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arxiv: 2605.22920 · v1 · pith:A2WP42DInew · submitted 2026-05-21 · 🪐 quant-ph

Estimating Green's functions with a robust quantum Arnoldi method

Jacob S. Nelson , Andrew B. Baczewski This is my paper

Pith reviewed 2026-05-25 05:46 UTC · model grok-4.3

classification 🪐 quant-ph
keywords Green's functionsquantum Arnoldi methodorthogonal polynomialsquantum impurity modelspectral functionKrylov subspacefinite-precision estimationnonzero temperature
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The pith

A robust quantum Arnoldi method estimates Green's functions over intervals or at multiple points with far lower resources than pointwise methods.

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

The paper introduces the robust quantum Arnoldi method (ROQAM) to return an efficiently computable functional representation of Green's functions rather than single-point estimates. Robustness comes from casting the iteration in terms of orthogonal polynomials, which keeps the projected matrices in upper-Hessenberg form even when matrix elements are known only to finite precision. The required estimation precision relaxes as the iteration depth grows, and a single Krylov subspace suffices for nonzero-temperature cases. Resource estimates for the spectral function of a quantum impurity model show ROQAM using multiple orders of magnitude fewer resources than pointwise quantum singular value transformation.

Core claim

ROQAM projects the Green's function problem onto a Krylov subspace using orthogonal polynomials so that the upper-Hessenberg structure of the projected matrices is preserved despite finite-precision matrix-element estimates; the precision needed for those estimates decreases with increasing iteration depth. Resource counts for the spectral function of a quantum impurity model indicate that this approach outperforms pointwise estimation via quantum singular value transformation by multiple orders of magnitude. The same single Krylov subspace also yields Green's functions at nonzero temperatures.

What carries the argument

Robust quantum Arnoldi method (ROQAM) formulated via orthogonal polynomials that preserves the upper-Hessenberg structure of projected matrices under finite-precision estimation.

If this is right

  • Resource estimates indicate multiple orders of magnitude improvement over pointwise estimation via quantum singular value transformation.
  • The precision required for matrix-element estimation can be reduced as iteration depth increases.
  • Green's functions at nonzero temperatures can be obtained from a single Krylov subspace.
  • The output is an efficiently computable functional representation usable at many points or over intervals.

Where Pith is reading between the lines

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

  • The approach could be tested on larger impurity models or lattice systems to check whether the orders-of-magnitude saving persists beyond the reported cases.
  • Hybrid algorithms might exploit the decreasing precision requirement at deeper iterations to allocate quantum measurements adaptively.
  • The single-subspace property at nonzero temperature may simplify calculations of thermal correlation functions in other condensed-matter settings.

Load-bearing premise

Formulation in terms of orthogonal polynomials preserves the upper-Hessenberg structure of the projected matrices despite finite-precision estimation.

What would settle it

Direct numerical comparison of total quantum resources required to reach a fixed accuracy on the spectral function of a concrete quantum impurity model when using ROQAM versus pointwise quantum singular value transformation.

Figures

Figures reproduced from arXiv: 2605.22920 by Andrew B. Baczewski, Jacob S. Nelson.

Figure 1
Figure 1. Figure 1: FIG. 1. In the quantum Arnoldi method (ROQAM) a quantum computer is used to estimate expectation values [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Estimating the Green’s function of a SIAM with 4 bath sites using the ROQAM. In a) we show the spectral function [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Aggregate T-gate count required to estimate the [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Estimates of the spectral function of SIAM with [PITH_FULL_IMAGE:figures/full_fig_p020_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The estimates Im( [PITH_FULL_IMAGE:figures/full_fig_p020_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. The mean relative error on the spectral function vs ROQAM iteration depth with three values of [PITH_FULL_IMAGE:figures/full_fig_p021_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Example of an artifact resulting from low iteration depth. The Arnoldi iteration is performed to depth 1 on a SIAM at [PITH_FULL_IMAGE:figures/full_fig_p022_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Dependence of the error on the choice of timestep for a SIAM with 4 bath sites on the real and imaginary axes for two [PITH_FULL_IMAGE:figures/full_fig_p022_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. The error resulting from ROQAM with two levels of sampling precision (triangles) compared to the noiseless case [PITH_FULL_IMAGE:figures/full_fig_p023_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Error scaling vs block-encoding subnormalization factor [PITH_FULL_IMAGE:figures/full_fig_p024_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Comparison of three different error budgets. Error budget one (EB1) sets all [PITH_FULL_IMAGE:figures/full_fig_p024_11.png] view at source ↗
read the original abstract

Many applications of Green's functions (GFs) require their evaluation over intervals or at multiple points, motivating quantum algorithms that return an efficiently computable functional representation rather than mere point estimates. We introduce a robust quantum Arnoldi method (ROQAM) that achieves this goal. Its robustness is derived from formulation in terms of orthogonal polynomials, which preserves the upper-Hessenberg structure of the projected matrices despite finite-precision estimation. We also show that as the iteration depth increases, the precision required for matrix-element estimation can be reduced. Resource estimates for the spectral function of a quantum impurity model indicate that ROQAM outperforms pointwise estimation via quantum singular value transformation by multiple orders of magnitude. Finally, we show that the ROQAM can be used to estimate GFs at nonzero temperatures using only a single Krylov subspace.

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

2 major / 2 minor

Summary. The paper introduces the robust quantum Arnoldi method (ROQAM) for estimating Green's functions. It formulates the problem using orthogonal polynomials to preserve the upper-Hessenberg structure of projected matrices under finite-precision matrix-element estimation, shows that required estimation precision decreases with iteration depth, provides resource estimates indicating orders-of-magnitude improvement over pointwise QSVT for the spectral function of a quantum impurity model, and demonstrates that nonzero-temperature Green's functions can be obtained from a single Krylov subspace.

