Multimode cubic phase gate for continuous variables quantum computing: a case of spectrally and spatially non-separable modes

Photonic quantum processors that encode quantum information in continuous variables (CV) – amplitude and momentum quadratures, are among the leading quantum computational platforms. The universal set of CV quantum gates [2] includes single-mode Gaussian operations, a two-mode Gaussian operation and a single-mode non-Gaussian operation. The cubic phase gate
$$ \hat{U}_{\text{cp}}(\gamma) = e^{i\gamma\hat{x}^3} $$
is the lowest-order non-Gaussian gate that has been intensively studied recently. The lack of strong optical nonlinearities makes a measurement-induced nonlinearity approach as in Refs. [3], based on an ancilla cubic phase state
$$ {\vert \gamma \rangle} = \int e^{i\gamma p^3}{\vert p \rangle}dp, $$
a proper quadrature measurement and feed-forward corrections, the leading one, highlighted by recent experimental measurement of a nonlinear quadrature [4].
Multimode light is a promising platform to realize scalable quantum computing. Therefore, it is important to realize non-Gaussian operations, such as the cubic phase gate, in multimode systems to make the computations both scalable and universal. In this talk, we discuss an extension of the measurement-induced cubic phase gate to multimode scenario. To demonstrate key properties of such a gate, we consider its two-mode realization in more details. We present analytical results for the output two-mode state considering pure two-mode input and pure two-mode ancilla states and show how the cubic phase gate can be realized for each mode without spectral or spatial mode separation. We also calculate the fidelity of the cubic phase gate for non-ideal cubic phase ancillas. Finally, we discuss how the scheme can be scaled to a higher number of modes.