  In this paper, we firstly briefly review the duality quantum computer.
Distinctly, the generalized quantum gates, the basic evolution operators in a
duality quantum computer are no longer unitary, and they can be expressed in
terms of linear combinations of unitary operators. All linear bounded operators
can be realized in a duality quantum computer, and unitary operators are just
the extreme points of the set of generalized quantum gates. A d-slits duality
quantum computer can be realized in an ordinary quantum computer with an
additional qudit using the duality quantum computing mode. Duality quantum
computer provides flexibility and clear physical picture in designing quantum
algorithms, serving as a useful bridge between quantum and classical
algorithms. In this review, we will show that duality quantum computer can
simulate quantum systems more efficiently than ordinary quantum computers by
providing descriptions of the recent efficient quantum simulation algorithms of
Childs et al [Quantum Information & Computation, 12(11-12): 901-924 (2012)] for
the fast simulation of quantum systems with a sparse Hamiltonian, and the
quantum simulation algorithm by Berry et al [Phys. Rev. Lett. 114, 090502
(2015)], which provides exponential improvement in precision for simulating
systems with a sparse Hamiltonian.
