pith. sign in

arxiv: 2605.18876 · v1 · pith:WU3O2TWXnew · submitted 2026-05-15 · 🪐 quant-ph

Statistical Quantum Phase Estimation: Extensions and Practical Considerations

Amit Surana , Brandon Allen This is my paper

Pith reviewed 2026-05-20 17:46 UTC · model grok-4.3

classification 🪐 quant-ph
keywords statistical quantum phase estimationground state energyrandom compilationchangepoint detectionFourier approximationPauli weightsearly fault-tolerant quantum computingcumulative distribution function
0
0 comments X

The pith

Generalizing random compilation to negative Pauli weights and adding changepoint detection lets statistical quantum phase estimation run without overlap estimates.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper refines the statistical quantum phase estimation framework to overcome practical barriers for early fault-tolerant quantum computers. It extends the random compilation procedure, originally limited to positive Pauli weights in the linear combination of unitaries decomposition, so that it works for Hamiltonians with negative coefficients as well. It replaces the need for a good lower-bound estimate of the trial-state overlap with the true ground state by using a changepoint detection algorithm that locates the first jump in the Fourier-approximated cumulative distribution function. It further shows that symmetry properties of the Fourier series allow the number of circuit executions to be cut in half without loss of accuracy. These changes are demonstrated through numerical simulations on a quantum simulator in Qiskit.

Core claim

By generalizing the random compilation procedure for negative Pauli weights, employing a changepoint detection method for determining ground state energy that does not rely on an estimate of the overlap, and exploiting symmetry of the Fourier series to reduce the number of circuit runs by a factor of two, the SQPE framework becomes applicable to realistic cases on early fault-tolerant quantum computers.

What carries the argument

Changepoint detection applied to the first jump discontinuity of the Fourier-approximated cumulative distribution function of the spectral density, together with a generalized random compilation procedure that accommodates arbitrary Pauli weights.

If this is right

  • Hamiltonians with negative Pauli coefficients can now be treated directly inside the SQPE workflow.
  • No prior estimate of the ground-state overlap is required to locate the ground state energy.
  • The number of circuit executions needed for a given accuracy is halved by using only the positive-frequency part of the Fourier series.
  • The method becomes practical for qubit- and depth-limited early fault-tolerant devices.
  • Numerical validation on a Qiskit simulator confirms that the refined procedure recovers correct ground state energies.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same changepoint technique might be combined with other statistical phase-estimation variants that also produce approximate distribution functions.
  • Symmetry reduction could be applied to related Fourier-based methods in quantum signal processing or Hamiltonian simulation.
  • Testing the approach on small molecular Hamiltonians with known spectra would give a direct check of robustness under realistic noise models.

Load-bearing premise

The changepoint detection method can reliably locate the first jump discontinuity in the Fourier-approximated CDF despite statistical noise from finite samples and the approximation error itself.

What would settle it

Run the changepoint detection procedure on a Hamiltonian with a known ground state energy and a trial state whose overlap is deliberately mis-estimated; if the detected energy deviates systematically from the true value under realistic finite-sample noise, the claim is falsified.

Figures

Figures reproduced from arXiv: 2605.18876 by Amit Surana, Brandon Allen.

Figure 1
Figure 1. Figure 1: SQPE binary search results for Case 1 with [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: CD Algo. 5 applied to Case 1 with η = p0 = 0.1. ACDF samples are shown by blue ‘x’ markers, while successive changepoints during the binary segmentation are shown by green ‘□’ markers. The GSE estimate is shown by red ‘+’ with an absolute error of ∆0 = 0.034 Ha. Also shown for reference is true CDF as the black curve. B. Case 2: H2 Molecule To evaluate SQPE on a more realistic model, we con￾sider molecular… view at source ↗
read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper extends the statistical quantum phase estimation (SQPE) framework with three main refinements: (1) generalization of the random compilation procedure based on LCU decomposition to accommodate negative Pauli weights, (2) replacement of the overlap-dependent GSE identification with a changepoint detection algorithm applied to the Fourier-approximated CDF of the spectral density, and (3) exploitation of Fourier-series symmetry to halve the number of circuit runs while preserving GSE accuracy. These extensions are illustrated via numerical experiments on the Qiskit simulator for a model Hamiltonian.

Significance. If the changepoint method and negative-weight compilation prove robust, the work would meaningfully broaden the applicability of statistical QPE to early fault-tolerant devices by removing two practical bottlenecks (overlap estimation and sign restriction) and cutting sample cost by 2x. The numerical demonstration on a simulator provides initial feasibility evidence, though stronger quantitative validation would be needed to establish the claimed practicality gains over prior SQPE variants.

