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Readout discriminator training (0-1-2 states)

This notebook trains a 0-1-2 discriminator

Description

The experiment measures the \(I\) and \(Q\) signals when the qubit is prepared in the ground state \(|g\rangle\), first excited state \(|e\rangle\), and second excited state \(|f\rangle\) (i.e., 0-1-2 states). Given the collected data, the discriminator fits using sklearn LDA (Linear Discriminant Analysis) or QDA (Quadratic Discriminant Analysis). LDA and QDA are two well-known supervised classification methods in statistical and probabilistic learning1. Without going into further details, given a dataset of instances \(\{(\textbf{x}_{i}, y_{i})\}^{n}_{i=1}\) with a sample size of \(n\) and dimensionality \(x{i} \in \mathbb{R}_{d}\) and \(y_{i} \in \mathbb{R}\), with \(y_{i}\) as the class labels, LDA and QDA can be used to classify the data space using these instances.

Experiment steps

  1. Measuring the resonator transmission and collecting the \(I\) and \(Q\) signals when the qubit is in \(|g\rangle\).

  2. Applying a \(\pi\)-pulse ( rx180 ), which prepares the qubit in the excited state \(|e\rangle\).

  3. Measuring the resonator transmission and collecting the \(I\) and \(Q\) signals when the qubit is in \(|e\rangle\).

  4. Applying a second \(\pi\)-pulse ( rx180_ef ), which prepares the qubit in the second excited state \(|f\rangle\). Note that the \(\pi\)-amplitude is represented as ef_x180_amplitude.

  5. Measuring the resonator transmission and collecting the \(I\) and \(Q\) signals when the qubit is in \(|f\rangle\).

Analysis steps

  1. Training the discriminator using LDA (or QDA), thus obtaining the conditional probability \(\text{P}(\text{state}_\text{measured}|\\ \text{state}_\text{prepared})\) for each state.

  2. Compute the Probability Distribution Function for each measured state along the axis that connects the centers of the \(I\)-\(Q\) cluster data. The axes are computed using pairs of adjacent centers. In this notebook, we compute two axes: one connecting the centers of the \(I\)-\(Q\) data for \(|g\rangle\) and \(|e\rangle\) states (named x_01), and a second connecting the centers of the \(I\)-\(Q\) data for \(|e\rangle\) and \(|f\rangle\) states (named x_12).

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  1. Benyamin Ghojogh and Mark Crowley. Linear and quadratic discriminant analysis: tutorial. 2019. arXiv:1906.02590. ↩