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. ↩