Optimal metrology with programmable quantum sensors

Quantum sensors are an established technology that has created new opportunities for precision sensing across the breadth of science. Using entanglement for quantum-enhancement will allow us to construct the next generation of sensors that can approach the fundamental limits of precision allowed by quantum physics. However, determining how state-of-the-art sensing platforms may be used to converge to these ultimate limits is an outstanding challenge. In this work we merge concepts from the field of quantum information processing with metrology, and successfully implement experimentally a *programmable quantum sensor* operating close to the fundamental limits imposed by the laws of quantum mechanics. We achieve this by using low-depth, parametrized quantum circuits implementing optimal input states and measurement operators for a sensing task on a trapped ion experiment. With 26 ions, we approach the fundamental sensing limit up to a factor of 1.45(1), outperforming conventional spin-squeezing with a factor of 1.87(3). Our approach reduces the number of averages to reach a given Allan deviation by a factor of 1.59(6) compared to traditional methods not employing entanglement-enabled protocols. We further perform on-device quantum-classical feedback optimization to `self-calibrate' the programmable quantum sensor with comparable performance. This ability illustrates that this next generation of quantum sensor can be employed without prior knowledge of the device or its noise environment.