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:2507.01361

Quantum phase estimation based filtering: performance analysis and application to low-energy spectral calculation

Rei Sakuma, Kaito Wada, Shu Kanno, Kimberlee Keithley, Kenji Sugisaki, Takashi Abe, Nakamura, Hajime, Naoki Yamamoto

Published Jul 2, 2025• 0 citations
arXivHTMLPDF

Abstract

Filtering is an important technique in quantum computing used for isolating or enhancing some specific states of quantum many-body systems. In this paper, we analyze the performance of filters based on the quantum phase estimation (QPE) algorithm, in which filtering removes states associated with bitstrings in the ancilla register above a given threshold. We show that when the conventional rectangular window function is used for the QPE input state, the resulting filter function exhibits an oscillating behavior known as the Gibbs phenomenon. We also show that in the case of the sine and Kaiser windows, this phenomenon is suppressed. Furthermore, we perform numerical simulations to compare the number of necessary queries to the Hamiltonian time evolution operation of for the QPE-based filtering algorithm and the quantum eigenvalue transformation of unitary matrices with real polynomials (QETU). We find that the number of queries required for Kaiser window-based filtering is comparable to that for QETU with optimized phase angles. As an application of the QPE-based filter, we also study a two-step algorithm for low-energy spectral simulations, composed of a coarse grid for filtering and a fine grid for obtaining final high-resolution spectra. As a benchmark of the proposed scheme for realistic continuous spectra, we present the density-of-states (DOS) calculation of antiferromagnetic type-II MnO in a one-particle approximation.

DOI
10.48550/arxiv.2507.01361

Related papers

Reducing quantum resources for observable estimation with window-assisted coherent QPE

Harriet Apel, Cristian L. Cortes, Jessica Lemieux +1 more

Quantum Phase Estimation (QPE) routines are known to fail probabilistically even with perfect gates and input states. This effect stems from an incompatibility of finite-sized quantum registers to capture a phase within QPE with phase angles of infinite precision, and the effect extend even beyond what would be reasonably expected from rounding. This effect can be partially mitigated by biasing the phase register with a window, or taper state, from classical signal processing. This paper focuses on how windowing a coherent QPE used as a subroutine can improve the accuracy of the overall algorithm. Specifically we study the quantum task of estimating observables where window-assisted coherent QPE is used as a subroutine to implement a reflection about an eigenstate. Quantum resource estimates show over 2-orders-of-magnitude reduction in Toffoli counts over the previous costed techniques -- also assisted by the use of improved block encoding techniques -- demonstrating an encouraging decrease in resources for quantum computation of molecular observables. Since QPE, as one of only a few quantum building blocks, appears as a subroutine in many algorithms; this analysis also provides a model for understanding how window functions propagate to an improved error in composite algorithms.

arXiv:2508.06677•Aug 8, 2025•0 citations

Quantum phase estimation based algorithms for machine learning

Oumayma Ouedrhiri, Oumayma Banouar, Salah El Hadaj +1 more

Quantum computing is certainly one of the greatest advances in the computer science field. Thanks to the parallelism and entanglement properties, it has proved to offer several advantages compared to the classical algorithms especially in the great reduction of the processing time. Quantum phase estimation (QPE) is one of the most important algorithms for quantum computing. It is known as the eigenvalue finding module for unitary operators. The Fourier transform is the key to this procedure. It has been researched and used to solve many problems such as the order finding problem, and the factoring problem. It was also applied for quantum sampling algorithms and the calculation of the eigenvalues of unitary matrices. In this paper, we study three important quantum algorithms for machine learning that use the QPE algorithm as a subroutine: the quantum principal components analysis (PCA) for data visualization, the Harrow-Hassidim-Lloyd (HHL) algorithm for solving linear systems, and the quantum singular value thresholding (SVT) for matrix completion in recommender systems. We also discuss the advantages and limits of such algorithms compared to their classical versions. Then we discuss potential ways of amelioration of such algorithms, and end with a proposed approach for further improvement.

