title: Continuous-Variable Instantaneous Quantum Computing is Hard to Sample.
abstract: Instantaneous quantum computing is a subuniversal quantum complexity class, whose circuits have proven to be hard to simulate classically in the discrete-variable realm. We extend this proof to the continuous-variable (CV) domain by using squeezed states and homodyne detection, and by exploring the properties of postselected circuits. In order to treat postselection 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 Gottesman-Kitaev-Preskill encoding of quantum information, which was previously shown to enable fault-tolerant CV quantum computation. Finally, we show that, in order to render postselected 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.
authors.affiliation: Laboratoire Matériaux et Phénomènes Quantiques, Sorbonne Paris Cité, Université Paris Diderot, CNRS UMR 7162, 75013 Paris, France.|Laboratoire d'Informatique de Paris 6, CNRS, UPMC-Sorbonne Universités, 4 place Jussieu, 75005 Paris, France.
authors.forename: T
authors.initials: T
authors.lastname: Douce
authors.affiliation: Laboratoire d'Informatique de Paris 6, CNRS, UPMC-Sorbonne Universités, 4 place Jussieu, 75005 Paris, France.|LTCI, CNRS, Télécom ParisTech, Université Paris-Saclay, 75013 Paris, France.
authors.forename: D
authors.initials: D
authors.lastname: Markham
authors.affiliation: Laboratoire d'Informatique de Paris 6, CNRS, UPMC-Sorbonne Universités, 4 place Jussieu, 75005 Paris, France.|LTCI, CNRS, Télécom ParisTech, Université Paris-Saclay, 75013 Paris, France.|School of Informatics, University of Edinburgh, 10 Crichton Street, Edinburgh, EH8 9AB.
authors.forename: E
authors.initials: E
authors.lastname: Kashefi
authors.affiliation: Laboratoire d'Informatique de Paris 6, CNRS, UPMC-Sorbonne Universités, 4 place Jussieu, 75005 Paris, France.|LTCI, CNRS, Télécom ParisTech, Université Paris-Saclay, 75013 Paris, France.
authors.forename: E
authors.initials: E
authors.lastname: Diamanti
authors.affiliation: Laboratoire Matériaux et Phénomènes Quantiques, Sorbonne Paris Cité, Université Paris Diderot, CNRS UMR 7162, 75013 Paris, France.
authors.forename: T
authors.initials: T
authors.lastname: Coudreau
authors.affiliation: Laboratoire Matériaux et Phénomènes Quantiques, Sorbonne Paris Cité, Université Paris Diderot, CNRS UMR 7162, 75013 Paris, France.
authors.forename: P
authors.initials: P
authors.lastname: Milman
authors.affiliation: Institute of Physics, Johannes-Gutenberg Universität Mainz, Staudingerweg 7, 55128 Mainz, Germany.
authors.forename: P
authors.initials: P
authors.lastname: van Loock
authors.affiliation: Laboratoire Matériaux et Phénomènes Quantiques, Sorbonne Paris Cité, Université Paris Diderot, CNRS UMR 7162, 75013 Paris, France.|Institute of Physics, Johannes-Gutenberg Universität Mainz, Staudingerweg 7, 55128 Mainz, Germany.
authors.forename: G
authors.initials: G
authors.lastname: Ferrini