Heisenberg limited metrology based on Hilbert space fragmentation in an interacting inhomogeneous system

In quantum metrology, an ensemble of entangled qubits can be used to enhance sensitivity in estimating external fields. One major challenge for quantum metrology is to achieve high sensitivity in many-body interacting systems. On the one hand, internal interactions among qubits are often helpful in preparing entangled states. On the other hand, complicated interactions, which are in general spatially inhomogeneous in actual experiments, make the many-body system thermalize and spoil the sensitivity. In this work, we propose an entanglement-enhanced sensing scheme that is robust against spatially inhomogeneous always-on Ising interactions. Our strategy is to tailor coherent quantum dynamics employing the Hilbert-space fragmentation (HSF), a recently recognized mechanism that evades thermalization in kinetically constrained many-body systems. Specifically, we analytically show that the emergent HSF caused by strong Ising interactions enables us to design a stable state where part of the qubits is effectively decoupled from the rest of the system. Using the decoupled qubits as a probe to measure a transverse field, we demonstrate that the Heisenberg limit is achieved without suffering from thermalization.