  Implementing quantum algorithms is essential for quantum computation. We
study the implementation of three quantum algorithms by performing homodyne
measurements on a two-dimensional temporal continuous-variable cluster state.
We first review the generation of temporal cluster states and the
implementation of gates using the measurement-based model. Alongside this we
discuss methods to introduce non-Gaussianity into the cluster states. The first
algorithm we consider is Gaussian Boson Sampling in which only Gaussian
unitaries need to be implemented. Taking into account the fact that input
states are also Gaussian, the errors due to the effect of finite squeezing can
be corrected, provided a moderate amount of online squeezing is available. This
helps to construct a large Gaussian Boson Sampling machine. The second
algorithm is the continuous-variable Instantaneous Quantum Polynomial circuit
in which one needs to implement non-Gaussian gates, such as the cubic phase
gate. We discuss several methods of implementing the cubic phase gate and fit
them into the temporal cluster state architecture. The third algorithm is the
continuous-variable version of Grover's search algorithm, the main challenge of
which is the implementation of the inversion operator. We propose a method to
implement the inversion operator by injecting a resource state into a
teleportation circuit. The resource state is simulated using the Strawberry
Fields quantum software package.
