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arXiv:2605.22770

Adiabatic Quantum Phase Estimation

Alexander Schmidhuber, Seth Lloyd

quant-phSubmitted May 21, 2026• Announced May 22, 2026• 0 citations
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Abstract

Quantum phase estimation (QPE) is a central algorithmic primitive that estimates eigenvalues of a Hamiltonian up to precision $ε$ in Heisenberg-limited time $T=Θ(1/ε)$. Standard gate-based implementations of QPE require deep controlled time-evolution circuits and are not native to analog hardware. Here, we present a simple adiabatic protocol for QPE that achieves (up to logarithmic factors) the optimal Heisenberg-limited scaling $T = O\left( \frac{1}ε \log\left(δ^{-1}\right)\right)$ in both the precision $ε$ and failure probability $δ$. By encoding eigenvalues in populations of computational basis states rather than complex phases, our approach is naturally robust against certain dephasing errors. The adiabatic protocol only requires the ability to couple a single ancilla qubit to the system Hamiltonian as well as pairwise couplings within the ancilla register.

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Quantum phase estimation (QPE) is a central algorithmic primitive that estimates eigenvalues of a Hamiltonian up to precision εεε in Heisenberg-limited time T=Θ(1/ε)T=Θ(1/ε)T=Θ(1/ε). Standard gate-based implementations of QPE require deep controlled time-evolution circuits and are not native to analog hardware. Here, we present a simple adiabatic protocol for QPE that achieves (up to logarithmic factors) the optimal Heisenberg-limited scaling T=O(1εlog⁡(δ−1))T = O\left( \frac{1}ε \log\left(δ^{-1}\right)\right)T=O(ε1​log(δ−1)) in both the precision εεε and failure probability δδδ. By encoding eigenvalues in populations of computational basis states rather than complex phases, our approach is naturally robust against certain dephasing errors. The adiabatic protocol only requires the ability to couple a single ancilla qubit to the system Hamiltonian as well as pairwise couplings within the ancilla register.
Eigenvalue estimation is a central problem for demonstrating quantum advantage, yet its implementation on digital quantum computers remains limited by circuit depth and operational overhead. We present an analog quantum phase estimation (aQPE) protocol that extracts the eigenenergies of a target Hamiltonian via continuous time evolution and single-mode cavity measurement. By encoding eigenvalue information as conditional cavity phase-space rotations, the scheme avoids deep quantum circuits and entangling gates, while enabling readout through established cavity tomography techniques. We further illustrate the feasibility of this approach by engineering a Hamiltonian that implements aQPE of the XY model, whose ground-state energy problem is QMA-complete, within a physical architecture compatible with existing circuit quantum electrodynamics technology. Our results provide a resource-efficient and scalable framework for implementing quantum phase estimation in near-term quantum platforms.
Quantum phase estimation is a cornerstone in quantum algorithm design, allowing for the inference of eigenvalues of exponentially-large sparse matrices. The maximum rate at which these eigenvalues may be learned, --known as the Heisenberg limit--, is constrained by bounds on the circuit depth required to simulate an arbitrary Hamiltonian. Single-control qubit variants of quantum phase estimation have garnered interest in recent years due to lower circuit depth and minimal qubit overhead. In this work we show that these methods can achieve the Heisenberg limit, {\em also} when one is unable to prepare eigenstates of the system. Given a quantum subroutine which provides samples of a `phase function' g(k)=∑jAjeiϕjkg(k)=\sum_j A_j e^{i \phi_j k}g(k)=∑j​Aj​eiϕj​k with unknown eigenvalue phases ϕj\phi_jϕj​ and probabilities AjA_jAj​ at quantum cost O(k)O(k)O(k), we show how to estimate the phases {ϕj}\{\phi_j\}{ϕj​} with accuracy (root-mean-square) error δ\deltaδ for total quantum cost T=O(δ−1)T=O(\delta^{-1})T=O(δ−1). Our scheme combines the idea of Heisenberg-limited multi-order quantum phase estimation for a single eigenvalue phase \cite{Higgins09Demonstrating,Kimmel15Robust} with subroutines with so-called dense quantum phase estimation which uses classical processing via time-series analysis for the QEEP problem \cite{Somma19Quantum} or the matrix pencil method. For our algorithm which adaptively fixes the choice for kkk in g(k)g(k)g(k) we prove Heisenberg-limited scaling when we use the time-series/QEEP subroutine. We present numerical evidence that using the matrix pencil technique the algorithm can achieve Heisenberg-limited scaling as well.
