# Quantum Support Vector Machines (QSVM)¶

Classiq's QSVM package allows the user to classify labeled data by using the relevant feature map.

## Introduction and Background¶

Quantum Support Vector Machines is the Quantum version of SVM - a data classification method which separates the data using a hyperplane.

This algorithm takes the following steps:

• Map the data into a different hyperspace (since the data may be non-linearly-separable in the original space)
• In the case of QSVM - mapping the classical data into Hilbert space.
• Calculate the kernel matrix
• The kernel entries are the fidelities between different feature vectors
• In the case of QSVM - this is done on a Quantum computer.
• Optimize the dual problem (this is always done classically)

$L_D(\alpha) = \sum_{i=1}^t \alpha_i - \frac{1}{2} \sum_{i,j=1}^t y_i y_j \alpha_i \alpha_k K(\vec{x}_i \vec{x}_j)$
• Where $$t$$ is the amount of data points
• the $$\vec{x}_i$$s are the data points
• $$y_i$$ is the label $$\in \{-1,1\}$$ of each data point
• $$K(\vec{x}_i \vec{x}_j)$$ is the kernel matrix element between the $$i$$ and $$j$$ data points
• and we optimize over the $$\alpha$$s
• We expect most of the $$\alpha$$s to be $$0$$. The $$\vec{x}_i$$s that correspond to non-zero $$\alpha_i$$ are called the Support Vectors.
• Finally, we may predict unlabeled data by calculating the kernel matrix of the new datum with respect to the support vectors. $$\text{Predicted Label}(\vec{s}) = \text{sign} \left( \sum_{i=1}^t y_i \alpha_i^* K(\vec{x}_i , \vec{s}) + b \right),$$ Where
• $$\vec{s}$$ is the data point to be classified
• $$\alpha_i^*$$ are the optimized $$\alpha$$s
• And $$b$$ is the bias.

Reference:

[1] Havlíček, V., Córcoles, A.D., Temme, K. et al. Supervised learning with quantum-enhanced feature spaces. Nature 567, 209-212 (2019). https://doi.org/10.1038/s41586-019-0980-2

## Creating a QSVM object¶

The Classiq API allows one to create a QSVM object, which handles the 3 steps of the algorithm:

1. Train
2. Test
3. Predict

First, let us create a QSVM instance.

When creating a QSVM instance, one must supply which feature map will be used.

A feature map is a way to encode classical data into quantum. Here, we chose to encode the data onto the surface of the bloch sphere. Behind the scenes, this can be translated (in 2 dimensions) to:

Where x is the 2D input vector, and the circuit takes a single qubit per data-point.

from classiq.applications.qsvm import QSVM

qsvm = QSVM("bloch_sphere")


### Training QSVM¶

In order to train, we pass data and labels, which may be either a list or a np.array.

qsvm.train(data, labels)


### Testing QSVM¶

In order to test, we pass data and labels, which may be either a list or a np.array. In return, we get an accuracy (float). between 0 to 1, indicating how good our training was (1 is better)

accuracy: float = qsvm.test(data, labels)


### Predicting QSVM¶

The predict method takes in data, and returns labels.

predicted_labels: np.ndarray = qsvm.predict(predict_input)


### Putting all the steps together¶

By passing all the data in the constructor, we may call a single method (.run) to handle all of the steps for us. For example:

from classiq.applications.qsvm import QSVM

qsvm = QSVM(
"bloch_sphere", train_data, train_labels, test_data, test_labels, predict_data
)
qsvm.run()


The above code will sequentially run train -> test -> predict.

If we were to pass only train and test data, then only these methods would run, and predict wouldn't.