International Informatics and Software Engineering Conference (IISEC)•Dec 16, 2021•3 citations

A case study against QSVT: assessment of quantum phase estimation improved by signal processing techniques

Sean Greenaway, William Pol, Sukin Sim

In recent years, quantum algorithms have been proposed which use quantum phase estimation (QPE) coherently as a subroutine without measurement. In order to do this effectively, the routine must be able to distinguish eigenstates with success probability close to unity. In this paper, we provide the first systematic comparison between two approaches towards maximizing this success probability, one using the quantum singular value transform and the other leveraging window functions, which have been previously studied as priors of the phase value distribution. We find that the quantum singular value transform is significantly outclassed by the window function approach, with the latter able to achieve between 3 and 5 orders of magnitude improvement in the success probability with approximately 1/4 the query cost. Our circuit simulation results indicate that QPE is not a domain which benefits from the integration of QSVT and we show that the use of the Kaiser window function is currently the most practical choice for realizing QPE with high success probability.

arXiv:2404.01396•Apr 1, 2024•4 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

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

Channel-based framework for phase estimation of multiple eigenvalues

Yuan-De Jin, Shiyu Zhang, Wen-Long Ma

Quantum phase estimation (QPE) is a crucial subroutine in many quantum algorithms. It is used to estimate the eigenvalues of a unitary operator acting on a target quantum system. Conventional QPE is often expensive to implement because it requires a large number of ancilla qubits and the ability to perform quantum Fourier transform. Recent developments in iterative QPE reduce the implementation cost by repetitive uses of a single ancilla and classical postprocessing. However, both conventional and iterative schemes often require preparation of the target system in an eigenstate of the unitary operator, while it remains ambiguous to achieve QPE of multiple eigenvalues with no need of initial-state preparation. Here we clarify this issue by developing a theoretical framework based on sequential quantum channels for iterative QPE. We find that QPE of multiple eigenvalues can be efficiently realized for arbitrary initial target system state by actively utilizing the measurement backaction of iterative QPE on the target system with a long coherence time. Specifically, we investigate two iterative QPE schemes based on sequential Ramsey interferometry measurements (RIMs) of an ancilla qubit: (a) the repetitive scheme, which conducts repetitive RIMs to achieve the standard quantum limit in estimating the eigenvalues; (b) the adaptive scheme, which adjusts the parameters of each RIM based on prior measurement outcomes to attain the Heisenberg limit. In both schemes, sequential ancilla measurements generate sequential quantum channels on the target system, gradually steering it to the eigenstates of the estimated unitary operator, while the measurement statistics of the ancilla can reveal the embedded information about its eigenvalues with proper postprocessing. We demonstrate the analysis by simulating a central spin model and evaluate the performance and noise resilience of both schemes.

