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arxiv: 2601.05788 · v3 · submitted 2026-01-09 · 🪐 quant-ph

On the robustness of Quantum Phase Estimation to compute ground properties of many-electron systems

Wassil Sennane , J\'er\'emie Messud This is my paper

Pith reviewed 2026-05-16 16:01 UTC · model grok-4.3

classification 🪐 quant-ph
keywords Quantum Phase Estimationmany-electron systemsground state energyTrotterizationquantum computingcomputational chemistryparameter settingdiscretization effects
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The pith

Constructive parameter setting for Quantum Phase Estimation makes its Trotterized complexity depend mostly on physical system properties rather than phase qubit count.

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

The paper gathers and refines conditions on QPE free parameters such as time step, phase qubit number, initial state, and shot count to achieve target success probability and precision when estimating ground energies of many-electron systems. It analyzes the blurring function arising from discretization and introduces a mitigation approach to reduce associated errors. Numerical illustrations on the H2 molecule show that, once these conditions are met, the dominant cost factors in the Trotterized implementation are the system's intrinsic physical scales rather than the chosen number of phase qubits. This framework is presented as a step toward more automated QPE use in predictive chemistry and materials calculations.

Core claim

We explicit a constructive method to set the QPE free parameters for ground energy estimation and ground state projection, gathering disseminated results from previous works, refining these results and developing new conditions for achieving target performance. We detail the impact of the QPE blurring function, related to discretization effects, and propose a method to overcome corresponding pathologies. We finally demonstrate that, using the conditions gathered here, the complexity of the Trotterized version of QPE tends to depend mostly on physical system properties and weakly on the number of phase qubits.

What carries the argument

The constructive method for setting QPE parameters together with blurring-function mitigation to control discretization effects.

If this is right

  • Ground-state energy estimates for molecules can be planned with resource costs predicted mainly from molecular properties.
  • Choice of phase qubit count becomes less critical to overall runtime once the other parameter conditions are satisfied.
  • Automation of QPE workflows for chemistry becomes more practical because fewer parameters require extensive tuning.
  • Discretization errors can be systematically reduced, improving precision without proportional increases in circuit depth.

Where Pith is reading between the lines

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

  • The same parameter conditions may help stabilize other phase-estimation-based quantum chemistry routines on near-term hardware.
  • Resource estimates for early fault-tolerant quantum computers applied to molecules could be tightened by incorporating this weak dependence on phase qubits.
  • Extending the analysis to open-shell or excited-state calculations would test whether the same dominance of physical properties holds.

Load-bearing premise

The constructive parameter-setting method and blurring mitigation developed from H2 examples will extend to larger many-electron systems without introducing new pathologies.

What would settle it

Numerical Trotterized QPE runs on a molecule larger than H2, such as LiH, that check whether the observed complexity remains dominated by physical properties or begins to scale strongly with phase qubit number.

Figures

Figures reproduced from arXiv: 2601.05788 by J\'er\'emie Messud, Wassil Sennane.

Figure 1
Figure 1. Figure 1: QPE algorithm, with outputs  l ∗ , [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: |f(θ (t) j − l 2N )| 2 as a function of the discrete variable l 2N with a fixed value of θ (t) j , for several values of N. The right figure is a zoom of the left figure around θ (t) j . Vertical lines denotes the l (N) j 2N value (with highest probability |f(θ (t) j − l (N) j 2N )| 2 ). Considering a larger interval around l (N) j , as in eq. (23), [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: System with 4 eigenstates and randomly generated θ (t) j . (Top) |f(θ (t) j − l 2N )| 2 as a function of l 2N for several N values. (Middle) P(l) as a function of l 2N for several N values. (Bottom) Overlap between each exact eigenstate of H and: the initial state (|cj | 2 , grey), and the final state after phase measurement (|c (l ∗ ) j | 2 , color), for several N values. 0 1 2 3 4 5 6 7 a 0.40 0.45 0.50 … view at source ↗
Figure 4
Figure 4. Figure 4: Probability to measure a value of l0 satisfying the relaxation defined by eqs. (23), (46) and (61), as a function of the a qubits added to Nmin. The system considers the same θ (t) j and cj than the case in [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: illustrates both the upper bound probability for several values of N in the same configuration as in [PITH_FULL_IMAGE:figures/full_fig_p019_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Implementation of U 2 q using first-order Trotterization (p = 1). Multiple publications have studied the Trotterization accuracy. Especially, Refs. [44, 41] bound ∥U 2 q − S(U 2 q , p)∥. Adapting these works to our case, we have: ∥U 2 q − S(U 2 q , p)∥ ≤ |Cp| (2πt2 q ) p+1 n(q, t) p , (80) where |Cp| has the unit of an energy power (p + 1) and is built from commutators of the Hβγβ 5 . We deduce using eq. (… view at source ↗
Figure 7
Figure 7. Figure 7: QPE features on H2 with t = 1 2 P β |γβ| choice. (Top) P(l) as a function of l 2N for several N values. (Middle) Overlap between each exact eigenstate of H with the initial state (|cj | 2 , grey) and the final state after phase measurement (|c (l ∗ ) j | 2 , color), for several N values. (Bottom left) Upper bound of the minimum number of shots mϵ required so that l ∗ is the most read phase with 1 − ϵ proba… view at source ↗
Figure 8
Figure 8. Figure 8: QPE result on H2 . The three values of t presented in [PITH_FULL_IMAGE:figures/full_fig_p024_8.png] view at source ↗
read the original abstract

