  We develop three new methods to implement any Linear Combination of Unitaries
(LCU), a powerful quantum algorithmic tool with diverse applications. While the
standard LCU procedure requires several ancilla qubits and sophisticated
multi-qubit controlled operations, our methods consume significantly fewer
quantum resources. The first method (Single-Ancilla LCU) estimates expectation
values of observables with respect to any quantum state prepared by an LCU
procedure while requiring only a single ancilla qubit, and no multi-qubit
controlled operations. The second approach (Analog LCU) is a simple, physically
motivated, continuous-time analogue of LCU, tailored to hybrid qubit-qumode
systems. The third method (Ancilla-free LCU) requires no ancilla qubit at all
and is useful when we are interested in the projection of a quantum state
(prepared by the LCU procedure) in some subspace of interest. We apply the
first two techniques to develop new quantum algorithms for a wide range of
practical problems, ranging from Hamiltonian simulation, ground state
preparation and property estimation, and quantum linear systems. Remarkably,
despite consuming fewer quantum resources they retain a provable quantum
advantage. The third technique allows us to connect discrete and
continuous-time quantum walks with their classical counterparts. It also
unifies the recently developed optimal quantum spatial search algorithms in
both these frameworks, and leads to the development of new ones that require
fewer ancilla qubits. Overall, our results are quite generic and can be readily
applied to other problems, even beyond those considered here.
