Measurement Error Mitigation Readout

Short answer: Readout error mitigation corrects for mistakes made when a qubit is measured, by characterizing how often each outcome is misread and then applying the inverse of that noise in classical postprocessing. A 2021 Physical Review A paper shows two schemes and reports that the overhead grows exponentially in the number of qubits for a given error rate. Software documentation from IBM and the Qiskit community describes practical versions. It reduces measurement noise but does not fix errors in the circuit itself.

What a readout error is

IBM's documentation says that if the sampled bitstring in classical memory differs from the bitstring characterizing the projected quantum state, a readout error has occurred. It says such errors are observed to be random and uncorrelated from sample to sample, and that it has proven useful to model them as a noisy classical channel: for every pair of bitstrings, there is a fixed probability that one true value is misread as another. The Qiskit Experiments documentation says readout errors affect computation during measurement, and that characterizing them lets one build a mitigator that gives a more accurate distribution of outputs and more accurate expectation values.

The assignment matrix

Both documentation pages describe an assignment matrix, with 2^n rows and columns for n qubits, whose entries give the probability of observing one outcome given a true outcome. IBM says the entries are determined experimentally by repeatedly preparing each basis state and measuring the frequencies of the sampled bitstrings. The matrix is then used to correct the measured distribution.

Local versus correlated models

The Qiskit Experiments documentation describes two kinds. A local mitigator assumes that readout errors are mostly local, meaning independent between qubits. The assignment matrix is then a tensor product of n small 2 by 2 matrices, one per qubit, which makes it practical to store. A correlated mitigator uses the full matrix, which the documentation says can be used only for small n. The default calibration experiment is local and uses two circuits, one preparing all zeros and one all ones.

The 2021 paper

A 2021 paper in Physical Review A says that reducing measurement errors in multiqubit devices is critical for any quantum algorithm. It shows how to mitigate them by classical postprocessing of measured outcomes, for experiments that use the outcomes to compute expected values of observables. It presents two schemes, based on tensor product and correlated Markovian noise models, with error rates extracted from calibration data using a simple formula. Mitigation applies the inverse noise matrix to the probability vector of noisy outcomes. The authors state that the overhead, including the number of measurements and the classical postprocessing cost, is exponential in epsilon times n, where epsilon is the maximum error rate and n the number of qubits. They report demonstrations on IBM Quantum devices using stabilizer measurements of graph states with up to 12 qubits and entangled 20-qubit states from low-depth random Clifford circuits.

Where it applies

IBM's tutorial says the Sampler primitive does not have built-in error mitigation, and that readout error mitigation is an exception that is highly effective and applicable there, using an addon called M3. That is the vendor's description of its own tool.

What this does not tell you

Mitigation does not remove errors from gates or decoherence during the circuit, and it is not error correction. The overhead growth means it suits small and moderate qubit counts. The sources do not quantify the improvement for any particular device.

FAQ

What is readout error?

A measured bitstring that differs from the true state of the projected qubits, per IBM's documentation.

How is it mitigated?

By measuring an assignment matrix and applying its inverse to the measured distribution.

What is the cost?

The 2021 paper says overhead is exponential in the error rate times the number of qubits.

Is it the same as error correction?

No. It is classical postprocessing of measurement results, and does not protect the computation itself.

Does it need correlations between qubits?

A local model assumes none and is cheap to store. A correlated model uses the full matrix and is limited to small n.