Introduction
The quantum kernel have many interesting aspect including speed and expressibility. Because of some practical issues, mapping the data into quantum kernel is easier than handling the data itself on the quantum computer. Therefore, for the contemporary quantum data science, using the quantum kernel is more easier way to handling the raw data.
In addition, the quantum kernel is fundamentally connected with the complex space. In the quantum circuit, there are a lot of gates handled at the complex fields. Therefore, the quantum kernel might handle larger space than the classical kernel. Moreover, according to the Euler’s formula \(e^{i \theta} = \cos \theta + i \sin \theta\), the quantum kernel have a periodic property. Therefore, the quantum kernel including such form might fit for some data with specific patterns. For example, any hermitian matrix \(H\) could be implemented into quantum gate which have to be unitary gates by using the form of matrix exponential \(e^{iH} (; (e^{ih})^{\dagger} e^{ih} = I)\)
However, there are only a few examples which shows the more detailed or visible potential of the quantum kernel. Therefore, we aim to figure out much more better examples which can show the power of the quantum kernel.
We’ll apply the quantum kernel into various dimensionality reduction techniques including Kernel Principal Components Analysis(KPCA), Generalized Sliced Inverse Regression(GSIR) and Generalized Sliced Average Variance Estimator(GSAVE). We choose the fields of dimension reduction to explore the potential of the quantum kernel because the the methods can be used to find the shape of the essential data and visualize complicated data.
There are various posting at my blog which explain the above concepts.
To handle the problem in detail, we’ll focus on solving the following specific questions.
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Quantum Kernel vs Quantum Kernel
What kind of the quantum kernels show consistently good performance?
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Quantum Kernel vs Classical Kernel
Can the quantum kernel show better performance than the classical kernel?
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Best Data Structure for the quantum kernel
What kind of the data could work well with the quantum kernel?
While solving the above questions, we’ll also handle the following subjects
- The effect of hyper parameters of the quantum kernels.
- Expressibility and trainability of the quantum kernels
- The periodic properties of the quantum kernels.
- The relation between the quantum kernel and wavelet kernel.
Keywords Map
While do the experiments, this posting is used as archive containing ideas, theories and results.