End-to-end Optimization of Single-Shot Quantum Machine Learning for Bayesian Inference

TL;DR

This work develops a data-driven, end-to-end framework for quantum sensing under finite measurement budgets by treating the quantum device as a noisy input–output map and co-optimizing state preparation, encoding, measurement, and classical post-processing within a Bayesian quantum metrology context. It introduces a six-component PNN sensing pipeline, links performance to the Resolvable Expressive Capacity (REC), and uses eigentasks to identify robust, low-dimensional feature combinations for single-shot function inference. The method achieves near-optimal single-shot performance (e.g., $\mathbb{E}_{u}[{\rm MSE}] \approx -19.1$ dB, close to $-20$ dB) for $L=32$ qubits and demonstrates clear computational-sensing advantages for direct function inference over indirect parameter estimation, including nonlinear targets like $\sin(10u)$. Overall, the paper provides a practical, hardware-aware route to on-device quantum sensing with finite resources, and suggests future work on noisy devices, hardware-aware ansätze, and nonlinear readouts.

Abstract

We introduce an end-to-end optimization strategy for quantum machine learning that directly targets performance under finite measurement resources, where learning objectives are defined directly at the level of task performance. The method is applied on a Bayesian quantum metrology task since it provides a natural testbed with known fundamental limits and scaling with system size. The sampling-aware hybrid algorithm achieves a single-shot risk within 1 dB of the -20 dB Bayesian limit using 32 qubits. We extend the Bayesian framework from parameter estimation to global function inference, where the task is to infer a target function of the sensor input drawn from an arbitrary prior, and we demonstrate a clear computational-sensing advantage for direct functional inference over indirect reconstruction. We relate the corresponding Bayesian risk to the Capacity metric and argue that the Resolvable Expressive Capacity provides a natural measure of the space of functions accessible in a single shot. The resulting eigentask analysis identifies noise-robust feature combinations that yield compact estimators with improved accuracy and reduced optimization cost in resource-limited or real-time on-device settings.

Figures (9)

• Figure 1: Physical neural network approach to quantum sensing. $L$ initially uncorrelated qubits are prepared in a metrologically useful state $\rho_{\rm en} = \mathcal{U}_{\rm {en}}(\bm{\theta_{ \rm en}})\rho_0$ via a parameterized entangling superoperator $\mathcal{U}_{\rm {en}}(\bm{\theta_{ \rm en}})$. In turn, the signal input $\bm{u}$, drawn from a training dataset, is embedded on the quantum system. A parameterized decoder $\mathcal{U}_{\rm {de}}(\bm{\theta_{\rm de}})$ is applied to the encoded state, followed by projective measurements in the computational basis. Collecting the measurement features allows to construct an estimation $f(\bm{u})$ of the target function $f^*(\bm{u})$ and calculate the objective loss function. Optimizing such loss function allow us to train the internal parameter of the quantum circuit as well as the estimator enabling the optimization of the whole sensing pipeline. We note that although our framework places no restriction on the form of the entangling, embedding and decoding operations and allows them to be arbitrary quantum maps, in present work we focus on unitary implementations.
• Figure 2: Results of the optimization for approximating the target function $f(u)=\sin(10u)$ where $u$ is sampled from a Gaussian prior of width $\sigma=0.7981$ and for a circuit with $n_{\rm en}=1$, $n_{\rm de}=0$, $L=32$, and $S=1$. (a) Mean squared error, ${\rm MSE}(u)$, as a function of the true input phase $u$ by indirect approximation of the target function, achieving $\mathbb{E}_{\bm{u}}\!\!\left[{\rm MSE_1}\right] \approx -2.9~{\rm dB}$ (b) Mean squared error, ${\rm MSE}(u)$, as a function of the true input phase $u$ by direct approximation of the target function achieving $\mathbb{E}_{\bm{u}}\!\!\left[{\rm MSE_2}\right] \approx -4.7~{\rm dB}$ (see Sec. \ref{['sec:results']} of the main text).
• Figure 3: Results of the optimization for approximating the target function $f(u)=u$ where $u$ is sampled from a symmetric mixture of Gaussian prior distributions of width $\sigma^2=0.05$ and for a circuit with $n_{\rm en}=1$, $n_{\rm de}=0$, $L=32$, and $S=1$. Mean squared error, ${\rm MSE}(u)$, as a function of the true input phase $u$. The dashed grey lines correspond to the standard deviation around the respective means. The achieved precision is $\mathbb{E}_{\bm{u}}\!\!\left[{\rm MSE}\right] \approx -18.3~{\rm dB}$ compared to the optimal allowed $\mathbb{E}_{\bm{u}}\!\!\left[{\rm MSE}\right]_{\rm ult} \approx -22.5~{\rm dB}$.
• Figure 4: Results of the optimization for $n_{\rm en}=1$, $n_{\rm de}=2$, $\sigma=0.7981$, $L=32$, and $S=1$. (a) Predicted output $u_{\rm pred}$ (orange dots) as a function of the input phase $u$ for a single experimental repetition. The target function $f^*(u) = u$ is shown in blue. Each dot corresponds to a distinct measurement outcome, highlighting the stochasticity of the single-shot regime. (b) Averaged prediction $\langle u_{\rm pred} \rangle$ over many repetitions, demonstrating convergence to the target function in a wide range around $u=0$. (c) Mean squared error ${\rm MSE}(u)$ as a function of the true phase $u$, quantifying the estimator's performance across the domain. Importantly, the achieved $\mathbb{E}_{\bm{u}}\!\!\left[{\rm MSE}\right] \approx -19.1~{\rm dB}$ is remarkably close to the ultimate bound of $\mathbb{E}_{\bm{u}}\!\!\left[{\rm MSE}\right]_{\rm ult} \approx -20~{\rm dB}$ allowed by quantum mechanics.
• Figure 5: Results of the optimization for various circuit ansätze. Orange squares correspond to a circuit with depth $n_{\rm en} = 0$ and $n_{\rm de} = 0$, green triangles to $n_{\rm en} = 1$ and $n_{\rm de} = 0$, purple crosses to $n_{\rm en} = 0$ and $n_{\rm de} = 2$, and red dots to $n_{\rm en} = 1$ and $n_{\rm de} = 2$. (a) Average predicted output as a function of $u$ for the different circuit depths. (b) Corresponding mean squared error for the same circuit configurations.