  Nagy's unitary dilation theorem in operator theory asserts the possibility of
dilating a contraction into a unitary operator. When used in quantum computing,
its practical implementation primarily relies on block-encoding techniques,
based on finite-dimensional scenarios. In this study, we delve into the
recently devised Schr\"odingerisation approach and demonstrate its viability as
an alternative dilation technique. This approach is applicable to operators in
the form of $V(t)=\exp(-At)$, which arises in wide-ranging applications,
particularly in solving linear ordinary and partial differential equations.
Importantly, the Schr\"odingerisation approach is adaptable to both finite and
infinite-dimensional cases, in both countable and uncountable domains. For
quantum systems lying in infinite dimensional Hilbert space, the dilation
involves adding a single infinite dimensional mode, and this is the
continuous-variable version of the Schr\"odingerisation procedure which makes
it suitable for analog quantum computing. Furthermore, by discretising
continuous variables, the Schr\"odingerisation method can also be effectively
employed in finite-dimensional scenarios suitable for qubit-based quantum
computing.
