Improved Quantum Sensing by Spectral Design

TL;DR

This work advances quantum sensing by treating the imprinting process itself as a controllable degree of freedom. It introduces the switching method, proving that the effective spectrum is a convex combination of the original eigenvalues with bi-stochastic weights, and that any relative spectrum can be achieved—subject to potential reductions in spectral range. By reducing spectral design to linear programming and Birkhoff decompositions, the authors provide a practical framework for spectrum engineering and quantify resource requirements, including qubit- and gate-level implementations. Across degeneracy lifting, non-linear and linear spectra, and priors, the method yields meaningful improvements in Bayesian single-parameter estimation, highlighting conditions where spectral design offers substantial benefits and where benefits are limited (e.g., linear spectra under flat priors). The approach holds promise for robust, adaptive quantum sensing, with potential extensions to multi-parameter tasks and noisy environments.

Abstract

We investigate how unitary control can improve parameter estimation by designing the effective spectrum of the imprinting Hamiltonian. We show that, for commuting Hamiltonians, the general problem of spectral manipulation via unitary control simplifies to a finite sequence of elementary switching operations. Furthermore, we demonstrate that any desired relative spacing of energy levels can be achieved, although this may come at the cost of a reduced spectral range. We also show that any modified spectrum can be expressed as a convex combination of the original eigenvalues, with the convex weights forming a bi-stochastic matrix. Through several single-parameter estimation examples, we demonstrate that our spectral engineering method substantially enhances estimation accuracy.

Figures (15)

• Figure 1: Illustration of quantum sensing protocols with and without unitary control. (a) Standard sensing process using unitary imprinting of parameter $\theta$. (b) Sensing with general unitary control implemented via an infinite sequence of intermediate control unitary operations $I_i$. (c) Sensing using the switching method, based on a finite sequence of permutation operations $P_i$.
• Figure 2: Illustration of the steps of the switching method to obtain from a given spectrum $\sigma(G)$ and a given target vector $\boldsymbol{t}$ the adapted spectrum $\sigma(G_\text{eff})$ and the corresponding control operations.
• Figure 3: Sketch of a possible switching-method lifting of a three-qubit degeneracy. Left: the degenerate spectrum defined by Eq. \ref{['eq:deg_ham']}. Right: an example non-degenerate spectrum obtained after applying the switching protocol.
• Figure 4: BMSE $\Delta^2 \tilde{\omega}$ of the optimal Bayesian estimation strategy for a Gaussian prior with unit width. Results are shown for the degenerate spectrum of the Hamiltonian in Eq. \ref{['eq:deg_ham']} (labeled $m_l$) and for the optimized, degeneracy-lifted spectrum obtained via the switching method (labeled $m$) across different system sizes $m$ ($m_l$).
• Figure 5: Sketch of a possible switching-method adaptation of the Rb III 4F-shell spectrum. Left: the reference energies measured by Sansonetti sansonetti2006wavelengths. Right: an example adjusted spectrum obtained after applying the switching protocol.