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  • Open Access

Trade-off Between Complexity and Energy in Quantum Phase Estimation

Yukuan Tao*, Mădălin Guţă†, and Gerardo Adesso‡

  • School of Mathematical Sciences and Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems, University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom

  • *Contact author: yukuan.tao@nottingham.ac.uk
  • †Contact author: madalin.guta@nottingham.ac.uk
  • ‡Contact author: gerardo.adesso@nottingham.ac.uk
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PRX Energy 5, 033016 – Published 8 September, 2026

DOI: https://doi.org/10.1103/zlhs-7m9m

Abstract

Quantifying the energetic cost of implementing quantum operations is essential for assessing the efficiency and scalability of quantum sensing and information-processing technologies. Here, we introduce a framework for analyzing the interplay between complexity and energy cost of quantum processes. In particular, we apply our framework to a sequential quantum phase estimation protocol, where a phase of physical significance is encoded in a quantum channel. The channel is applied to a probe state repeatedly until the probe is measured and the outcome leads to an estimate on the phase. We establish a trade-off relation between the total energy cost of the protocol and the number of times the channel is applied (complexity), while reaching a desired estimation precision. A sweet spot is located where the two quantities are co-optimized. The principles of our analysis can be adapted to benchmark the energetic requirements in other quantum protocols and devices.

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Popular Summary

Quantum technologies can achieve extremely precise measurements, but this precision comes at a physical cost: energy is needed to implement the quantum operations that extract information. In this work, the authors develop a framework to study the trade-off between two key resources in a quantum protocol: the complexity, meaning how many operations are required, and the overall energy cost of the protocol. The authors apply this idea to phase estimation, a basic building block of quantum sensing in which an unknown physical quantity is learned from the imprint it leaves during a sequence of interactions with a quantum probe system. Improving the quality of each interaction generally makes the estimation more effective, so that fewer operations are needed to achieve a target precision. But higher-quality operations also tend to require more energy. Conversely, reducing the energy invested in each step can make the overall protocol more complex. This study shows that these competing effects can give rise to a sweet spot, where neither the least complex implementation nor the one consuming the least energy per operation is optimal, but the total energy expenditure is minimized by balancing the two. This challenges the conventional perspective that quantum protocols should always be pushed as close as possible to ideal operation. Instead, the most useful implementation may depend on how efficiently physical resources are spent to reach the desired performance. This approach provides a route towards energy-aware benchmarking of quantum sensing and, more broadly, towards designing quantum technologies that combine high performance with real-world sustainability.

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PRX Energy 5, 033016– Published 8 September, 2026
  • Received 2 December 2025
  • Revised 7 August 2026
  • Accepted 13 August 2026
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