The process of choosing the optimal QEM method and/or the optimal parameter settings of a method for a user’s specific situation (problem type, circuit structure, resource constraints, etc.). It is analogous to choosing a machine learning method and its optimal hyperparameters. (Not to be confused with “noise calibration” in the sense of tuning a physical device so that it better approximates some ideal property or operation.)

Expectation Value

The expectation value of an observable \(A\) on state \(\rho\) is the average readout value when \(A\) is measured on \(\rho\). Mathematically, this is \(\text{Tr}[A\rho]\) and usually denoted \(\langle A \rangle\) (when this notation is used, the state \(\rho\) that \(A\) is being measured on should be clear from context). Expectation values are important for near-term quantum computing because in variational quantum algorithms, the only role of the quantum processor is to repeatedly compute expectation values, which a classical processor then uses to perform some overall useful computational task. In Mitiq, Executors are used to calculate error-mitigated expectation values.

Gate Fidelity

A number between 0 and 1 measuring how closely a particular device’s (noisy) physical implementation of a gate approximates the ideal gate’s action on quantum states. Mitiq implements a noise-scaling method for ZNE in which each gate of the input circuit is sampled for unitary folding with probability proportional to its infidelity (1 - fidelity), described here and here in the documentation.


A Hermitian operator whose eigenvalues and eigenvectors represent, respectively, a quantum system’s possible energy levels and corresponding energy states. Most variational quantum algorithms work by encoding the objective of an optimization problem (e.g. finding the maximum cut in a graph) as the task of minimizing the expectation value of a problem-specific Hamiltonian, which physically corresponds to finding the ground-state energy of that Hamiltonian. For an example of how error mitigation helps such algorithms, see Solving MaxCut with Mitiq-improved QAOA.

Sampling Overhead

The basic resource-cost measure used to evaluate QEM methods—how many more circuit executions (“runs,” “shots”) does a method need to achieve the same level of statistical precision in estimating an expectation value, compared to the naive (i.e. unmitigated) method of running the same noisy input circuit \(N\) times and returning the sample mean of the measurement outcomes. Also called sampling cost, it is usually reported as a multiplicative factor \(C\), defined as the ratio of the QEM estimator’s variance to the sample-mean estimator’s variance, and meaning that the method needs \(C \cdot N\) circuit shots to obtain the same precision as the sample-mean estimator would with only \(N\) shots.

QEM Methods#

Clifford Data Regression (CDR)

An error mitigation model is trained with quantum circuits that resemble the circuit of interest, but which are easier to classically simulate.

Digital Dynamical Decoupling (DDD)

Sequences of gates are applied to slack windows (single-qubit idle windows) in a quantum circuit to reduce the coupling between the qubits and the environment, mitigating the effects of noise.

Probabilistic Error Cancellation (PEC)

Ideal operations are represented as quasi-probability distributions over noisy implementable operations, and unbiased estimates of expectation values are obtained by averaging over circuits sampled according to this representation.

Readout Error Mitigation (REM)

Inverted transition/confusion matrices are applied to noisy measurement results to mitigate errors in the estimation of expectation values.

Zero Noise Extrapolation (ZNE)

An expectation value is computed at different noise levels and then the ideal expectation value is inferred by extrapolating the measured results to the zero-noise limit.