  Duality quantum computing (DQC) offers the use of linear combination of
unitaries (LCU), or generalized quantum gates, in designing quantum algorithms.
DQC contains wave divider and wave combiner operations. The wave function of a
quantum computer is split into several subwaves after the wave division
operation. Then different unitary operations are performed on different
subwaves in parallel. A quantum wave combiner combines the subwaves into a
final wave function, so that a linear combination of the unitaries are
performed on the final state. In this paper, we study of the properties of
duality quantum computer with projections on subwaves. In subwave-projection
DQC (SWP-DQC), we can realize the linear combinations of non-unitaries, and
this not only gives further flexibility for designing quantum algorithms, but
also offers additional speedup in the expected time complexity. Specifically,
SWP-DQC offers an O(M) acceleration over DQC with only final-wave-projection in
the mean time complexity, where M is the number of projections. As an
application, we show that the ground state preparation algorithm recently
proposed by Ge, Tura, and Cirac is actually an DQC algorithm, and we further
optimized the algorithm using SWP-DQC, which can save up to $\log_2 N$ qubits
compared DQC without subwave projection, where N is the dimension of the
system's Hilbert Space.
