Robust Quantum Algorithmic Binary Decision-Making on Gaussian Signals

A relevant signal in the quantum domain may manifest as a displacement or a squeezing operator in the bosonic phase space. For a real parameter $\beta$ embedded in such a Gaussian operator, the task of determining if $\beta \in [\beta_{-th}, \beta_{+th}]$ for real asymmetric thresholds $(\beta_{-th} \ne -\beta_{+th})$ is a binary decision problem. We propose a framework, the \emph{generalized quantum signal processing interferometry} (GQSPI), to solve this parameter detection problem by recasting the practical task of active binary hypothesis testing on quantum systems to a polynomial approximation problem. We achieve a small decision error probability $p_{\text{err}}$ on the order of $\mathcal{O}(\frac{1}{d}\log{(d)})$, with $d$ as the circuit depth. We analyze the protocol when (i) $\beta$ is a deterministic parameter, and (ii) when $\beta$ is drawn randomly from a known prior distribution. The GQSPI protocol is also shown to be robust under oscillator dephasing noise. We further extend our protocol from two thresholds to more general multi-threshold cases. Overall, the proposed framework enables decision-making over arbitrary thresholds for any general Gaussian signal in a single or a few shots.