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  • Open Access

Exponential Distillation of Dominant Eigenproperties

Bence Bakó1,2,3, Tenzan Araki1,4, and Bálint Koczor1,*

  • *Contact author: balint.koczor@maths.ox.ac.uk
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PRX Quantum 7, 010334 – Published 18 February, 2026

DOI: https://doi.org/10.1103/bglh-9snd

Abstract

Estimating observable expectation values in eigenstates of quantum systems has a broad range of applications and is an area where early fault-tolerant quantum computers may provide practical quantum advantage. We develop a hybrid quantum-classical algorithm that enables the estimation of an arbitrary observable expectation value in an eigenstate, given an initial state is supplied that has dominant overlap with the targeted eigenstate—but may overlap with any other eigenstates. Our approach builds on and is conceptually similar to purification-based error mitigation techniques; however, it achieves exponential suppression of algorithmic errors using only a single copy of the quantum state. The key innovation is that random time evolution is applied in the quantum computer to create an average mixed quantum state, which is then virtually purified with exponential efficacy. We prove rigorous performance guarantees and conclude that the complexity of our approach depends directly on the energy gap in the problem Hamiltonian and remarkably, can be compared to phase estimation combined with amplitude estimation in terms of its scaling with respect to a target precision. We demonstrate in a broad range of numerical simulations the applicability of our framework in near-term and early fault-tolerant settings. Furthermore, we demonstrate in a 100-qubit example that direct classical simulation of our approach enables the prediction of ground and excited state properties of quantum systems using tensor-network techniques, which we recognize as a quantum-inspired classical approach.

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Popular Summary

The most natural application of quantum computers is the simulation of quantum many-body systems, with practical relevance to quantum chemistry, materials science, and beyond. Fully fault-tolerant algorithms offer rigorous performance guarantees, but typically require deep circuits that exceed current and near-term hardware capabilities. This has spurred the development of early fault-tolerant algorithms that use fewer quantum resources and offload some computation to classical postprocessing. A typical—and particularly relevant—task here is estimating specific properties of Hamiltonian eigenstates, which yields deeper, essential physical insights beyond energy estimation alone but incurs higher algorithmic costs.

Our distillation of dominant eigenproperties (DDE) approach enables the precise estimation of arbitrary observable expectation values in a targeted eigenstate. DDE proceeds by first applying time evolution to an initial state that has dominant overlap with the target eigenstate, and then measuring temporal correlations with respect to an observable. These correlations are then classically processed via Monte Carlo integration—the approach conceptually builds on the error mitigation technique virtual distillation and therefore achieves exponential suppression of algorithmic errors. However, DDE requires only a single copy. Furthermore, we prove that the approach requires shallow circuits and its scaling with respect to a desired target precision is comparable to fully coherent, textbook quantum phase estimation combined with amplitude estimation.

Additionally, we demonstrate that DDE can be applied as a powerful quantum-inspired classical technique via tensor network simulation. Therefore, by establishing a fascinating connection between purification-based error mitigation and quantum phase estimation algorithms, this work unlocks eigenstate property estimation using shallow circuits, paving the way for potential future work in various application domains.

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PRX Quantum 7, 010334– Published 18 February, 2026
  • Received 23 June 2025
  • Revised 26 November 2025
  • Accepted 9 December 2025
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