Composing techniques: Readout Error Mitigation and Zero Noise Extrapolation#

Noise in quantum computers can arise from a variety of sources, and sometimes applying multiple error mitigation techniques can be more beneficial than applying a single technique alone. Here we apply a combination of Readout Error Mitigation (REM) and Zero Noise Extrapolation (ZNE) to a randomized benchmarking (RB) task. In REM, the inverse transition / confusion matrix is generated and applied to the noisy measurement results. In ZNE, the expectation value of the observable of interest is computed at different noise levels, and subsequently the ideal expectation value is inferred by extrapolating the measured results to the zero-noise limit. More information on the REM and ZNE techniques can be found in the corresponding sections of the user guide (linked above).


We begin by importing the relevant modules and libraries required for the rest of this tutorial.

import cirq
import numpy as np
from mitiq.benchmarks import generate_rb_circuits
from mitiq import MeasurementResult, Observable, PauliString, raw


We will demonstrate using REM + ZNE on RB circuits, which are generated using Mitiq’s built-in benchmarking circuit generation function, generate_rb_circuits(). More information on the RB protocol is available in the Randomized Benchmarking section of the Qiskit Experiments Manual. In this example we use a two-qubit RB circuit with a Clifford depth (number of Clifford groups) of 10.

circuit = generate_rb_circuits(2, 10)[0]

Noise model and executor#

The noise in this example is a combination of depolarizing and readout errors, the latter of which are modeled as bit flips immediately prior to measurement. We use an executor function to run the quantum circuit with the noise model applied.

def execute(circuit: cirq.Circuit, noise_level: float = 0.002, p0: float = 0.05) -> MeasurementResult:
    """Execute a circuit with depolarizing noise of strength ``noise_level`` and readout errors ...
    measurements = circuit[-1]
    circuit =  circuit[:-1]
    circuit = circuit.with_noise(cirq.depolarize(noise_level))

    simulator = cirq.DensityMatrixSimulator()

    result =, repetitions=10000)
    bitstrings = np.column_stack(list(result.measurements.values()))
    return MeasurementResult(bitstrings)


In this example, the observable of interest is \(ZI + IZ\). For the circuit defined above, the ideal (noiseless) expectation value of the \(ZI + IZ\) observable is 2, but as we will see, the unmitigated (noisy) result is impacted by depolarizing and readout errors.

obs = Observable(PauliString("ZI"), PauliString("IZ"))
noisy = raw.execute(circuit, execute, obs)
from functools import partial

ideal = raw.execute(circuit, partial(execute, noise_level=0, p0=0), obs)
print("Unmitigated value:", "{:.5f}".format(noisy.real))
Unmitigated value: 1.34080

Next we generate the inverse confusion matrix and apply readout error mitigation (REM). More information on generating the inverse confusion matrix is available in the REM theory section of the user guide.

from mitiq import rem

p0 = p1 = 0.05
icm = rem.generate_inverse_confusion_matrix(2, p0, p1)
rem_executor = rem.mitigate_executor(execute, inverse_confusion_matrix=icm)

rem_result = obs.expectation(circuit, rem_executor)
print("Mitigated value obtained with REM:", "{:.5f}".format(rem_result.real))
Mitigated value obtained with REM: 1.48300

We can see that REM improves the results, but errors remain. For comparison, we then apply ZNE without REM.

from mitiq import zne

zne_executor = zne.mitigate_executor(execute, observable=obs, scale_noise=zne.scaling.folding.fold_global)
zne_result = zne_executor(circuit)
print("Mitigated value obtained with ZNE:", "{:.5f}".format(zne_result.real))
Mitigated value obtained with ZNE: 1.87380

Finally, we apply a combination of REM and ZNE. REM is applied first to minimize the impact of measurement errors on the extrapolated result in ZNE.

combined_executor = zne.mitigate_executor(rem_executor, observable=obs, scale_noise=zne.scaling.folding.fold_global)

combined_result = combined_executor(circuit)
print("Mitigated value obtained with REM + ZNE:", "{:.5f}".format(combined_result.real))
Mitigated value obtained with REM + ZNE: 2.03620

From this example we can see that each technique affords some improvement, and the combination of REM and ZNE is more effective in mitigating errors than either technique alone.