Figure 3

Photonic number-dependent expressive power of the variational linear quantum photonic circuit. (a) Optimal fits of a degree three Fourier series \(g(x) = \sum_{n=-3}^{3} c_{n} e^{-nix}\) with coefficients \(c_{0} = 0.2\), \(c_{1} = 0.69+0.52i\), \(c_{2} = 0.81+0.41i\) and \(c_{3} = 0.68+0.82i\) using a three mode circuit with photon number-resolving (PNR) detectors and different input Fock states, i.e. , , and . A perfect fit is achieved using at least three input photons, since a sufficient number of non-zero frequencies are encoded, i.e. \(D_{n} = 3\). (b) The number of real degrees of freedom of the three mode parameterized quantum photonic circuit with PNR detectors (black) and threshold detectors (red). The former is always larger than the minimum number of parameters \(M_{\mathrm{min}}\) required to control all the circuit’s Fourier coefficients (blue), and hence, arbitrary Fourier coefficients can be realized by this circuit. In contrast, the expressive power of the threshold detectors (red) can only be enhanced for up to 9 input photons