Quantum Kernel t-Distributed Stochastic Neighbor Embedding
Yoshiaki Kawase Kosuke Mitarai Keisuke Fujii
Data visualization is important in understanding the characteristics of data
that are difficult to see directly. It is used to visualize loss landscapes and
optimization trajectories to analyze optimization performance. Popular
optimization analysis is performed by visualizing a loss landscape around the
reached local or global minimum using principal component analysis. However,
this visualization depends on the variational parameters of a quantum circuit
rather than quantum states, which makes it difficult to understand the
mechanism of optimization process through the property of quantum states. Here,
we propose a quantum data visualization method using quantum kernels, which
enables us to offer fast and highly accurate visualization of quantum states.
In our numerical experiments, we visualize hand-written digits dataset and
apply $k$-nearest neighbor algorithm to the low-dimensional data to
quantitatively evaluate our proposed method compared with a classical kernel
method. As a result, our proposed method achieves comparable accuracy to the
state-of-the-art classical kernel method, meaning that the proposed
visualization method based on quantum machine learning does not degrade the
separability of the input higher dimensional data. Furthermore, we visualize
the optimization trajectories of finding the ground states of transverse field
Ising model and successfully find the trajectory characteristics. Since quantum
states are higher dimensional objects that can only be seen via observables,
our visualization method, which inherits the similarity of quantum data, would
be useful in understanding the behavior of quantum circuits and algorithms.