Robust Quantum State Generation in Symmetric Spin Networks

TL;DR

This work addresses the robust generation of symmetric entangled states in a long-range Ising spin network for quantum metrology. It introduces a parameterized ensemble model and a Legendre moment kernel that collapses the infinite-dimensional ensemble dynamics to a finite set of moment equations, enabling an iterative quadratic programming approach to design robust control pulses. The method achieves high-fidelity preparation of target Dicke-profile states—$|W\rangle$, $|HEDS\rangle$, and $|GHZ\rangle$—across amplitude-uncertainty inhomogeneities and for different network sizes, with single- and dual-parameter robustness demonstrated. The results show strong resilience to control imperfections and highlight the approach’s potential for reliable quantum metrology, while remaining extensible to a broader class of symmetric quantum states.

Abstract

In this work, we consider a parameterized Ising model with long-range symmetric pairwise interactions on a network of spin $\frac{1}{2}$ particles. The system is designed with symmetric dynamics, allowing for the reduction of the state space to a subspace defined by the set of Dicke states. We propose a method for designing robust electromagnetic amplitude pulses based on a moment quantization approach. The introduced parameter accounts for uncertainties in the electromagnetic field, resulting in a family of distinct Hamiltonians. By employing a discretized moment-based quantization technique, we design a control pulse capable of simultaneously steering an infinite collection of dynamical systems to compensate for parameter variations. This approach benefits from the duality between the infinite-dimensional parameterized system and its finite-dimensional trucnated moment dynamics. Simulation results demonstrate the efficacy of this method in achieving states of significant interest in quantum sensing, including the GHZ and W states.

Figures (4)

• Figure 1: Fidelity metric values for designed robust pulses. Designs are for $\xi\in[1-\delta_{\xi},1+\delta_{\xi}]$ with $\delta_{\xi} = 0.2$ for the first and second columns of plots, and $\zeta\in[1-\delta_{\zeta},1+\delta_{\zeta}]$ with $\delta_{\zeta} = 0.2$ for the latter columns. Desired states to achieve are, from top row to bottom, $|W\rangle$, $|HEDS\rangle$ and $|GHZ\rangle$. The experiments are also differentiated by the population of atoms, being equal to 5 for the first and third columns and equal to 10 for the second and fourth.
• Figure 2: Simulated results for the robust design of quantum states for a network with 10 atoms for a system with $\delta_{\xi} = 0.2$ and $\delta_{\zeta} = 0$. Results refer to the design of (from top to bottom) $|W\rangle$, $|HEDS\rangle$ and $|GHZ\rangle$ states. The left plots refer to the final achieved states as a function of the probability amplitude of $|S, m \rangle$ (represented here as $\lVert|S, m\rangle\rVert ^{2}_{2}$). The second plot is the same as the first plot viewed from above. The right plot shows the control profiles obtained for $u_{x}(t)$ and $u_{z}(t)$ for the total time $T=9$.
• Figure 3: Fidelity metric evaluated for designed robust pulses for a population of $N=5$ particles. Designs are for $\xi\in[1-\delta_{\xi},1+\delta_{\xi}]$ and $\zeta\in[1-\delta_{\zeta},1+\delta_{\zeta}]$ with $\delta_{\zeta} = 0.2$ with $\delta_{\zeta} = 0.2$. Desired states to achieve are, from left to right, $|W\rangle$, $|HEDS\rangle$ and $|GHZ\rangle$.
• Figure 4: Simulated results for the robust design of quantum states for a network with 5 atoms for a system with $\delta_{\xi} = 0.2$ and $\delta_{\zeta} = 0.2$. Results refer to the design of (from top to bottom) $|W\rangle$, $|HEDS\rangle$ and $|GHZ\rangle$ states. The left plots refer to the final achieved states by displaying the density of measured particles in quantum states with eigenvalue $m$ (i.e. ion the $|S, m\rangle$ state) as a function of $\xi$ and $\zeta$, while the second plot displays the total counting of particles in the former. The right plot shows the control profiles obtained for $u_{x}(t)$ and $u_{z}(t)$ for the total time $T=9$.