  In contexts where relevant problems can easily attain configuration spaces of
enormous sizes, solving Linear Differential Equations (LDEs) can become a hard
achievement for classical computers; on the other hand, the rise of quantum
hardware can conceptually enable such high-dimensional problems to be solved
with a foreseeable number of qubits, whilst also yielding quantum advantage in
terms of time complexity. Nevertheless, in order to bridge towards experimental
realizations with several qubits and harvest such potential in a short-term
basis, one must dispose of efficient quantum algorithms that are compatible
with near-term projections of state-of-the-art hardware, in terms of both
techniques and limitations. As the conception of such algorithms is no trivial
task, insights on new heuristics are welcomed. This work proposes a novel
approach by using the Quantum Amplitude Damping operation as a resource, in
order to construct an efficient quantum algorithm for solving homogeneous LDEs.
As the intended implementation involves performing Amplitude Damping
exclusively via a simple equivalent quantum circuit, our algorithm shall be
given by a gate-level quantum circuit (predominantly composed of elementary
2-qubit gates) and is particularly nonrestrictive in terms of connectivity
within and between some of its main quantum registers. We show that such an
open quantum system-inspired circuitry allows for constructing the real
exponential terms in the solution in a non-interferometric way; we also provide
a guideline for guaranteeing a lower bound on the probability of success for
each realization, by exploring the decay properties of the underlying quantum
operation.
