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  • Open Access

Quantum Phase Estimation Without Controlled Unitaries

Laura Clinton1,*, Toby S. Cubitt1,2, Raul Garcia-Patron1,3, Ashley Montanaro1,4, Stasja Stanisic1, and Maarten Stroeks1,5,†

  • *Contact author: laura@phasecraft.io
  • †Contact author: m.e.h.m.stroeks@tudelft.nl
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PRX Quantum 7, 010345 – Published 5 March, 2026

DOI: https://doi.org/10.1103/7qcr-znl2

Abstract

In this work we demonstrate the use of adapted classical phase retrieval algorithms to perform control-free quantum phase estimation. We eliminate the costly controlled time evolution and Hadamard test commonly required to access the complex time-series needed to reconstruct the spectrum. This significant reduction of the number of coherent controlled-operations lowers the circuit depth and considerably simplifies the implementation of statistical quantum phase estimation in near-term devices. This seemingly impossible task can be achieved by extending the problem that one wishes to solve to one with a larger set of input signals while exploiting natural constraints on the signal and/or the spectrum. We leverage well-established algorithms that are widely used in the context of classical signal processing, demonstrating two complementary methods to do this, vectorial phase retrieval and two-dimensional phase retrieval. We numerically investigate the feasibility of both approaches for estimating the spectrum of the Fermi-Hubbard model and discuss their resilience to inherent statistical noise.

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Popular Summary

Our paper “Quantum Phase Estimation without Controlled Unitaries” proposes a new way to perform quantum phase estimation (QPE) without the need for controlled time evolution and Hadamard tests. This reduces the gate-count of the requisite quantum circuits, particularly for quantum devices with constrained connectivity.

We adapt classical phase-retrieval algorithms from signal processing to the quantum setting. By constructing an enlarged set of input states and exploiting constraints on the time-series and underlying spectrum, we show that one can reconstruct the full complex signal required for QPE from measurements of absolute values alone. We adapt two complementary techniques to the quantum setting: vectorial phase retrieval and two-dimensional phase retrieval with Fienup’s hybrid input-output algorithm. These methods reconstruct the spectrum using only standard time evolution circuits on an enlarged set of inputs, thereby eliminating the need for ancillary qubits and controlled unitaries.

Our work includes numerical tests on the Fermi-Hubbard model to evaluate feasibility and resilience to shot noise, demonstrating that these phase-retrieval approaches can recover spectral features with potentially promising performance on near-term devices. Our work reframes QPE as a signal reconstruction problem, trading controlled quantum operations for increased classical postprocessing and sampling complexity, potentially enabling simpler implementations on noisy intermediate-scale quantum hardware.

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Outline

Information

PRX Quantum 7, 010345– Published 5 March, 2026
  • Received 18 August 2025
  • Revised 19 November 2025
  • Accepted 7 January 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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