Readout Error Mitigation with FiQCI EMS

Readout Error Mitigation with FiQCI EMS#

What is Readout Error Mitigation?#

Readout errors occur when quantum measurements incorrectly identify the state of a qubit. For example, a qubit prepared in state |0⟩ might be measured as |1⟩, and vice versa. These errors are typically characterized by:

  • P(0|0): Probability of correctly measuring 0 when qubit is in state |0⟩

  • P(1|0): Probability of incorrectly measuring 1 when qubit is in state |0⟩

  • P(0|1): Probability of incorrectly measuring 0 when qubit is in state |1⟩

  • P(1|1): Probability of correctly measuring 1 when qubit is in state |1⟩

The M3 (Matrix-free Measurement Mitigation) Method#

M3 is a readout error mitigation technique that:

  1. Calibrates by running circuits that prepare known computational basis states

  2. Characterizes the confusion matrix describing measurement errors

  3. Corrects measured distributions by inverting the error model

Unlike traditional methods that explicitly compute and invert large matrices, M3 uses tensor network methods to efficiently handle multi-qubit systems, making it scalable to larger quantum computers.

Key advantages of M3:

  • Scales efficiently to many qubits

  • Handles correlated and uncorrelated readout errors

  • Provides quasi-probability distributions (can have negative values)

  • Can convert to nearest valid probability distribution

Quasi-probabilities, counts and error bars#

M3’s correction produces a quasi-probability distribution, whose entries may be negative. That is an unbiased estimate but not a physical distribution, so EMS uses it two different ways depending on what you asked for:

  • FiQCISampler counts are the quasi-probabilities projected onto the nearest physical distribution (negative entries clipped to zero) and scaled back to the shot count, because counts have to be non-negative integers for the rest of Qiskit.

  • FiQCIEstimator expectation values are computed from the unprojected quasi-probabilities. Clipping removes exactly the low-probability outcomes an expectation value is most sensitive to, which biases ⟨P⟩ towards ±1 and lands it exactly on ±1 whenever the clipped outcomes are the only ones contributing the opposite sign, as for a two-qubit ZZ observable. A mitigated expectation value can therefore fall slightly outside [-1, 1]; that is the honest estimate for an over-subtracting calibration, and a value far outside means the readout calibration is too noisy (raise calibration_shots).

Mitigation trades bias for variance, so standard_errors() reports the shot noise of the raw counts inflated by sqrt(mitigation_overhead), mthree’s own error bound. A mitigated value’s error bar is therefore always larger than the unmitigated one it came from.

Both quantities, and how many outcomes the projection clipped, travel in each result’s header:

result = FiQCISampler(backend, mitigation_level=1).run(circuit, shots=4096).result()
metadata = result.results[0].header["fiqci_ems"]

metadata["quasi_probabilities"]   # unprojected M3 output, keyed like the counts
metadata["mitigation_overhead"]   # M3's variance amplification factor (>= 1)
metadata["clipped_outcomes"]      # outcomes the projection zeroed
metadata["raw_counts"]            # pre-mitigation counts

A warning is raised when the projection had to clip anything, since the counts from those circuits are biased towards the physical boundary.

Advanced: Direct M3IQM Control#

For fine-grained control over the mitigation process, you can use the M3IQM class directly. This allows you to:

  • Choose calibration strategies ("balanced", "independent", "marginal")

  • Inspect per-qubit calibration matrices

  • Apply correction manually and access quasi-probability distributions

  • Calculate expectation values and standard deviations from quasi-distributions

See the Advanced Readout Error Mitigation example for a full walkthrough.

Examples#

References:#