major comments (3)
  1. [§4] §4 (Changepoint detection): The central claim that the method reliably locates the first jump discontinuity without an overlap estimate is load-bearing, yet the numerical results provide no error bars, success-rate statistics over repeated trials, or analysis of how Fourier truncation error interacts with finite-shot noise; this leaves the weakest assumption untested at the level required for the applicability claim.
  2. [§3.2] §3.2 (Negative-weight LCU generalization): The variance or bias introduced by the generalized random compilation is not bounded or compared against the positive-weight baseline; without such analysis it is unclear whether the procedure preserves the original SQPE sample complexity, which directly affects the practicality argument.
  3. [§5] §5 (Symmetry reduction): The 2x reduction in circuit runs is asserted to maintain GSE accuracy, but no direct side-by-side comparison of estimation error (with vs. without symmetry exploitation) is shown when the changepoint detector is active; this interaction is central to the combined resource-saving claim.
minor comments (2)
  1. [Abstract and §6] The abstract and §6 numerical section should report concrete accuracy metrics (e.g., mean absolute GSE error, success probability) rather than qualitative statements about “illustration.”
  2. [Figures] Figure captions and axis labels for CDF plots should explicitly mark the detected changepoint and indicate the number of shots used.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed feedback. The comments highlight important areas where additional analysis would strengthen the manuscript's claims about the robustness and practicality of the proposed SQPE extensions. We address each major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [§4] §4 (Changepoint detection): The central claim that the method reliably locates the first jump discontinuity without an overlap estimate is load-bearing, yet the numerical results provide no error bars, success-rate statistics over repeated trials, or analysis of how Fourier truncation error interacts with finite-shot noise; this leaves the weakest assumption untested at the level required for the applicability claim.

    Authors: We agree that the current numerical results lack the statistical rigor needed to fully substantiate the reliability of the changepoint detection method. In the revised manuscript we will expand §4 with error bars obtained from repeated independent trials, empirical success-rate statistics across multiple random seeds, and a quantitative analysis of the combined effects of Fourier truncation error and finite-shot noise. Additional simulations will be performed at varying shot counts and truncation orders to report the probability of correctly identifying the ground-state energy. revision: yes

  2. Referee: [§3.2] §3.2 (Negative-weight LCU generalization): The variance or bias introduced by the generalized random compilation is not bounded or compared against the positive-weight baseline; without such analysis it is unclear whether the procedure preserves the original SQPE sample complexity, which directly affects the practicality argument.

    Authors: We acknowledge that a variance or bias analysis is required to confirm that the negative-weight generalization does not degrade the original sample complexity. We will revise §3.2 to include either an analytical bound on the variance of the generalized LCU procedure or direct numerical comparisons of variance and bias against the positive-weight baseline under equivalent conditions. This will clarify the impact on overall resource requirements. revision: yes

  3. Referee: [§5] §5 (Symmetry reduction): The 2x reduction in circuit runs is asserted to maintain GSE accuracy, but no direct side-by-side comparison of estimation error (with vs. without symmetry exploitation) is shown when the changepoint detector is active; this interaction is central to the combined resource-saving claim.

    Authors: We agree that a direct side-by-side comparison is necessary to validate the symmetry reduction in the presence of the changepoint detector. In the revised §5 we will present explicit comparisons of estimation error (mean absolute deviation and success rate) for the full pipeline with and without symmetry exploitation, keeping the changepoint detection active in both cases. This will demonstrate that GSE accuracy is preserved while achieving the claimed factor-of-two reduction in circuit runs. revision: yes

Circularity Check

0 steps flagged

No significant circularity in SQPE extensions and refinements

full rationale

The paper describes practical extensions to the existing SQPE framework: generalizing the LCU-based random compilation to negative Pauli weights, replacing overlap-dependent GSE detection with changepoint detection on the Fourier-approximated CDF, and halving circuit runs via Fourier symmetry. None of these steps are shown to reduce by construction to prior fitted quantities, self-citations, or ansatzes imported from the authors' own prior work. The abstract and reader's summary present the new procedures as independent additions that address stated limitations without re-deriving the core CDF estimation from itself. No equations or claims equate a prediction to its own input parameters. This is the common case of an honest extension paper whose central claims remain externally falsifiable.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard quantum computing assumptions and statistical estimation techniques; no new free parameters, invented entities, or ad-hoc axioms are introduced at the level of the abstract.

axioms (1)
  • domain assumption Fourier approximation of the CDF is sufficiently accurate for changepoint detection to identify the ground-state jump.
    Invoked when replacing overlap-based GSE identification with changepoint detection.

pith-pipeline@v0.9.0 · 5772 in / 1187 out tokens · 96202 ms · 2026-05-20T17:46:53.659733+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    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, and by exploiting symmetry of the Fourier series

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages · 4 internal anchors

  1. [1]

    Quantum measurements and the Abelian Stabilizer Problem

    A. Y . Kitaev, “Quantum measurements and the abelian stabilizer prob- lem,”arXiv preprint quant-ph/9511026, 1995

    work page internal anchor Pith review Pith/arXiv arXiv 1995

  2. [2]

    M. A. Nielsen and I. L. Chuang,Quantum computation and quantum information. Cambridge university press, 2010

    work page 2010

  3. [3]

    The Trotter Step Size Required for Accurate Quantum Simulation of Quantum Chemistry

