Readout as a Confusion Matrix

qc/noise qc/hardware

A qubit is read out via dispersive coupling to a resonator: the qubit’s state shifts the resonator’s frequency slightly, and that shift is inferred from the reflected microwave signal’s I/Q (in-phase/quadrature) components, classified against a decision boundary as $0$ or $1$. This classification step isn’t perfect — readout error is exactly the rate at which it gets it wrong.

The confusion matrix formalism

For $n$ qubits, readout error is fully described by a $2^n\times 2^n$ confusion matrix $M$:

${\vec{p}}_{\text{meas}}=M{\vec{p}}_{\text{true}}$

where ${\vec{p}}_{\text{true}}$ is the true outcome-probability vector and ${\vec{p}}_{\text{meas}}$ is what you actually observe after readout error mixes things up. Key insight: this is a real matrix, not just an error rate — it captures which wrong outcomes a given true outcome tends to be confused with, not just how often readout is wrong overall. In principle, if $M$ is known and invertible, $M^{-1}{\vec{p}}_{\text{meas}}$ recovers an estimate of ${\vec{p}}_{\text{true}}$ — this is the formal object TREX is built to correct for, more efficiently than inverting the full $2^n\times 2^n$ matrix directly.

Related

• Backend Properties
• EstimatorOptions and the Five Mitigation Knobs — TREX, the readout-error-correction technique
• Coherent vs Incoherent Gate Errors
• Measurement and Collapse

Self-Check

• Could you explain what dispersive readout physically measures, and why classification can go wrong?
• Why is a full confusion matrix more informative than a single readout error rate?
• What would $M^{-1}{\vec{p}}_{\text{meas}}$ give you, in principle, and why is that only “in principle”?