QuantumCore

Hands-on lab · 1 step

Readout-error mitigation by confusion-matrix inversion

A qubit's readout flips 0→10\to1 with probability p01=0.05p_{01}=0.05 and 1→01\to0 with probability p10=0.10p_{10}=0.10. Undo this classical noise on a measured distribution by inverting the 2×22\times2 confusion matrix.

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Step 1: Invert the confusion matrix

Calibration of a qubit's readout gives the error rates

p01=P(read 1∣true 0)=0.05,p10=P(read 0∣true 1)=0.10,p_{01}=P(\text{read }1\mid\text{true }0)=0.05,\qquad p_{10}=P(\text{read }0\mid\text{true }1)=0.10,

so the measured distribution is p~=Ap\tilde{\mathbf p}=A\mathbf p with the column-stochastic confusion matrix (Art=P(read r∣true t)A_{rt}=P(\text{read }r\mid\text{true }t))

A=(1−p01p10p011−p10).A=\begin{pmatrix}1-p_{01}&p_{10}\\p_{01}&1-p_{10}\end{pmatrix}.

A circuit is measured many times and the observed frequencies are p~=(0.78, 0.22)\tilde{\mathbf p}=(0.78,\,0.22). Compute the mitigated distribution p=A−1p~\mathbf p=A^{-1}\tilde{\mathbf p} and enter p(0)p(0) and p(1)p(1). Each value is graded with absolute tolerance 10−310^{-3}; exact expressions such as (0.90.78-0.10.22)/0.85 are accepted.

Enter exact expressions or decimals, e.g. 1/sqrt(2), pi/3. Values must be real.

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