Exact Fock Amplitudes of the Cubic-Phase Gate on Arbitrary Pure Gaussian States
Abstract
The cubic phase gate is the only non-Gaussian element of universal continuous-variable quantum computation. Its Fock amplitudes are known in closed form only when the input is a Fock state. For a Gaussian input the usual route is to build the cubic quadrature in a truncated Fock space and exponentiate it, which alters the generator before the exponential is taken. A closed form is obtained here for every pure single-mode Gaussian input, with the squeezing along any quadrature axis and with arbitrary displacement and boost. It is a finite combination of the Airy function and its derivative, and the argument is shared by all Fock indices, so a whole Fock profile costs two Airy evaluations. A recurrence is also derived that builds each coefficient from the four preceding ones, together with the regime in which it is stable. Evaluating the closed form at the weak nonlinearities of current experiments requires care, since its terms grow large and alternate in sign. The precision this demands is quantified, and the truncated construction is benchmarked against the exact result, whose error at cutoffs in common use is found to be of the same size as the quantity being computed.
Related
- Exact Ionization Amplitudes for a Delta-Function Well in an Arbitrary-Strength DC Electric Field
- Tradeoff between Wigner negativity and decoherence time for cubic Gaussian states
- Exact Entanglement Swapping through Single-Occupancy Measurements in Gaussian Fermion States
- Exact factorization of finite-range three-body matrix elements between Gaussian wave packets and its application to the 0+ states of 12C
- Closed-form weak-wave propagation pressure amplitudes for arbitrary power-law attenuation
BibTeX
@misc{cordshooli2026exact,
title = {{Exact Fock Amplitudes of the Cubic-Phase Gate on Arbitrary Pure Gaussian States}},
author = {Ghasem Asadi Cordshooli},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13077},
eprint = {2609.13077},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13077},
}
RIS
TY - UNPB
TI - Exact Fock Amplitudes of the Cubic-Phase Gate on Arbitrary Pure Gaussian States
AU - Ghasem Asadi Cordshooli
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.13077
UR - https://arxiv.org/abs/2609.13077
AB - The cubic phase gate is the only non-Gaussian element of universal continuous-variable quantum computation. Its Fock amplitudes are known in closed form only when the input is a Fock state. For a Gaussian input the usual route is to build the cubic quadrature in a truncated Fock space and exponentiate it, which alters the generator before the exponential is taken. A closed form is obtained here for every pure single-mode Gaussian input, with the squeezing along any quadrature axis and with arbitrary displacement and boost. It is a finite combination of the Airy function and its derivative, and the argument is shared by all Fock indices, so a whole Fock profile costs two Airy evaluations. A recurrence is also derived that builds each coefficient from the four preceding ones, together with the regime in which it is stable. Evaluating the closed form at the weak nonlinearities of current experiments requires care, since its terms grow large and alternate in sign. The precision this demands is quantified, and the truncated construction is benchmarked against the exact result, whose error at cutoffs in common use is found to be of the same size as the quantity being computed.
ER -