  Quantum algorithm involves the manipulation of amplitudes and computational
basis, of which manipulating basis is largely a quantum analogue of classical
computing that is always a major contributor to the complexity. In order to
make full use of quantum mechanical speedup, more transformation should be
implemented on amplitudes. Here we propose the notion of quantum amplitude
arithmetic (QAA) that intent to evolve the quantum state by performing
arithmetic operations on amplitude. Based on the basic design of multiplication
and addition operations, QAA can be applied to solve the black-box quantum
state preparation problem and the quantum linear system problem with fairly low
complexity, and evaluate nonlinear functions on amplitudes directly. QAA is
expected to find applications in a variety of quantum algorithms.
