Dilation Theorem Via Schrödingerization, With Applications to the Quantum Simulation of Differential Equations

Nagy's unitary dilation theorem in the 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 Schrödingerization approach and demonstrate its viability as an alternative dilation technique. This approach is applicable to operators in the form of $V\left(t\right)=\exp (-At)$, which arises in wide-ranging applications, particularly in solving linear ordinary and partial differential equations. Importantly, the Schrödingerization 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 Schrödingerization procedure which makes it suitable for analog quantum computing. Furthermore, by discretizing continuous variables, the Schrödingerization method can also be effectively employed in finite-dimensional scenarios suitable for qubit-based quantum computing.