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

TL;DR

This paper introduces Schr"odingerisation as a versatile dilation technique for operators like $V(t)= ext{exp}(-At)$, applicable to both finite and infinite-dimensional quantum systems, enhancing quantum simulation of differential equations.

Contribution

It presents Schr"odingerisation as an alternative to Nagy's dilation theorem, applicable to infinite-dimensional operators, and suitable for both analog and digital quantum computing.

Findings

Schr"odingerisation effectively dilates operators in infinite-dimensional spaces.

The method is adaptable to both continuous-variable and qubit-based quantum systems.

It enables quantum simulation of differential equations in diverse settings.

Abstract

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.