Gradient-descent methods for quantum detector tomography

TL;DR

This work addresses scalable quantum detector tomography (QDT) for phase-insensitive detectors by reframing the POVM reconstruction as gradient-descent optimization. It demonstrates that diagonal, phase-insensitive POVMs can be learned by minimizing $||P - F\Pi||_F^2$ with rows of $\Pi$ kept as probability vectors via a softmax, using Adam with minibatching to achieve comparable or superior fidelity to constrained convex optimization while reducing time and memory costs. The study also introduces a Riemannian-geometry-based extension to phase-sensitive detectors through a Stiefel-manifold parameterization, enabling rank-controlled POVMs and constrained gradient updates. Together, these results establish a scalable framework for detector tomography and motivate GPU-accelerated and manifold-based approaches for larger quantum systems and datasets.

Abstract

We present a technique for performing quantum detector tomography (QDT) of phase insensitive quantum detectors using gradient descent-based optimization to learn the positive operator-valued measure (POVM) that best describes the data collected using the detector under study. We numerically benchmark our method against constrained convex optimization (CCO) and show that it reaches higher or comparable reconstruction fidelity in much less time even in the presence of noise and limited probe state resources. We also present a possible extension of our approach to the phase sensitive case via a parametrization of POVMs on the complex Stiefel manifold which enables gradient based optimization restricted to this manifold.

Figures (4)

• Figure 1: Wall clock time (in seconds) and wall clock time per iteration for QDT methods as Hilbert space size increases.
• Figure 2: Average reconstruction fidelity of QDT methods when using different Hilbert space truncations. Gradient descent curve shows mean value of $\bar{\mathcal{F}}$ over 20 different runs of the algorithm each with different random parameter initialization. Shaded region shows one standard deviation. (a) Average POVM reconstruction fidelity for ideal PNR detector. (b) Average POVM reconstruction fidelity for PNR detector with $85\%$ efficiency.
• Figure 3: Average reconstruction fidelity of QDT methods under varying levels of Gaussian noise $\mathcal{N}(0, \sigma^2)$ in the coherent state amplitudes $|\alpha|^2$. (a) Average POVM reconstruction fidelity for ideal PNR detector. (b) Average POVM reconstruction fidelity for PNR detector with $85\%$ efficiency.
• Figure 4: Average reconstruction fidelity of QDT methods when using datasets of varying size. (a) Average POVM reconstruction fidelity for ideal PNR detector. (b) Average POVM reconstruction fidelity for PNR detector with $85\%$ efficiency.