  Computing finite temperature properties of a quantum many-body system is key
to describing a broad range of correlated quantum many-body physics from
quantum chemistry and condensed matter to thermal quantum field theories.
Quantum computing with rapid developments in recent years has a huge potential
to impact the computation of quantum thermodynamics. To fulfill the potential
impacts, it is crucial to design quantum algorithms that utilize the
computation power of the quantum computing devices. Here we present a quantum
kernel function expansion (QKFE) algorithm for solving thermodynamic properties
of quantum many-body systems. In this quantum algorithm, the many-body density
of states is approximated by a kernel-Fourier expansion, whose expansion
moments are obtained by random state sampling and quantum interferometric
measurements. As compared to its classical counterpart, namely the kernel
polynomial method (KPM), QKFE has an exponential advantage in the cost of both
time and memory. In computing low temperature properties, QKFE becomes
inefficient, as similar to classical KPM. To resolve this difficulty, we
further construct a thermal ensemble and approaches the low temperature regime
step-by-step. For quantum Hamiltonians, whose ground states are preparable with
polynomial quantum circuits, THEI has an overall polynomial complexity. We
demonstrate its efficiency with applications to one and two-dimensional quantum
spin models, and a fermionic lattice. With our analysis on the realization with
digital and analogue quantum devices, we expect the quantum algorithm is
accessible to current quantum technology.
