Formulate continuous-space quantum-walk and transducer theory
Develop a theory of quantum walks, transducers, and associated electrical-network constructions directly for continuous state spaces, and determine how to discretize the resulting continuous-variable quantum algorithms while controlling the effect of discretization.
References
An alternative approach would be to do discretization after quantization. I.e., develop the theory of quantum walks, transducers and electrical networks directly in the continuous-space setting,\footnote{For quantum walks this was already done to some extent in .} and then discretize the resulting continuous-variable quantum algorithm into a finite-dimensional one. While this is bound to yield an elegant quantization, it seems less clear how to properly handle the discretization and its impact.
An open question concerns the conversion of the transducer over those infinite Hilbert spaces into practical quantum algorithms. Either we will need to consider quantum computing architectures able to represent those Hilbert spaces faithfully (there exists many physical systems realizing infinite dimensional Hilbert spaces, such as bosonic systems where a mode is precisely described by Hilbert space \ell2(\mathbb{N})), or we could design techniques to embed the necessary unitaries inside finite dimensional Hilbert spaces, either exactly or via truncation, which could result in trade-offs between the space complexity and the error introduced by the truncation.