoufalc and @a-matsuo it seems that the conclusion is to change the documentation and not the code, because

1. differentiating left or right is equivalent; we just have to fix a definition
2. this way, we don’t have to break the code behavior and just update the docs

Yes, sounds good to me. I will adapt the PR that I opened already.