Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2603.21873

Accurate ground state energy estimation with noise and imperfect state preparation

Alicja Dutkiewicz, Thomas E. O'Brien, Stefano Polla

quant-phSubmitted Mar 23, 2026• Announced Mar 24, 2026• 0 citations• Revised Aug 3, 2026 (v2)
v1 · Mar 23, 2026v2 · Aug 3, 2026
arXivHTMLPDF

Abstract

We introduce a classical estimator for the post-processing of quantum phase estimation (QPE) data when a single target phase is isolated within a known interval, as is typical of ground state energy estimation of gapped systems. Our estimator filters the QPE signal within this promise region and recovers the phase through a moment-projection routine, which is robust to both external spurious phases and experimental noise. In the noiseless case this achieves an exponential suppression of bias with respect to a naive mean estimator. In the presence of global depolarizing noise the bias is exponentially small in the circuit depth $t$, and the variance is $O(t^{-2}F^{-2})$ for circuit fidelity $F$. This improves by a factor of $t^2$ over a naive shifted-and-rescaled-mean approach. To mitigate realistic circuit-level noise, we combine our method with the explicit unbiasing scheme described in [Dutkiewicz et al., 2025]. This yields an overhead interpolating between the $F^{-4}$ scaling typical of explicitly unbiased error mitigation and a reduced $F^{-2}$ scaling when the noise samples fall outside the promise interval. We validate our estimators on a small-scale simulation of the Ising model, observing better-than-expected performance for a global depolarizing noise approximation. This robustness to both multiple eigenvalues and realistic noise makes limited-depth phase estimation practical for early fault tolerant quantum experiments.

Comments
28 pages + 11 page appendix, 6 figures

Related papers

Quantum phase estimation of multiple eigenvalues for small-scale (noisy) experiments

Thomas E. O’Brien, Brian Tarasinski, Barbara M. Terhal

Quantum phase estimation (QPE) is the workhorse behind any quantum algorithm and a promising method for determining ground state energies of strongly correlated quantum systems. Low-cost QPE techniques make use of circuits which only use a single ancilla qubit, requiring classical post-processing to extract eigenvalue details of the system. We investigate choices for phase estimation for a unitary matrix with low-depth noise-free or noisy circuits, varying both the phase estimation circuits themselves as well as the classical post-processing to determine the eigenvalue phases. We work in the scenario when the input state is not an eigenstate of the unitary matrix. We develop a new post-processing technique to extract eigenvalues from phase estimation data based on a classical time-series (or frequency) analysis and contrast this to an analysis via Bayesian methods. We calculate the variance in estimating single eigenvalues via the time-series analysis analytically, finding that it scales to first order in the number of experiments performed, and to first or second order (depending on the experiment design) in the circuit depth. Numerical simulations confirm this scaling for both estimators. We attempt to compensate for the noise with both classical post-processing techniques, finding good results in the presence of depolarizing noise, but smaller improvements in 9-qubit circuit-level simulations of superconducting qubits aimed at resolving the electronic ground state of a H 4 -molecule.

arXiv:1809.09697•New Journal of Physics•Jan 2, 2019•175 citations

Robust ground-state energy estimation under depolarizing noise

Zhiyan Ding, Yulong Dong, Yu Tong +1 more

Abstract We present a novel ground-state energy estimation algorithm that is robust under global depolarizing error channels. Building upon the recently developed Quantum Exponential Least Squares (QCELS) algorithm [Ding, Lin, PRX Quantum, 4, 020331, 2023], our new approach incorporates significant advancements to ensure robust estimation while maintaining a polynomial cost in precision. By leveraging the spectral gap of the Hamiltonian effectively, our algorithm overcomes limitations observed in previous methods like quantum phase estimation (QPE) and robust phase estimation (RPE). Going beyond global depolarizing error channels, our work underscores the significance and practical advantages of utilizing randomized compiling techniques to tailor quantum noise towards depolarizing error channels. Our research demonstrates the feasibility of ground-state energy estimation in the presence of depolarizing noise, offering potential advancements in error correction and algorithmic-level error mitigation for quantum algorithms.

arXiv:2307.11257•APL Computational Physics•Jun 1, 2026•5 citations

Control Free Quantum Phase Estimation for Efficient Eigenphase and Energy Estimation without Quantum Fourier Transform

Yalla Jnan Devi Satya Prasad

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.

