  We present strictly efficient schemes for scalable measurement-based quantum
computing using continuous-variable systems: These schemes are based on
suitable non-Gaussian resource states, ones that can be prepared using
interactions of light with matter systems or even purely optically. Merely
Gaussian measurements such as optical homodyning as well as photon counting
measurements are required, on individual sites. These schemes overcome
limitations posed by Gaussian cluster states, which are known not to be
universal for quantum computations of unbounded length, unless one is willing
to scale the degree of squeezing with the total system size. We establish a
framework derived from tensor networks and matrix product states with infinite
physical dimension and finite auxiliary dimension general enough to provide a
framework for such schemes. Since in the discussed schemes the logical encoding
is finite-dimensional, tools of error correction are applicable. We also
identify some further limitations for any continuous-variable computing scheme
from which one can argue that no substantially easier ways of
continuous-variable measurement-based computing than the presented one can
exist.

Quantum Physics