Utilities for measurement error mitigation using confusion matrix (aka calibration matrix)

[tanujkhattar]

Is your feature request related to a use case or problem? Please describe. Measurement error is one of the major sources of error when running circuits on the device. A common technique to compensate for the measurement error is to first compute a confusion matrix (aka calibration matrix) by running 2 ** n circuits to prepare all basis. Measurement outcomes from the i'th circuit becomes the i'th row of the matrix.

The resulting confusion matrix would be the transpose of a stochastic map from final states to measurement outcomes, s.t. M_obs = CM.T @ M_act. For 2 qubits, this corresponds to:

[ 00o ]         ⎡ Pr(00o|00a) Pr(01o|00a) Pr(10o|00a) Pr(11o|00a) ⎤T      [ 00a ] 
[ 01o ]         ⎢ Pr(00o|01a) Pr(01o|01a) Pr(10o|01a) Pr(11o|01a) ⎥       [ 01a ]
[ 10o ]    =    ⎢ Pr(00o|10a) Pr(01o|10a) Pr(10o|10a) Pr(11o|10a) ⎥  @    [ 10a ] 
[ 11o ]         ⎣ Pr(00o|11a) Pr(01o|11a) Pr(10o|11a) Pr(11o|11a) ⎦       [ 11a ]

We can then use the computed confusion matrix to calibrate the measurement results of another experiment on the same qubits as:

M_compensated = Inv(CM.T) @ M_obs

Describe the solution you'd like We should have utility methods / classes to

• Compute the confusion matrix given a sampler and list of qubits.
• Use the confusion matrix for compensating the results of an observed experiment. The inverse of the confusion matrix can be computed either as a Pseudo-Inverse or using least squares method (simple inverse can lead to negative probabilities due to sampling error)
• Be able to compensate for measurement errors for experiments done on subsets of n qubits.
• Compute tensored confusion matrix on smaller groups of qubits assuming the measurement error is local (this would need Sum(2 ** k) circuits instead of 2 ** n).
• Write tutorials on using these newly added features, similar to https://qiskit.org/documentation/tutorials/noise/3_measurement_error_mitigation.html

What is the urgency from your perspective for this issue? Is it blocking important work? P2 - we should do it in the next couple of quarters

[crazy4pi314]

Folks on the Mitiq project are also working on implementing measurement error mitigation: https://github.com/unitaryfund/mitiq/issues/815 that might be a helpful example

[tanujkhattar]

We should add this to cirq/experiments. Look at potential collaboration opportunities with Unitary Fund.