Quantum Machine Learning over Infinite Dimensions

Machine learning is a fascinating and exciting field within computer science. Recently, this excitement has been transferred to the quantum information realm. Currently, all proposals for the quantum version of machine learning utilize the finite-dimensional substrate of discrete variables. Here we generalize quantum machine learning to the more complex, but still remarkably practical, infinite-dimensional systems. We present the critical subroutines of quantum machine learning algorithms for an all-photonic continuous-variable quantum computer that can lead to exponential speedups in situations where classical algorithms scale polynomially. Finally, we also map out an experimental implementation which can be used as a blueprint for future photonic demonstrations.

Figure 1

All-photonic implementation schematic of the operator $\exp (i\theta \mathcal{S})=C_{\mathcal{S}}^{c{c}^{\prime }}R(\theta )C_{\mathcal{S}}^{c{c}^{\prime }}$. We initially have an ancillary input mode $|+⟩=(|01⟩+|10⟩)/\sqrt{2}$ with two (swap) modes $C$ and $C^{\prime }$ used to implement the operators given in Eq. (4). The method for generating $|+⟩$ is one of many possibilities, e.g., preparing a heralded superposition of polarization states is illustrated. ${\chi }^{(2)}$, nonlinear crystal source; APD, avalanche photodiode detector; $50/50$, balanced beam splitter; $\lambda /2$, half wave plate; PBS, polarizing beam splitter; $C_S^{C{C}^{\prime }}$, controlled-swap operator; $R(\theta )$, rotation operator; see the text for an explanation of the operators. Note that $C_{\mathcal{S}}^{c{c}^{\prime }}$ can be implemented with the quartic gate [38, 47] and $R(\theta )$ can be efficiently implemented using linear optics.