QMetro++ -- Python optimization package for large scale quantum metrology with customized strategy structures

TL;DR

QMetro++ introduces a Python optimization framework tailored for large-scale quantum metrology, using quantum Fisher information $F_Q$ as the figure of merit and allowing arbitrary protocol structures. It combines two core approaches: Minimization over Purifications (MOP) for small-scale exact optimization and Iterative See-Saw (ISS) with tensor-network representations (MPS/MPO) to scale to $N$ up to about $100$, including standard (single-channel, parallel, adaptive) and customized collisional strategies. The package provides efficient computation of fundamental QFI bounds for benchmarking in both uncorrelated and correlated noise settings, enabling rapid assessment of optimality and scaling regimes. By exposing high-level functions for standard tasks and low-level tensor-network tools for arbitrary structures, QMetro++ facilitates systematic comparison of strategies against fundamental limits and supports exploration of noise effects and environment-assisted metrology in a flexible, scalable workflow.

Abstract

QMetro++ is a Python package that provides a set of tools for identifying optimal estimation protocols that maximize quantum Fisher information (QFI). Optimization can be performed for arbitrary configurations of input states, parameter-encoding channels, noise correlations, control operations, and measurements. The use of tensor networks and an iterative see-saw algorithm allows for an efficient optimization even in the regime of a large number of channel uses ($N\approx100$). Additionally, the package includes implementations of the recently developed methods for computing fundamental upper bounds on QFI, which serve as benchmarks for assessing the optimality of numerical optimization results. All functionalities are wrapped up in a user-friendly interface which enables the definition of strategies at various levels of detail.

Figures (11)

• Figure 1: Different classes of metrological strategies that one can optimize using the package: (A) optimization of a single channel input probe $\rho_0$ (potentially entangled with an ancillary system $\mathcal{A}$), (B) optimization of an entangled state $\rho_0$ of $N$ input probes in a parallel strategy (potentially additionally entangled with an ancillary system $\mathcal{A}$), (C) optimization of an input probe $\rho_0$ and control operations $C_i$ in an adaptive metrological scheme, (D) a customized protocol structure, here inspired by quantum collisional models, where multi-partite entangled ancillary state is sent piece-by-piece to interact with a common sensing system via interaction gates $C_i$. Here the optimization may affect either the input state $\rho_0$ or interaction gates $C_i$ or both. The above strategies correspond to functions listed in Table \ref{['tab:summary']}.
• Figure 2: File structure of the package with a selection of the most important functions. Classes from tensors.py file are displayed in more detail in Fig. \ref{['fig:classes']}. All functions, classes and methods are listed and described in the package documentation Dulian2025.
• Figure 3: Final state resulting from the action of an adaptive protocol on $N$ parameter encoding channels, written via a formal link product operation between the corresponding CJ operators---quantum combs.
• Figure 4: (A) Diagram of a strategy with a single parameter-dependent channel $\Lambda_\theta$ and ancilla $\mathcal{A}$. Values of QFI for (B) dephasing and (C) amplitude damping, for various values of $p$ and for different methods: MOP - black $\times$, ISS with $d_\mathcal{A}=1$ - blue dashed line, ISS with $d_\mathcal{A}=2$ - red dotted line.
• Figure 5: (A) Diagram of a parallel strategy with multiple parametrized channels $\Lambda_\theta$ and ancilla $\mathcal{A}$. Values of QFI normalized by the number of channel uses, $N$, for: (B) dephasing ($p=0.75$), (C) amplitude damping ($p=0.75$) and different methods: MOP - black $\times$, simple ISS with $d_\mathcal{A}=2$ - black +, tensor network ISS with $d_\mathcal{A}=2, \; r_\mathrm{MPS} = \sqrt{r_\mathfrak{L}}=2$ - blue dashed line, tensor network ISS with $d_\mathcal{A}=2, \; r_\mathrm{MPS} = \sqrt{r_\mathfrak{L}}=4$ - red dotted line, upper bound - black solid line.