Optimally learning functions in interacting quantum sensor networks

TL;DR

These results unify and extend existing bounds for non-interacting qubits and multimode interferometers, providing a general framework for distributed sensing and Hamiltonian learning in realistic many-body systems.

Abstract

Estimating extensive combinations of local parameters in distributed quantum systems is a central problem in quantum sensing, with applications ranging from magnetometry to timekeeping. While optimal strategies are known for sensing non-interacting Hamiltonians in quantum sensor networks, fundamental limits in the presence of uncontrolled interactions remain unclear. Here, we establish optimal bounds and protocols for estimating a linear combination of local parameters of Hamiltonians with arbitrary, unknown interactions. In the process, we more generally establish bounds for learning any linear combination of Hamiltonian coefficients for arbitrary, commuting terms. Our results unify and extend existing bounds for non-interacting qubits and multimode interferometers, providing a general framework for distributed sensing and Hamiltonian learning in realistic many-body systems.

Figures (1)

• Figure 1: (a) Schematic of an interacting sensor network, where each of $n$ qubits couples to a unique phase $\theta_i$, with unknown, arbitrary multi-body interactions between them, $\hat{H}_{\rm int}$. The task of the Interacting Sensing problem is to learn a linear function $q = \pmb{\alpha}\cdot\pmb{\theta}$ of the phases for some real $\pmb{\alpha}$. (b) To convert the Interacting Sensing problem to the Diagonal Learning problem, we first utilize randomized Trotter evolution under the non-diagonal Hamiltonian $\hat{H}_0$, conjugated by random Pauli-$Z$ strings. This engineers an effective, diagonal Hamiltonian $\hat{H}_{\rm eff}$, consisting of a sum over $\rm{poly}(n)$ Pauli-$Z$ strings $\hat{s}_j\in \{\hat{I},\hat{Z}\}^{\otimes n}$, so we have effective evolution $\hat{U}_{\rm eff}(t) = \exp\{-i\hat{H}_{\rm eff}t\}$. (c) Sketch of our optimal GHZ protocol for the Diagonal Learning problem, consisting of evolution via $\hat{U}_{\rm eff}(t)$, where $\hat{H}_{\rm eff}$ more generally may include any number of Pauli-$Z$ strings, interspersed with coherent switches between computational basis states for either the left (blue) or right (red) half of our initial superposition. The sequence of computational basis states $\ket{x^\pm(\mu)}$ and corresponding time intervals $\Delta t_\mu^\pm$ for the left/right halves of the superposition are selected to engineer a maximal total phase difference $\varphi \propto q \times t$, which can then be measured.

Theorems & Definitions (3)

• theorem 1: Sparsity
• proof : Proof sketch
• proof