Singular-value decomposition using quantum annealing

In the present study, we demonstrate how to perform, using quantum annealing, the singular value decomposition and the principal component analysis. Quantum annealing gives a way to find a ground state of a system, while the singular value decomposition requires the maximum eigenstate. The key idea is to transform the sign of the final Hamiltonian, and the maximum eigenstate is obtained by quantum annealing. Furthermore, the adiabatic time scale is obtained by the approximation focusing on the maximum eigenvalue.

Figure 1

The logo of Tokyo University of Science (left-hand side). The binary image of the logo of TUS (right-hand side) is regarded as the information matrix $A$.

Figure 2

Singular value decomposition of the logo of TUS given by the quantum annealing method. All figures show decomposed $1328\times 1324$ matrices. The left-hand sides show each component $|u_j\rangle \langle v_j|$ while the right-hand sides show the summation $\sum \sqrt{{\lambda }_j}|u_j\rangle \langle v_j|$.

Figure 3

Eigenvalue properties of the ground state and the excited state. The parameters $K$ and $\alpha$ are assumed as $K=\alpha =0.5$. $\mathrm{\Delta }E\left(x\right)$ denotes the energy gap.

Figure 4

The functions $a\left(x\right)$ and $b\left(x\right)$.