Exact boson sampling using Gaussian continuous-variable measurements

Boson sampling is a quantum mechanical task involving Fock basis state preparation and detection and evolution using only linear interactions. A classical algorithm for producing samples from this quantum task cannot be efficient unless the polynomial hierarchy of complexity classes collapses, a situation believed to be highly implausible. We present a method for constructing a device which uses Fock state preparations, linear interactions, and Gaussian continuous-variable measurements for which one can show that exact sampling would be hard for a classical algorithm in the same way as boson sampling. The detection events used from this arrangement do not allow a similar conclusion to be drawn for the classical hardness of approximate sampling. We discuss the details of this result outlining some specific properties required by approximate sampling hardness.

Figure 1

CV-n measurement device. The outer green dashed box encloses the whole CV-n device, which measures the input signal in the displaced Fock state basis. The inner red boxes represent two homodyne detections performed simultaneously that give two CV outcomes $x_{1,\theta }$ and $p_{2,\theta }$ from the measurement device.

Figure 2

Comparison between $\eta \left(t\right)$ (solid line) and $p_D\left(t\right)$ (dashed line) for the CV-1 detector.

Figure 3

Boson sampling using CV-1 measurements.