Expectation value of Y and variances are wrong in SnapshotExpectationValue

[ikkoham]

Informations

• Qiskit Aer version: master
• Python version: 3.8.5
• Operating system: macOS

What is the current behavior?

from qiskit import QuantumCircuit, execute, Aer
from qiskit.providers.aer.extensions import SnapshotExpectationValue

qc = QuantumCircuit(1)
qc.snapshot_expectation_value('x', [(1, 'X')], qubits=[0], variance=True)
qc.snapshot_expectation_value('y', [(1, 'Y')], qubits=[0], variance=True)
qc.snapshot_expectation_value('z', [(1, 'Z')], qubits=[0], variance=True)

result = execute(qc, Aer.get_backend('qasm_simulator'), shots=1024).result()
print(result.data()['snapshots'])

The output is

{'expectation_value': {'x': [{'value': 0j, 'variance': 0j}],
  'y': [{'value': (1+0j), 'variance': 0j}],
  'z': [{'value': (1+0j), 'variance': 0j}]}}

Are these behaviors as expected? I'm concerned about the following points:

1. The expectation value of Y is 1, but should be 0.
2. All variances are 0.

[ikkoham]

I just found the expectaiton value of Y is correct for statevector_simulator.

from qiskit import QuantumCircuit, execute, Aer

qc = QuantumCircuit(1)
qc.snapshot_expectation_value('x', [(1, 'X')], qubits=[0], variance=True)
qc.snapshot_expectation_value('y', [(1, 'Y')], qubits=[0], variance=True)
qc.snapshot_expectation_value('z', [(1, 'Z')], qubits=[0], variance=True)

result = execute(qc, Aer.get_backend('statevector_simulator')).result()
print(result.data())

the output is

{'snapshots': {'expectation_value': {
'x': [{'value': 0j, 'variance': 0j}], 
'y': [{'value': 0j, 'variance': 0j}],
'z': [{'value': (1+0j), 'variance': 0j}]}},
'statevector': array([1.+0.j, 0.+0.j])}

I have checked this for other input states.

However, variances are still zero.

[yaelbh]

Value: I guess that there is a bug in the stabilizer simulator. In the previous execution you were probably automatically directed to the stabilizer simulator, whereas in the second execution you explicitly asked for the non-stabilizer simulator.

Variance: as long as there is no noise it is expected to be 0, because we compute <psi|Y|psi> (we don't perform average of measurement results).

[ikkoham]

Thank you for the comment.

I checked about value.

from qiskit import QuantumCircuit, execute, Aer

qc = QuantumCircuit(1)
qc.t(0)
qc.snapshot_expectation_value('x', [(1, 'X')], qubits=[0], variance=True)
qc.snapshot_expectation_value('y', [(1, 'Y')], qubits=[0], variance=True)
qc.snapshot_expectation_value('z', [(1, 'Z')], qubits=[0], variance=True)

result = execute(qc, Aer.get_backend('qasm_simulator')).result()
print(result.data()['snapshots'])

the output is

{'expectation_value': {'x': [{'value': 0j, 'variance': 0j}],
  'y': [{'value': 0j, 'variance': 0j}],
  'z': [{'value': (1+0j), 'variance': 0j}]}}

so I obtained correct value. Your guess seems correct.

For variance, we should return the population variance for (at least) the statevector simulator.
This is a non-zero value. (because <X^2>-<X>^2 = 1 - <X>^2 ≠ 0 in general)

[chriseclectic]

The variance in snapshots is only the sample variance over noise for each shot of the snapshotted value: eg for 5-shots you have shot_values = [val0, val1, val2, val3, val4], then the returned quantity is mean(shot_values), variance(shot_values) and is (usually) 0 for ideal simulations since the values are all identical. This is somewhat confusing so I am planning on removing it and just returning the mean for average snapshot types.

[ikkoham]

The first issue was fixed in #849 .

For the second issue, I discussed with @chriseclectic , and it's good to remove the variance parameter and my proposed variance should be done in other instructions.
So I think this issue is closable.

Thanks a lot.