arXiv:2308.02307•Physical Review A•Feb 6, 2025•2 citations
Filtering is an important technique in quantum computing used for isolating or enhancing some specific states of quantum many-body systems. In this paper, we analyze the performance of filters based on the quantum phase estimation (QPE) algorithm, in which filtering removes states associated with bitstrings in the ancilla register above a given threshold. We show that when the conventional rectangular window function is used for the QPE input state, the resulting filter function exhibits an oscillating behavior known as the Gibbs phenomenon. We also show that in the case of the sine and Kaiser windows, this phenomenon is suppressed. Furthermore, we perform numerical simulations to compare the number of necessary queries to the Hamiltonian time evolution operation of for the QPE-based filtering algorithm and the quantum eigenvalue transformation of unitary matrices with real polynomials (QETU). We find that the number of queries required for Kaiser window-based filtering is comparable to that for QETU with optimized phase angles. As an application of the QPE-based filter, we also study a two-step algorithm for low-energy spectral simulations, composed of a coarse grid for filtering and a fine grid for obtaining final high-resolution spectra. As a benchmark of the proposed scheme for realistic continuous spectra, we present the density-of-states (DOS) calculation of antiferromagnetic type-II MnO in a one-particle approximation.
Quantum Phase Estimation (QPE) routines are known to fail probabilistically even with perfect gates and input states. This effect stems from an incompatibility of finite-sized quantum registers to capture a phase within QPE with phase angles of infinite precision, and the effect extend even beyond what would be reasonably expected from rounding. This effect can be partially mitigated by biasing the phase register with a window, or taper state, from classical signal processing. This paper focuses on how windowing a coherent QPE used as a subroutine can improve the accuracy of the overall algorithm. Specifically we study the quantum task of estimating observables where window-assisted coherent QPE is used as a subroutine to implement a reflection about an eigenstate. Quantum resource estimates show over 2-orders-of-magnitude reduction in Toffoli counts over the previous costed techniques -- also assisted by the use of improved block encoding techniques -- demonstrating an encouraging decrease in resources for quantum computation of molecular observables. Since QPE, as one of only a few quantum building blocks, appears as a subroutine in many algorithms; this analysis also provides a model for understanding how window functions propagate to an improved error in composite algorithms.
Quantum computing is certainly one of the greatest advances in the computer science field. Thanks to the parallelism and entanglement properties, it has proved to offer several advantages compared to the classical algorithms especially in the great reduction of the processing time. Quantum phase estimation (QPE) is one of the most important algorithms for quantum computing. It is known as the eigenvalue finding module for unitary operators. The Fourier transform is the key to this procedure. It has been researched and used to solve many problems such as the order finding problem, and the factoring problem. It was also applied for quantum sampling algorithms and the calculation of the eigenvalues of unitary matrices. In this paper, we study three important quantum algorithms for machine learning that use the QPE algorithm as a subroutine: the quantum principal components analysis (PCA) for data visualization, the Harrow-Hassidim-Lloyd (HHL) algorithm for solving linear systems, and the quantum singular value thresholding (SVT) for matrix completion in recommender systems. We also discuss the advantages and limits of such algorithms compared to their classical versions. Then we discuss potential ways of amelioration of such algorithms, and end with a proposed approach for further improvement.
In recent years, quantum algorithms have been proposed which use quantum phase estimation (QPE) coherently as a subroutine without measurement. In order to do this effectively, the routine must be able to distinguish eigenstates with success probability close to unity. In this paper, we provide the first systematic comparison between two approaches towards maximizing this success probability, one using the quantum singular value transform and the other leveraging window functions, which have been previously studied as priors of the phase value distribution. We find that the quantum singular value transform is significantly outclassed by the window function approach, with the latter able to achieve between 3 and 5 orders of magnitude improvement in the success probability with approximately 1/4 the query cost. Our circuit simulation results indicate that QPE is not a domain which benefits from the integration of QSVT and we show that the use of the Kaiser window function is currently the most practical choice for realizing QPE with high success probability.
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.
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.
Quantum phase estimation (QPE) is a crucial subroutine in many quantum algorithms. It is used to estimate the eigenvalues of a unitary operator acting on a target quantum system. Conventional QPE is often expensive to implement because it requires a large number of ancilla qubits and the ability to perform quantum Fourier transform. Recent developments in iterative QPE reduce the implementation cost by repetitive uses of a single ancilla and classical postprocessing. However, both conventional and iterative schemes often require preparation of the target system in an eigenstate of the unitary operator, while it remains ambiguous to achieve QPE of multiple eigenvalues with no need of initial-state preparation. Here we clarify this issue by developing a theoretical framework based on sequential quantum channels for iterative QPE. We find that QPE of multiple eigenvalues can be efficiently realized for arbitrary initial target system state by actively utilizing the measurement backaction of iterative QPE on the target system with a long coherence time. Specifically, we investigate two iterative QPE schemes based on sequential Ramsey interferometry measurements (RIMs) of an ancilla qubit: (a) the repetitive scheme, which conducts repetitive RIMs to achieve the standard quantum limit in estimating the eigenvalues; (b) the adaptive scheme, which adjusts the parameters of each RIM based on prior measurement outcomes to attain the Heisenberg limit. In both schemes, sequential ancilla measurements generate sequential quantum channels on the target system, gradually steering it to the eigenstates of the estimated unitary operator, while the measurement statistics of the ancilla can reveal the embedded information about its eigenvalues with proper postprocessing. We demonstrate the analysis by simulating a central spin model and evaluate the performance and noise resilience of both schemes.