Under suitable assumptions, the quantum-phase-estimation (QPE) algorithm is able to achieve Heisenberg-limited precision scaling in estimating the ground-state energy.However, QPE requires a large number of ancilla qubits and a large circuit depth, as well as the ability to perform inverse quantum Fourier transform, making it expensive to implement on an early fault-tolerant quantum computer.We propose an alternative method to estimate the ground-state energy of a Hamiltonian with Heisenberglimited precision scaling, which employs a simple quantum circuit with one ancilla qubit, and a classical postprocessing procedure.Besides the ground-state energy, our algorithm also produces an approximate cumulative distribution function of the spectral measure, which can be used to compute other spectral properties of the Hamiltonian.
Quantum phase estimation is a cornerstone algorithm for determining eigenvalues of unitary operators with Heisenberg-limited precision. Conventional implementations rely on digital controlled-unitary operations together with phase-extraction circuits, which generally results in substantial circuit depth and hardware overhead. Here, we propose a hardware-native alternative that replaces digital controlled-unitary operations by analog bosonic interactions, naturally available in circuit quantum electrodynamics. The protocol extracts the phase through a sequence of binary threshold tests. A bosonic mode serves as an efficient quantum memory where the binary digits of the phase are encoded into the direction of phase-space rotations. These digits are then read out sequentially via high-fidelity homodyne measurements. We show that the protocol preserves Heisenberg scaling in estimation precision while simultaneously providing exponentially suppressed failure probability. By exploiting bosonic degrees of freedom and analog dispersive interactions, the scheme provides a hardware-efficient realization of quantum phase estimation and establishes a natural route toward implementing high-precision phase estimation on circuit quantum electrodynamics platforms.
Quantum phase estimation is a cornerstone in quantum algorithm design, allowing for the inference of eigenvalues of exponentially-large sparse matrices.The maximum rate at which these eigenvalues may be learned, –known as the Heisenberg limit–, is constrained by bounds on the circuit complexity required to simulate an arbitrary Hamiltonian. Single-control qubit variants of quantum phase estimation that do not require coherence between experiments have garnered interest in recent years due to lower circuit depth and minimal qubit overhead. In this work we show that these methods can achieve the Heisenberg limit, a l s o when one is unable to prepare eigenstates of the system. Given a quantum subroutine which provides samples of a `phase function' g ( k ) = ∑ j A j e i ϕ j k with unknown eigenphases ϕ j and overlaps A j at quantum cost O ( k ) , we show how to estimate the phases { ϕ j } with (root-mean-square) error δ for total quantum cost T = O ( δ − 1 ) . Our scheme combines the idea of Heisenberg-limited multi-order quantum phase estimation for a single eigenvalue phase [Higgins et al (2009) and Kimmel et al (2015)] with subroutines with so-called dense quantum phase estimation which uses classical processing via time-series analysis for the QEEP problem [Somma (2019)] or the matrix pencil method. For our algorithm which adaptively fixes the choice for k in g ( k ) we prove Heisenberg-limited scaling when we use the time-series/QEEP subroutine. We present numerical evidence that using the matrix pencil technique the algorithm can achieve Heisenberg-limited scaling as well.
Quantum phase estimation (QPE) is fundamental to quantum algorithms for eigenvalue problems, chemistry simulation, and number-theoretic applications. However, canonical QPE implementations employ controlled unitary operations and quantum Fourier transform, incurring substantial circuit depth and control overhead that severely limit applicability on nearterm quantum hardware. This work develops a novel class of control-free phase estimation algorithms that eliminate both controlled unitaries and QFT by leveraging signal processing principles. Our approach applies uncontrolled powers of the unitary U = e −iHτ to an eigenstate or eigenstate-rich state, measures in multiple bases to capture interference patterns, and reconstructs eigenphases via classical post-processing techniques including Prony's method, compressed sensing, and harmonic inversion. We establish that for eigenstates or states with sparse eigendecomposition, our algorithms achieve phase estimation precision ϵ using Õ(1/ϵ) applications of U , matching standard QPE query complexity up to logarithmic factors while dramatically reducing circuit depth and eliminating control overhead. We provide detailed error analysis characterizing robustness to measurement noise, finite sampling, and state preparation errors. Applications to electronic structure estimation in small molecules and variational quantum eigensolvers demonstrate the practical viability of control-free QPE on both NISQ and early fault-tolerant quantum computers. Our results suggest that signal-processing-based phase estimation may offer more efficient implementations of quantum algorithms on realistic hardware.