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Nagy's unitary dilation theorem in operator theory asserts the possibility ofdilating 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 therecently devised Schr\"odingerisation approach and demonstrate its viability asan alternative dilation technique. This approach is applicable to operators inthe 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 andinfinite-dimensional cases, in both countable and uncountable domains. Forquantum systems lying in infinite dimensional Hilbert space, the dilationinvolves adding a single infinite dimensional mode, and this is thecontinuous-variable version of the Schr\"odingerisation procedure which makesit suitable for analog quantum computing. Furthermore, by discretisingcontinuous variables, the Schr\"odingerisation method can also be effectivelyemployed in finite-dimensional scenarios suitable for qubit-based quantumcomputing.

Dilation theorem via Schr\"odingerisation, with applications to the quantum simulation of differential equations