We propose an analysis of the Quantum Phase Estimation (QPE) algorithm applied to many-electron systems by investigating its free parameters such as the time step, number of phase qubits, initial state preparation, number of measurement shots, and other parameters related to the unitary operators implementation. A deep understanding of these parameters and their impact on QPE probability of success and precision of the results is important to pave the way towards more automation of QPE applied to predictive computational chemistry and material science. We here explicit a constructive method to set the QPE free parameters for ground energy estimation and ground state projection, gathering disseminated results from previous works, refining these results and developing new conditions for achieving target performance. We detail the impact of the QPE `blurring function', related to discretization effects, and propose a method to overcome corresponding pathologies. We finally demonstrate that, using the conditions gathered here, the complexity of the Trotterized version of QPE tends to depend mostly on physical system properties and weakly on the number of phase qubits. Various numerical results illustrate the impact of QPE free parameters on success probability and discretization effects. The impact of Trotterization and other features on the precision of the results are illustrated by first numerical simulations on the H2 molecule, that allows us to derive useful insights.

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

1 major / 2 minor

Summary. The manuscript analyzes free parameters of Quantum Phase Estimation (QPE) for ground-energy and ground-state estimation in many-electron systems. It gathers, refines, and extends conditions on time step, phase-qubit count, initial-state preparation, shot number, and unitary implementation to achieve target success probability and precision; it examines discretization-induced blurring and proposes mitigation; and it claims that, once these conditions are applied, Trotterized QPE complexity depends primarily on physical system properties and only weakly on the number of phase qubits, with supporting numerical illustrations performed on H2.

Significance. If the constructive parameter rules and the reported weak phase-qubit dependence generalize, the work would supply practical, automatable guidelines for QPE in quantum chemistry, reducing empirical tuning and focusing complexity estimates on intrinsic molecular properties rather than register size.

major comments (1)
  1. [Abstract / numerical illustrations] Abstract and numerical-results section: the claim that Trotterized QPE complexity 'tends to depend mostly on physical system properties and weakly on the number of phase qubits' rests exclusively on H2 simulations. These occur in a regime of tiny Hilbert space, near-unity initial-state overlap, and easily controlled Trotter error; the manuscript provides no analysis or additional data showing that the gathered conditions continue to suppress phase-qubit dependence once state-preparation fidelity drops or Trotter-step count scales with larger electron numbers and basis sets.
minor comments (2)
  1. [Abstract] Abstract: success-probability and discretization claims are stated without error bars, full data-exclusion rules, or explicit derivation of the new conditions, reducing immediate verifiability.
  2. [Blurring-function discussion] Notation for the blurring function and its mitigation procedure should be introduced with a dedicated equation or algorithm box to improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback. We address the major comment below and will revise the manuscript to strengthen the presentation of our claims.

read point-by-point responses

  1. Referee: [Abstract / numerical illustrations] Abstract and numerical-results section: the claim that Trotterized QPE complexity 'tends to depend mostly on physical system properties and weakly on the number of phase qubits' rests exclusively on H2 simulations. These occur in a regime of tiny Hilbert space, near-unity initial-state overlap, and easily controlled Trotter error; the manuscript provides no analysis or additional data showing that the gathered conditions continue to suppress phase-qubit dependence once state-preparation fidelity drops or Trotter-step count scales with larger electron numbers and basis sets.

    Authors: We acknowledge that the numerical illustrations are performed exclusively on H2. The constructive conditions on time step, phase-qubit count, initial-state overlap, shot number, and unitary implementation are nevertheless derived from general theoretical bounds that apply to arbitrary many-electron systems; they ensure that the success probability and target precision are met by controlling the blurring function and the effective spectral gap independently of register size. Once these conditions are satisfied, the leading complexity term arises from the Trotterized unitary cost, which is governed by the molecular Hamiltonian spectrum and the required precision rather than the number of phase qubits. We will revise the abstract and the numerical-results section to make this theoretical grounding explicit, add a dedicated paragraph discussing how the same conditions suppress phase-qubit dependence when state-preparation fidelity is lower and Trotter-step counts grow with system size, and include a brief scaling argument based on the gathered bounds. We will also add a short numerical check on LiH (still within the same code framework) to illustrate the behavior beyond H2. revision: yes

Circularity Check

0 steps flagged

No significant circularity; central claim follows from gathered conditions and H2 numerics without reduction to self-defined inputs

full rationale

The paper gathers and refines QPE parameter conditions from prior literature, develops new conditions for success probability and blurring mitigation, then demonstrates via explicit formulas and H2 simulations that Trotterized complexity depends mostly on physical properties and weakly on phase-qubit count. No step equates a derived prediction to a fitted parameter by construction, imports uniqueness via self-citation, or smuggles an ansatz. The derivation is self-contained against standard QPE theory and external Trotter bounds; H2 numerics serve only as illustration, not as the sole support for the general claim.

Axiom & Free-Parameter Ledger

3 free parameters · 2 axioms · 0 invented entities

The claims rest on standard quantum phase estimation correctness, Trotter approximation validity for the unitary, and the assumption that H2 results generalize; no new entities are postulated.

free parameters (3)
  • time step
    Investigated as a tunable parameter affecting discretization and success probability.
  • number of phase qubits
    Analyzed for its weak impact on overall complexity once other conditions are met.
  • number of measurement shots
    Studied for effect on precision and success rate.
axioms (2)
  • standard math Quantum Phase Estimation correctly estimates eigenvalues of the unitary operator when parameters are properly set.
    Invoked throughout the analysis of success probability and precision.
  • domain assumption Trotterization provides a controllable approximation to the time-evolution unitary for many-electron Hamiltonians.
    Used in the complexity scaling argument.

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