Instantaneous quantum computing is a sub-universal quantum complexity class, whose circuits have proven to be hard to simulate classically in the Discrete-Variable (DV) realm. We extend this proof to the Continuous-Variable (CV) domain by using squeezed states and homodyne detection, and by exploring the properties of post-selected circuits. In order to treat post-selection in CVs we consider finitely-resolved homodyne detectors, corresponding to a realistic scheme based on discrete probability distributions of the measurement outcomes. The unavoidable errors stemming from the use of finitely squeezed states are suppressed through a qubit-into-oscillator GKP encoding of quantum information, which was previously shown to enable fault-tolerant CV quantum computation. Finally, we show that, in order to render post-selected computational classes in CVs meaningful, a logarithmic scaling of the squeezing parameter with the circuit size is necessary, translating into a polynomial scaling of the input energy.
@article{arxiv.1607.07605,
title = {Continuous-Variable Instantaneous Quantum Computing is hard to sample},
author = {T. Douce and D. Markham and E. Kashefi and E. Diamanti and T. Coudreau and P. Milman and P. van Loock and G. Ferrini},
journal= {arXiv preprint arXiv:1607.07605},
year = {2017}
}