    D. Poulin, M. B. Hastings, D. Wecker, N. Wiebe, A. C. Doherty, and M. Troyer, “The trotter step size required for accurate quantum simulation of quantum chemistry,”arXiv preprint arXiv:1406.4920, 2014

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  4. [4]

    Chemical basis of trotter-suzuki errors in quantum chemistry simula- tion,

    R. Babbush, J. McClean, D. Wecker, A. Aspuru-Guzik, and N. Wiebe, “Chemical basis of trotter-suzuki errors in quantum chemistry simula- tion,”Physical Review A, vol. 91, no. 2, p. 022311, 2015

    work page 2015

  5. [5]

    Quantum simulation of chemistry with sublinear scaling in basis size,

    R. Babbush, D. W. Berry, J. R. McClean, and H. Neven, “Quantum simulation of chemistry with sublinear scaling in basis size,”npj Quantum Information, vol. 5, no. 1, p. 92, 2019

    work page 2019

  6. [6]

    Qubitization of arbitrary basis quantum chemistry leveraging sparsity and low rank factorization,

    D. W. Berry, C. Gidney, M. Motta, J. R. McClean, and R. Babbush, “Qubitization of arbitrary basis quantum chemistry leveraging sparsity and low rank factorization,”Quantum, vol. 3, p. 208, 2019

    work page 2019

  7. [7]

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

    T. E. O’Brien, B. Tarasinski, and B. M. Terhal, “Quantum phase estimation of multiple eigenvalues for small-scale (noisy) experiments,” New Journal of Physics, vol. 21, no. 2, p. 023022, 2019

    work page 2019

  8. [8]

    Dutkiewicz, B

    A. Dutkiewicz, B. M. Terhal, and T. E. O’Brien, “Heisenberg-limited quantum phase estimation of multiple eigenvalues with a single control qubit,”arXiv preprint arXiv:2107.04605, 2021

    work page arXiv 2021

  9. [9]

    Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers,

    L. Lin and Y . Tong, “Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers,”PRX quantum, vol. 3, no. 1, p. 010318, 2022

    work page 2022

  10. [10]

    Randomized quantum algorithm for statistical phase estimation,

    K. Wan, M. Berta, and E. T. Campbell, “Randomized quantum algorithm for statistical phase estimation,”Physical Review Letters, vol. 129, no. 3, p. 030503, 2022

    work page 2022

  11. [11]

    Scalable quantum circuits for exponential of pauli strings and hamiltonian simulations,

    R. S. Sarkar, S. Chakraborty, and B. Adhikari, “Scalable quantum circuits for exponential of pauli strings and hamiltonian simulations,” arXiv preprint arXiv:2405.13605, 2024

    work page arXiv 2024

  12. [12]

    Selective review of offline change point detection methods,

    C. Truong, L. Oudre, and N. Vayatis, “Selective review of offline change point detection methods,”Signal processing, vol. 167, p. 107299, 2020

    work page 2020

  13. [13]

    A survey of methods for time series change point detection,

    S. Aminikhanghahi and D. J. Cook, “A survey of methods for time series change point detection,”Knowledge and information systems, vol. 51, no. 2, pp. 339–367, 2017

    work page 2017

  14. [14]

    Optimal detection of changepoints with a linear computational cost,

    R. Killick, P. Fearnhead, and I. A. Eckley, “Optimal detection of changepoints with a linear computational cost,”Journal of the American Statistical Association, vol. 107, no. 500, pp. 1590–1598, 2012

    work page 2012

  15. [15]

    Using penalized contrasts for the change-point problem,

    M. Lavielle, “Using penalized contrasts for the change-point problem,” Signal processing, vol. 85, no. 8, pp. 1501–1510, 2005

    work page 2005

  16. [16]

    The Python-based Simulations of Chemistry Framework (PySCF)

    Q. Sun, T. C. Berkelbach, N. S. Blunt, G. H. Booth, S. Guo, Z. Li, J. Liu, J. McClain, E. R. Sayfutyarova, S. Sharma, S. Wouters, and G. K.-L. Chan, “The python-based simulations of chemistry framework (pyscf),” 2017. [Online]. Available: https://arxiv.org/abs/1701.08223

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  17. [17]

    OpenFermion: The Electronic Structure Package for Quantum Computers

    J. R. McClean, K. J. Sung, I. D. Kivlichan, Y . Cao, C. Dai, E. S. Fried, C. Gidney, B. Gimby, P. Gokhale, T. H ¨aner, T. Hardikar, V . Havl´ıˇcek, O. Higgott, C. Huang, J. Izaac, Z. Jiang, X. Liu, S. McArdle, M. Neeley, T. O’Brien, B. O’Gorman, I. Ozfidan, M. D. Radin, J. Romero, N. Rubin, N. P. D. Sawaya, K. Setia, S. Sim, D. S. Steiger, M. Steudtner, Q...

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  18. [18]

    ¨Uber das paulische ¨aquivalenzverbot,

    P. Jordan and E. Wigner, “ ¨Uber das paulische ¨aquivalenzverbot,” Zeitschrift f ¨ur Physik, vol. 47, no. 9-10, pp. 631–651, 1928

    work page 1928