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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. Here 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 ± 0.01, outperforming conventional spin-squeezing with a factor of 1.87 ± 0.03. Our approach reduces the number of averages to reach a given Allan deviation by a factor of 1.59 ± 0.06 compared with traditional methods not using 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 used without previous knowledge of the device or its noise environment.

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Fig. 1: Measurement and feedback concept for variational quantum Ramsey interferometry circuits.
Fig. 2: Generalized Ramsey sequence performance measurements.
Fig. 3: On-device hybrid quantum-classical optimization performance with 26 ions at \({\boldsymbol{\delta \phi }}{\boldsymbol{\approx }}{\bf{0.74}}\) (minimum BMSE versus δϕ).
Fig. 4: Frequency measurement using 12 ions with a standard and variationally optimized Ramsey sequence.

Data availability

All data obtained in the study are available from the corresponding author upon request. Source data are provided with this paper.


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We acknowledge funding from the EU H2020-FETFLAG-2018-03 under grant agreement no. 820495. We also acknowledge support by the Austrian Science Fund (FWF), through the SFB BeyondC (FWF Project No. F7109), and the IQI GmbH. P.S. acknowledges support from the Austrian Research Promotion Agency (FFG) contract 872766. P.S., T.M. and R.B. acknowledge funding by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA), through US ARO grant no. W911NF-16-1-0070 and W911NF-20-1-0007, and the US Air Force Office of Scientific Research (AFOSR) via IOE grant no. FA9550-19-1-7044 LASCEM. R.K., D.V.V. and P.Z. are supported by the US Air Force Office of Scientific Research (AFOSR) through IOE grant no. FA9550-19-1-7044 LASCEM, D.V.V by a joint-project grant from the FWF (grant no. I04426, RSF/Russia 2019), R.v.B and P.Z. by the European Union’s Horizon 2020 research and innovation programme under grant agreement no. 817482 (PASQuanS) and R.v.B by the Austrian Research Promotion Agency (FFG) contract 884471 (ELQO). P.Z. acknowledges funding by the the European Union’s Horizon 2020 research and innovation programme under grant agreement no. 731473 (QuantERA through QTFLAG), and by the Simons Collaboration on Ultra-Quantum Matter, which is a grant from the Simons Foundation (651440). Innsbruck theory is a member of the NSF Quantum Leap Challenge Institute Q-Sense. The computational results presented here have been achieved (in part) using the LEO HPC infrastructure of the University of Innsbruck. All statements of fact, opinions or conclusions contained herein are those of the authors and should not be construed as representing the official views or policies of the funding agencies.

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Ch.D.M. was the lead writer of the manuscript with assistance from R.K., D.V.V., R.v.B. and P.Z., and input from all coauthors. Ch.D.M., T.F. and I.P. built the experiment. Ch.D.M. and T.F. performed measurements. R.K., D.V.V. and P.Z. conceived of the method and provided theory. R.K. and R.v.B. developed the optimizer routines and implementation. Ch.D.M. and R.K. analysed the data. P.S., R.B. and T.M. supervised the experiment.

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Correspondence to Thomas Monz.

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Marciniak, C.D., Feldker, T., Pogorelov, I. et al. Optimal metrology with programmable quantum sensors. Nature 603, 604–609 (2022).

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