On Classifying Images using Quantum Image Representation

FRQI encodes the image data into a quantum state given by:

Here, $\left|0\right>$ and $\left|1\right>$ are single qubit computational basis states and $\left|i\right>,i=0,1,\dots,2^{2n}-1$ are $2n$ qubit computational basis states. The $\cos(\theta_{i})\left|0\right>+\sin(\theta_{i})\left|1\right>$ part is used to encode the pixel values while $\left|i\right>$ encodes the pixel location.

The circuit to encode the image can be constructed using Hadamard ($H$) and Control Rotation, $C^{2n}R_{y}(2\theta)$, gates. It needs to be measured multiple times to get back the image from the state. The image retrieval process is probabilistic, and the result will depend on the number of shots used.

The number of qubits used in this representation is $2n+1$ with $2n$ qubits to encode the pixel location and 1 qubit to encode the pixel values. Pixel values are encoded as angles and are thus scaled to fit in the range $\left[0,\frac{\pi}{2}\right]$.

MCQI representation uses $2n+3$ qubits to encode colour images while using $2n$ qubits to encode the pixel location like FRQI and the 3 remaining qubits to encode the pixel values of the RGB channels. This encoding is inspired by FRQI.
MCQI encodes the image into a quantum state given by:

The color information is encoded in:

Colour encoding angle is applied to the R channel qubit using Control Rotation ($C^{2}R_{y}(2\theta)$) gates where it is controlled by the G and B channel qubits, and for each pixel, 3 $C^{2n}(C^{2}R_{y}(2\theta))$ gates are applied to encode the position and value information. The $\theta$ values are calculated from pixel values:

where, $p$ is the pixel values $\in[0,1]$. We get in this range by dividing the integer pixel values $\in[0,255]$ by 255.

As before, the image retrieval process is probabilistic and depends on the number of shots.

To classify the images, we need some value to distinguish the two classes. The Z expectation value $(ez)$ of the first qubit gives a natural split. We apply a variational ansatz on the quantum image and measure the expectation value of the colour qubit for FRQI and the R channel qubit for MCQI. The expectation value lies in the range $[-1,1]$, and thus a split can be formed such that:

where $s$ is the split set to 0 by default but can be trained to get optimal results.

An autoencoder is a tool to reduce the dimension of data. We use the autoencoder’s quantum analogue [4] to compress the image state into a single qubit state. To use it as a classifier, we train the autoencoder to only compress the positive class. The autoencoder is trained by maximising the fidelity of the trash qubits with the zero state (${\left|0\right>}^{\otimes T}$) where $T$ is the number of trash qubits. Once the autoencoder is trained on the positive class of the training data, we then use it to compress the validation data. We then measure the fidelity of the trash qubits again with the zero state and classify them depending on the resulting fidelity. The positive class should have higher fidelity values than the negative class. A single layer of the autoencoder is shown in Fig 2.

This dataset contains black and white images of dimension $2^{n}\times 2^{n}$. Example images are shown in Fig 3 for $n=5$. These images are randomly generated. Horizontal stripes are one class, and vertical bars are another class. This dataset is used for binary classification.

This is the famous dataset of handwritten digits [5]. We have used this for both binary (0 and 1) and multi-class (0, 1 and 2) classification. The original images are of $28\times 28$ dimension, which first needs to be squared into $2^{n}\times 2^{n}$. Bilinear interpolation is used to transform the data for different $n$. There also exists another version of this data which contains 15 different corruption variations [6]. We also classify these corrupted datasets. Fig 4 shows the MNIST images, and Fig 5 shows the corrupted images.

This is randomly generated $2\times 2$ colour image data for classification. The image has 4 pixels of random colours. For positive class we change the pixel values of the $4^{th}$ pixel to $(0,0,0)$. This makes the pixel black. The classification problem is then to differentiate between images with and without a black pixel. The pixel can also be modified to have dark shades instead of absolute 0 values. Fig 6 shows example images for different shades.

Table 1 shows accuracies on the validation set using different training dataset sizes and $n$ values for the BAS data.

Table 2 shows accuracies on the validation set using different training dataset sizes and $n$ values for the MNIST data. For AC we have used 100 epochs for $n<3$ with MNIST data. Table 3 shows accuracies on the validation set for the MNIST Corruption data for $n=4$. Table 4 shows the results for Multi-Class classification between 0, 1 and 2 digit images using the VQC. For this, we split the $ez$ range into 3 parts.

Table 5 shows accuracy on the validation set using different training dataset sizes and shade values for the $2\times 2$ colour image data. As expected the accuracy decreases as we increase the shade (see Fig 7). This is because, higher shade values imply smaller difference between the classes with a value of 255 implying no difference between the two classes.