Continuous-Variable Instantaneous Quantum Computing is hard to sample

TL;DR

This paper proves that continuous-variable instantaneous quantum computing is computationally hard to simulate classically, extending previous discrete-variable results by incorporating realistic measurement and error correction techniques.

Contribution

It extends the hardness proof of instantaneous quantum computing to the continuous-variable domain using squeezed states, homodyne detection, and GKP encoding, considering realistic measurement resolutions.

Findings

CV IQP is classically hard to simulate

Finitely-resolved homodyne detection models realistic measurements

Logarithmic squeezing scaling is necessary for meaningful CV post-selection

Abstract

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.