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Error Mitigation and Circuit Division for Early Fault-Tolerant Quantum Phase Estimation

Alicja Dutkiewicz1,*,†, Stefano Polla1,2,*,‡, Maximilian Scheurer3, Christian Gogolin1,3, William J. Huggins2, and Thomas E. O’Brien1,2,§

  • *These authors contributed equally to this work.
  • †Contact author: dutkiewicz@lorentz.leidenuniv.nl
  • ‡Contact author: polla@lorentz.leidenuniv.nl
  • §Contact author: teobrien@google.com
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PRX Quantum 6, 040318 – Published 27 October, 2025

DOI: https://doi.org/10.1103/mlmy-yskj

Abstract

As fully fault-tolerant quantum computers capable of solving useful problems remain a distant goal, we anticipate an era of early fault tolerance during which limited error correction is available. We propose a framework for designing early fault-tolerant algorithms by trading between error-correction overhead and residual logical noise, and apply it to quantum phase estimation (QPE). We develop a quantum-Fourier-transform- (QFT) based QPE technique that is robust to global depolarizing noise and outperforms the previous state of the art at low and moderate noise rates. We further introduce the explicitly unbiased maximum-likelihood estimator (EUMLE), a data-processing technique that mitigates arbitrary errors in QFT-based QPE schemes. EUMLE provides consistent asymptotically normal error-mitigated estimates, addressing the open problem of extending error mitigation beyond expectation-value estimation. Applying this scheme to the ground-state problem of the two-dimensional Hubbard model and various molecular Hamiltonians, we find that we can roughly halve the number of physical qubits with a ∼10 times wall-clock time overhead, but that further reduction causes a steep run-time increase. This work provides an end-to-end analysis of early fault-tolerance cost reductions and space-time trade-offs, and identifies which areas can be improved in the future.

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Popular Summary

Quantum computers have the potential to revolutionize fields such as chemistry, materials science, and cryptography, but current hardware is not yet powerful or reliable enough to tackle practical problems. We are entering a transitional era called “early fault tolerance,” in which quantum error correction becomes possible but remains too costly to apply at full strength. In this work, we develop a new framework to help quantum algorithms function effectively during this in-between stage, balancing the benefits of error correction with smarter algorithm design and error-mitigation strategies.

We focus on quantum phase estimation, a central algorithm for calculating energy levels in molecules and materials. Traditionally, this algorithm assumes perfect hardware or requires full error correction, which is still out of reach. We introduce a new method that splits complex quantum circuits into smaller more manageable pieces that can tolerate some errors. We also develop a novel data-processing technique that removes noise-induced bias from results, even when the noise model is complicated. We use this to estimate the resources needed for real-world problems in chemistry and physics, showing where this hybrid strategy helps and where it hits fundamental limits.

This research provides practical tools and a guideline for running meaningful quantum computations on the next generation of devices. Looking ahead, our framework can be extended and refined as quantum hardware improves, helping researchers extract the most value out of limited resources.

Article Text

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Outline

Information

PRX Quantum 6, 040318– Published 27 October, 2025
  • Received 23 May 2025
  • Accepted 11 September 2025
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