Noise

A Bell pair under a Qiskit Aer noise model: depolarizing errors on the gates, amplitude damping before measurement, and a readout error. MENOS computes the expectation values exactly, and the script checks them against Aer's own density-matrix estimator.

Needs pip install "menos[qiskit]" qiskit-aer==0.17.2.

"""A noisy Bell pair: a Qiskit Aer noise model simulated exactly by MENOS.

Needs qiskit-aer (pip install 'menos[noise]'). The expectation values are
checked against Aer's own density-matrix estimator, and the counts against the
distribution they imply.
"""

from qiskit import QuantumCircuit
from qiskit.quantum_info import SparsePauliOp
from qiskit_aer.noise import NoiseModel, ReadoutError, amplitude_damping_error, depolarizing_error
from qiskit_aer.primitives import EstimatorV2 as AerEstimator

from menos.qiskit import MENOSBackend, MENOSEstimator

noise_model = NoiseModel()
noise_model.add_all_qubit_quantum_error(depolarizing_error(0.02, 1), ["h"])
noise_model.add_all_qubit_quantum_error(depolarizing_error(0.05, 2), ["cx"])
noise_model.add_all_qubit_quantum_error(amplitude_damping_error(0.1), ["measure"])
noise_model.add_all_qubit_readout_error(ReadoutError([[0.98, 0.02], [0.04, 0.96]]))

bell = QuantumCircuit(2)
bell.h(0)
bell.cx(0, 1)

# Exact expectation values under noise.
observables = [SparsePauliOp("ZZ"), SparsePauliOp("XX"), SparsePauliOp("ZI")]
menos_evs = MENOSEstimator(noise_model=noise_model).run([(bell, observables)]).result()[0].data.evs
aer = AerEstimator(
    options={"backend_options": {"noise_model": noise_model, "method": "density_matrix"}}
)
aer_evs = aer.run([(bell, observables)]).result()[0].data.evs
print("<ZZ>, <XX>, <ZI> =", [round(float(v), 6) for v in menos_evs])
assert max(abs(m - a) for m, a in zip(menos_evs, aer_evs)) < 1e-12

# Counts, with the measurement error and readout error applied.
measured = bell.copy()
measured.measure_all()
counts = (
    MENOSBackend()
    .run(measured, shots=10_000, seed_simulator=7, noise_model=noise_model)
    .result()
    .get_counts()
)
print("counts:", dict(sorted(counts.items())))
assert sum(counts.values()) == 10_000
# Noise makes the odd-parity outcomes possible but rare.
odd = counts.get("01", 0) + counts.get("10", 0)
assert 0 < odd < 2_000

Output

<ZZ>, <XX>, <ZI> = [0.95, 0.931, -0.0]
counts: {'00': 4828, '01': 733, '10': 776, '11': 3663}

Without noise, ⟨ZZ⟩ and ⟨XX⟩ would both be 1, and only 00 and 11 would ever be recorded. The two-qubit depolarizing error on cx scales both correlations by 0.95. ⟨XX⟩ is also scaled by the one-qubit error on h, 0.98, giving 0.931. Amplitude damping, which acts just before each measurement, pulls population toward 0, so 00 outnumbers 11. Damping one qubit of 11 records 01 or 10, and readout error flips further bits, so about 15% of outcomes have odd parity. The expectation values are exact, which is why they can be compared with Aer to 1e-12. Readout error is not part of them.

See also

Noise Models · Density Matrix