Robust Quantum Algorithmic Binary Decision-Making on Displacement Signals

TL;DR

This work addresses binary decision-making for quantum displacement signals encoded by $S_\beta = I \otimes e^{i\beta \hat p}$, asking whether $\beta$ lies in $[\beta_{-th},\beta_{+th}]$ using a hybrid qubit-oscillator platform. It introduces Generalized Quantum Signal Processing Interferometry (GQSPI), which sandwiches a degree-$d$ GQSP sequence around the displacement to realize a polynomial response $P(M=\downarrow|\beta)=\sum_{s=-d}^{d} c_s e^{i(2\kappa)\beta s}$, enabling asymmetric and multi-threshold detection with a target error $p_{err} \in \mathcal{O}(\frac{1}{\kappa d}\log d)$. The framework remains robust under oscillator dephasing, accommodates stochastic prior distributions for $\beta$, and scales to multiple threshold bands, as supported by analytic bounds and simulation results. By reframing quantum detection as a polynomial-approximation task, the paper connects quantum sensing with quantum algorithmic techniques to enable efficient, few-shot decision-making in realistic noisy settings.

Abstract

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

Figures (6)

• Figure 1: Quantum circuit for Generalized Quantum Signal Processing Interferometry (GQSPI). Note that for the qubit rotation gate in Eq. \ref{['eq:r']}, $R^{\dagger}(\theta, \phi, \lambda) = R(\theta, -\lambda, -\phi)$.
• Figure 2: Error analysis of the GQSPI protocol.
• Figure 3: Qubit repsonse function for asymmetric thresholding problem. $P(M = \downarrow | \beta)$ plotted against $\beta$ for $\kappa = \frac{1}{2048}$ for degrees $1,3, 6, 9, \text{and }13$ with the blue dotted lines representing the thresholds at $\beta_{-th} = -\frac{\pi}{8\kappa}$ and $\beta_{+th} = \frac{\pi}{4\kappa}$.
• Figure 4: Total probability of decision error $p_{err}$ vs GSQPI degree obtained from loss calculated for simulation results in Fig. \ref{['fig:GQSPI-Sim-Results']}. The red dots on log-log plot indicate the $p_{err}$ value for the given GQSP degree. The trendlines for $\frac{1}{d}, \frac{1}{d^\alpha}$ and fitting line for $\frac{1}{d}\log{d}$ are plotted against the given GQSP degrees to show the general trend of non-linear error suppression with increasing degree.
• Figure 5: Qubit response function for the multi-thresholding problem. $P(M = \downarrow | \beta)$ plotted against $\beta$ for $\kappa = \frac{1}{2048}$ for degrees $9, 13, \text{ and }15$ with the blue dotted lines representing the thresholds at $\beta_{1} = -0.75\frac{\pi}{2\kappa}$, $\beta_{2} = -0.45\frac{\pi}{2\kappa}, \beta_{3} = -0.125\frac{\pi}{2\kappa}$, $\beta_{4} = 0.75\frac{\pi}{2\kappa}$. The region for which we wish to detect $\beta$ is $[\beta_{1},\beta_{2}] \cup [\beta_{3},\beta_{4}]$.

Theorems & Definitions (2)

• Theorem 1
• Theorem 2