Assignment Matrix Shape Scan - Quantum Hardware Readout: Signals and Calibration Step by Step

Build each assignment row from its own prepared-state calibration counts. The scan compares a skew matrix from 9/1 and 2/8 counts, an identity matrix from 10/0 and 0/10 counts, and a balanced ambiguous matrix; real matrices also carry shot noise, threshold drift, state-preparation error, and calibration uncertainty.

Calibration count pairs can make skew, identity, or balanced assignment matrices, but each displayed row must close. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Assignment rows begin as prepared-state counts

The first matrix is built from prepared-zero counts 9 and 1, then prepared-one counts 2 and 8.

(N_{zz},N_{zo})=(9,1),\quad (N_{oz},N_{oo})=(2,8)

Three calibration shapes expose matrix behavior

The scan includes a skew matrix, an identity matrix, and a balanced ambiguous matrix. Each probability row is normalized from its own prepared-state counts.

\begin{array}{c|c|c|c|c|c|c|c}N_{zz}&N_{zo}&N_{oz}&N_{oo}&A_{zz}&A_{zo}&A_{oz}&A_{oo}\\9&1&2&8&\frac{9}{10}&\frac{1}{10}&\frac{1}{5}&\frac{4}{5}\\10&0&0&10&1&0&0&1\\6&4&4&6&\frac{3}{5}&\frac{2}{5}&\frac{2}{5}&\frac{3}{5}\\\end{array}

Perfect assignment is an extreme checked shape

The identity row uses counts 10 and 0 for prepared zero, then 0 and 10 for prepared one.

A_{\text{identity}}=\begin{bmatrix}1&0\\0&1\end{bmatrix}

A balanced matrix keeps rows closed but less decisive

The balanced matrix still has rows that sum to 1, but each prepared state has both readout outcomes present.

\frac{3}{5}+\frac{2}{5}=1,\quad \frac{2}{5}+\frac{3}{5}=1

The matrix shape is calibration arithmetic

The chapter does not infer microscopic measurement dynamics from these rows. It only checks count normalization and matrix shape.

\text{row sums}=1