Configure error mitigation for Qiskit Runtime

No error mitigation is applied to the user program.

Level 1 applies error mitigation methods that particularly address readout errors. In the Estimator, we apply a model-free technique known as Twirled Readout Error eXtinction (TREX). It reduces measurement error by diagonalizing the noise channel associated with measurement by randomly flipping qubits through X gates immediately before measurement, and flipping the corresponding measured bit if an X gate was applied. A rescaling term from the diagonal noise channel is learned by benchmarking random circuits initialized in the zero state. This allows the service to remove bias from expectation values that result from readout noise. This approach is described further in Model-free readout-error mitigation for quantum expectation values (opens in a new tab).

Level 2 uses the Zero Noise Extrapolation method (ZNE) which computes an expectation value of the observable for different noise factors (amplification stage) and then uses the measured expectation values to infer the ideal expectation value at the zero-noise limit (extrapolation stage). This approach tends to reduce errors in expectation values, but is not guaranteed to produce an unbiased result.

The overhead of this method scales with the number of noise factors. The default settings sample the expectation value at three noise factors, leading to a roughly 3x overhead when employing this resilience level.

Level 3 enables the Probabilistic Error Cancelation (PEC) method. This approach mitigates error by learning and inverting a sparse noise model that is able to capture correlated noise. PEC returns an unbiased estimate of an expectation value so long as learned noise model faithfully represents the actual noise model at the time of mitigation. In practice, the experimental procedure for learning the noise model has ambiguities due to certain error terms that cannot be independently distinguished. These are resolved by a symmetry assumption, which depending on the true underlying noise may lead a biased estimate of the mitigated expectation values due to using an imperfect noise model.

The Qiskit Runtime primitive implementation of PEC specifically addresses noise in self-inverse two-qubit gates, so it first stratifies each input circuit into an alternating sequence of simultaneous 1-qubit gates followed by a layer of simultaneous 2-qubit gates. Then it learns the noise model associated with each unique 2-qubit gate layer.

The overhead of this method scales with the number of noise factors. The default settings sample the expectation value at three noise factors, leading to a roughly 3x overhead when employing this resilience level.

PEC uses a quasi-probability method to mimic the effect of inverting the learned noise. This requires sampling from a randomized circuit family associated with the user's original circuit. Applying PEC will increase the variability of the returned expectation value estimates unless the number of samples per circuit is also increased for both input and characterization circuits. The amount of samples required to counter this variability scales exponentially with the noise strength of the mitigated circuit.

How this works:

When estimating an unmitigated Pauli observable $\langle P\rangle$ the standard error in the estimated expectation value is given by

$\frac{1}{\sqrt{N_{\text{shots}}}}\left(1- \langle P\rangle^2\right)$

where $N_{\text{shots}}$ is the number of shots used to estimate $\langle P\rangle$. When applying PEC mitigation, the standard error becomes $\sqrt{\frac{S}{N_{\text{samples}}}}\left(1- \langle P\rangle^2\right)$ where $N_{\text{samples}}$ is the number of PEC samples.

The sampling overhead scales exponentially with a parameter that characterizes the collective noise of the input circuit. As the Qiskit Runtime primitive learns the noise of your circuit, it will return metadata about the sampling overhead associated with that particular layer. Let's label the overhead of layer $l$ as $\gamma_l$. Then the total sampling overhead for mitigating your circuit is the product of all the layer overheads, that is:

$S = \prod_l \gamma_l$

When the Estimator completes the model-learning phase of the primitive query, it will return metadata about the total sampling overhead for circuit.

Depending on the precision required by your application, you will need to scale the number of samples accordingly. The following plot illustrates the relationship between estimator error and number of circuit samples for different total sampling overheads.

Note that the number of samples required to deliver a desired accuracy is not known before the primitive query because the mitigation scaling factor is discovered during the learning phase of PEC.

We suggest starting with short depth circuits to get a feel for the scaling of the sampling overhead of PEC before attempting larger problems.

Level 1 uses matrix-free measurement mitigation (M3) routine to mitigate readout error. M3 works in a reduced subspace defined by the noisy input bit strings that are to be corrected. Because the number of unique bit strings can be much smaller than the dimensionality of the full multi-qubit Hilbert space, the resulting linear system of equations is nominally much easier to solve.