•Feb 5, 2026•0 citations

Statistical Quantum Phase Estimation: Extensions and Practical Considerations

Amit Surana, Brandon Allen

We present several refinements and extensions of the statistical quantum phase estimation (SQPE) framework to address some of its key practical limitations, improving its applicability to realistic cases. Recently, a family of statistical approaches for QPE have been proposed where each run uses only a few ancillae and shorter circuits than standard QPE and thus is better suited for early fault-tolerant quantum computers that are qubit-and depth-limited. SQPE method within that family estimates the cumulative distribution function (CDF) associated with spectral density of the Hamiltonian for a given trial state by using its Fourier approximation and then identifies the first jump discontinuity of the CDF to determine the ground state energy (GSE) of the Hamiltonian. It relies on random compilation procedure based on linear combination of unitaries (LCU) decomposition of the Hamiltonian assuming positive Pauli weights and requires a good estimate of lower bound on the overlap between the trial and true ground state, both of which may be difficult to achieve in practice. We address these limitations by generalizing the random compilation procedure for negative Pauli weights and employing a changepoint detection method for determining GSE which does not rely on an estimate of this overlap. We also show that by exploiting symmetry of the Fourier series one can reduce number of circuit runs/samples by a factor of 2x while keeping the GSE estimation accuracy the same. We illustrate these new developments numerically via a quantum simulator in Qiskit.

quant-pharXiv:2605.18876•May 19, 2026•0 citations

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

Alicja Dutkiewicz, Stefano Polla, Maximilian Scheurer +3 more

As fully fault-tolerant quantum computers capable of solving useful problems remain a distant goal, we anticipate an era of 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 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.

arXiv:2410.05369•PRX Quantum•Sep 11, 2025•5 citations

Even Shorter Quantum Circuit for Phase Estimation on Early Fault-Tolerant Quantum Computers with Applications to Ground-State Energy Estimation

Zhiyan Ding, Lin Lin

We develop a phase-estimation method with a distinct feature: its maximal run time (which determines the circuit depth) is δ/ , where is the target precision, and the preconstant δ can be arbitrarily close to 0 as the initial state approaches the target eigenstate.The total cost of the algorithm satisfies the Heisenberg-limited scaling O( -1 ).As a result, our algorithm may significantly reduce the circuit depth for performing phase-estimation tasks on early fault-tolerant quantum computers.The key technique is a simple subroutine called quantum complex exponential least squares (QCELS).Our algorithm can be readily applied to reduce the circuit depth for estimating the ground-state energy of a quantum Hamiltonian, when the overlap between the initial state and the ground state is large.If this initial overlap is small, we can combine our method with the Fourier-filtering method developed in [Lin and Tong, PRX Quantum 3, 010318, 2022], and the resulting algorithm provably reduces the circuit depth in the presence of a large relative overlap compared to .The relative-overlap condition is similar to a spectral-gap assumption but it is aware of the information in the initial state and is therefore applicable to certain Hamiltonians with small spectral gaps.We observe that the circuit depth can be reduced by around 2 orders of magnitude in numerical experiments under various settings.

