  Hamiltonian diagonalization is at the heart of understanding physical
properties and practical applications of quantum systems. It is highly desired
to design quantum algorithms that can speedup Hamiltonian diagonalization,
especially those can be implemented on near-term quantum devices. In this work,
we propose a variational algorithm for Hamiltonians diagonalization (VQHD) of
quantum systems, which explores the important physical properties, such as
temperature, locality and correlation, of the system. The key idea is that the
thermal states of the system encode the information of eigenvalues and
eigenstates of the system Hamiltonian. To obtain the full spectrum of the
Hamiltonian, we use a quantum imaginary time evolution algorithm with high
temperature, which prepares a thermal state with a small correlation length.
With Trotterization, this then allows us to implement each step of imaginary
time evolution by a local unitary transformation on only a small number of
sites. Diagonalizing these thermal states hence leads to a full knowledge of
the Hamiltonian eigensystem. We apply our algorithm to diagonalize local
Hamiltonians and return results with high precision. Our VQHD algorithm sheds
new light on the applications of near-term quantum computers.