arXiv:2211.11973•PRX Quantum•May 26, 2023•80 citations
We introduce a classical estimator for the post-processing of quantum phase estimation (QPE) data when a single target phase is isolated within a known interval, as is typical of ground state energy estimation of gapped systems. Our estimator filters the QPE signal within this promise region and recovers the phase through a moment-projection routine, which is robust to both external spurious phases and experimental noise. In the noiseless case this achieves an exponential suppression of bias with respect to a naive mean estimator. In the presence of global depolarizing noise the bias is exponentially small in the circuit depth ttt, and the variance is O(t−2F−2)O(t^{-2}F^{-2})O(t−2F−2) for circuit fidelity FFF. This improves by a factor of t2t^2t2 over a naive shifted-and-rescaled-mean approach. To mitigate realistic circuit-level noise, we combine our method with the explicit unbiasing scheme described in [Dutkiewicz et al., 2025]. This yields an overhead interpolating between the F−4F^{-4}F−4 scaling typical of explicitly unbiased error mitigation and a reduced F−2F^{-2}F−2 scaling when the noise samples fall outside the promise interval. We validate our estimators on a small-scale simulation of the Ising model, observing better-than-expected performance for a global depolarizing noise approximation. This robustness to both multiple eigenvalues and realistic noise makes limited-depth phase estimation practical for early fault tolerant quantum experiments.
Quantum phase estimation (QPE) is the workhorse behind any quantum algorithm and a promising method for determining ground state energies of strongly correlated quantum systems. Low-cost QPE techniques make use of circuits which only use a single ancilla qubit, requiring classical post-processing to extract eigenvalue details of the system. We investigate choices for phase estimation for a unitary matrix with low-depth noise-free or noisy circuits, varying both the phase estimation circuits themselves as well as the classical post-processing to determine the eigenvalue phases. We work in the scenario when the input state is not an eigenstate of the unitary matrix. We develop a new post-processing technique to extract eigenvalues from phase estimation data based on a classical time-series (or frequency) analysis and contrast this to an analysis via Bayesian methods. We calculate the variance in estimating single eigenvalues via the time-series analysis analytically, finding that it scales to first order in the number of experiments performed, and to first or second order (depending on the experiment design) in the circuit depth. Numerical simulations confirm this scaling for both estimators. We attempt to compensate for the noise with both classical post-processing techniques, finding good results in the presence of depolarizing noise, but smaller improvements in 9-qubit circuit-level simulations of superconducting qubits aimed at resolving the electronic ground state of a H 4 -molecule.
Abstract We present a novel ground-state energy estimation algorithm that is robust under global depolarizing error channels. Building upon the recently developed Quantum Exponential Least Squares (QCELS) algorithm [Ding, Lin, PRX Quantum, 4, 020331, 2023], our new approach incorporates significant advancements to ensure robust estimation while maintaining a polynomial cost in precision. By leveraging the spectral gap of the Hamiltonian effectively, our algorithm overcomes limitations observed in previous methods like quantum phase estimation (QPE) and robust phase estimation (RPE). Going beyond global depolarizing error channels, our work underscores the significance and practical advantages of utilizing randomized compiling techniques to tailor quantum noise towards depolarizing error channels. Our research demonstrates the feasibility of ground-state energy estimation in the presence of depolarizing noise, offering potential advancements in error correction and algorithmic-level error mitigation for quantum algorithms.
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.
We present several refinements and extensions of the statistical quantum phase estimation (SQPE) framework to address some of its key practical limitations, improving its applicability to realistic cases. Recently, a family of statistical approaches for QPE have been proposed where each run uses only a few ancillae and shorter circuits than standard QPE and thus is better suited for early fault-tolerant quantum computers that are qubit-and depth-limited. SQPE method within that family estimates the cumulative distribution function (CDF) associated with spectral density of the Hamiltonian for a given trial state by using its Fourier approximation and then identifies the first jump discontinuity of the CDF to determine the ground state energy (GSE) of the Hamiltonian. It relies on random compilation procedure based on linear combination of unitaries (LCU) decomposition of the Hamiltonian assuming positive Pauli weights and requires a good estimate of lower bound on the overlap between the trial and true ground state, both of which may be difficult to achieve in practice. We address these limitations by generalizing the random compilation procedure for negative Pauli weights and employing a changepoint detection method for determining GSE which does not rely on an estimate of this overlap. We also show that by exploiting symmetry of the Fourier series one can reduce number of circuit runs/samples by a factor of 2x while keeping the GSE estimation accuracy the same. We illustrate these new developments numerically via a quantum simulator in Qiskit.
As fully fault-tolerant quantum computers capable of solving useful problems remain a distant goal, we anticipate an era of 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 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.
We develop a phase-estimation method with a distinct feature: its maximal run time (which determines the circuit depth) is δ/ , where is the target precision, and the preconstant δ can be arbitrarily close to 0 as the initial state approaches the target eigenstate.The total cost of the algorithm satisfies the Heisenberg-limited scaling O( -1 ).As a result, our algorithm may significantly reduce the circuit depth for performing phase-estimation tasks on early fault-tolerant quantum computers.The key technique is a simple subroutine called quantum complex exponential least squares (QCELS).Our algorithm can be readily applied to reduce the circuit depth for estimating the ground-state energy of a quantum Hamiltonian, when the overlap between the initial state and the ground state is large.If this initial overlap is small, we can combine our method with the Fourier-filtering method developed in [Lin and Tong, PRX Quantum 3, 010318, 2022], and the resulting algorithm provably reduces the circuit depth in the presence of a large relative overlap compared to .The relative-overlap condition is similar to a spectral-gap assumption but it is aware of the information in the initial state and is therefore applicable to certain Hamiltonians with small spectral gaps.We observe that the circuit depth can be reduced by around 2 orders of magnitude in numerical experiments under various settings.