Significance. If the claimed robustness property holds with supporting analysis, ROQAM would represent a meaningful advance in quantum algorithms for Green's functions by enabling efficient functional representations rather than pointwise estimates, with substantial resource savings for impurity models and extensions to finite temperature. The orthogonal-polynomial approach and single-subspace finite-T feature are potentially useful if rigorously established.

major comments (2)
  1. [Abstract] Abstract (robustness paragraph) and the formulation of ROQAM: the central claim that the orthogonal-polynomial representation preserves upper-Hessenberg structure despite finite-precision estimation is load-bearing for the resource estimates, yet the manuscript provides no explicit stability theorem, error bound on the Hessenberg property, or numerical validation at the noise levels used in the impurity-model calculations. Without this, the claimed relaxation of precision with depth and the orders-of-magnitude advantage cannot be confirmed.
  2. [Resource estimates] Resource estimates section (impurity model): the reported outperformance versus pointwise QSVT by multiple orders of magnitude rests directly on the unverified Hessenberg-preservation property. If the structure is not maintained at the finite precisions employed, the polynomial representation cannot be constructed reliably and the complexity comparison does not hold.
minor comments (2)
  1. [Method] Notation for the orthogonal polynomials and the projected matrices should be introduced with explicit definitions before the robustness argument is presented.
  2. [Finite-temperature extension] The single-Krylov-subspace claim for nonzero temperature would benefit from a short explicit statement of the temperature dependence in the relevant theorem or algorithm box.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback. We address the major comments point by point below, agreeing that additional formal support is warranted and outlining the revisions we will make.

read point-by-point responses

  1. Referee: [Abstract] Abstract (robustness paragraph) and the formulation of ROQAM: the central claim that the orthogonal-polynomial representation preserves upper-Hessenberg structure despite finite-precision estimation is load-bearing for the resource estimates, yet the manuscript provides no explicit stability theorem, error bound on the Hessenberg property, or numerical validation at the noise levels used in the impurity-model calculations. Without this, the claimed relaxation of precision with depth and the orders-of-magnitude advantage cannot be confirmed.

    Authors: We agree that an explicit stability theorem and targeted numerical validation would strengthen the presentation. The preservation follows from the three-term recurrence that defines the orthogonal polynomial basis: by construction, the projected operator remains strictly upper Hessenberg when inner products are exact, and finite-precision estimates perturb only the existing nonzero entries while leaving the structural zeros intact to within the estimation error. In the revised manuscript we will add a formal theorem that bounds the deviation from exact Hessenberg form in terms of the per-element estimation precision, together with numerical experiments performed at the noise levels used in the impurity-model resource estimates. These additions will also make the depth-dependent precision relaxation explicit. revision: yes

  2. Referee: [Resource estimates] Resource estimates section (impurity model): the reported outperformance versus pointwise QSVT by multiple orders of magnitude rests directly on the unverified Hessenberg-preservation property. If the structure is not maintained at the finite precisions employed, the polynomial representation cannot be constructed reliably and the complexity comparison does not hold.

    Authors: The resource comparison indeed depends on reliable construction of the polynomial representation, which in turn relies on the Hessenberg property. With the stability theorem and numerical validation added as described above, the orders-of-magnitude advantage will be placed on a rigorous footing. In the revised version we will update the resource-estimates section to cite the new theorem and validation results, and we will clarify how the required estimation precision scales with iteration depth. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via new formulation

full rationale

The paper presents ROQAM as a new method whose robustness follows from reformulating the Arnoldi iteration in terms of orthogonal polynomials. This is asserted to preserve the upper-Hessenberg structure under finite-precision estimation, with the resource advantage then following from relaxed precision requirements at greater depth. No equations, fitted parameters, or predictions are shown reducing to their own inputs by construction. No self-citation chains or ansatzes imported from prior author work are visible in the supplied text. The central claims rest on the properties of the introduced formulation rather than re-deriving or renaming existing fitted quantities, making the chain self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only; no explicit free parameters, ad-hoc axioms, or invented entities are stated. Relies on standard quantum linear algebra and orthogonal polynomial properties.

axioms (1)
  • domain assumption Orthogonal polynomials preserve upper-Hessenberg structure under finite-precision matrix-element estimates
    Invoked to derive robustness of the Arnoldi iteration.

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    Arnoldi quadrature After [M] is found, we estimate the action of some matrix functionf(M) on the starting vector|χ 0⟩by evaluating f([M]). It is considerably cheaper to evaluatef([M]) thanf(M), because the dimension of [M] is much smaller than that ofM. Formally, we approximate the quadratic form off(M) as ⟨χ0|f(M)|χ 0⟩ ≈ ⟨χ 0|Q†f([M])Q|χ 0⟩.(A23) This ex...

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  62. [64]

    Inputs In order to perform the Arnoldi iteration on a quantum computer we require access toHand the initial vector |χ0⟩. We assume efficient access to a (λ, m, ϵ) block-encoding [19, 20, 47] ofHdefined as (⟨0m| ⊗I)U M(|0m⟩ ⊗I)− H λ ≤ϵ.(B1) Efficient initialization of|χ 0⟩is a more stringent requirement that generally cannot be guaranteed. State preparatio...

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  63. [65]

    The standard formulation requires storage of the entire Arnoldi basis, which would requirercopies of the system register forrlevels of iteration

    Quantum Arnoldi iteration While we could perform the Arnoldi iteration on a quantum computer using the standard formulation in Ap- pendix A 2, the orthogonal polynomial formulation in Appendix A 2 is more efficient in both space and time. The standard formulation requires storage of the entire Arnoldi basis, which would requirercopies of the system regist...

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  64. [66]

    Choice of timestep In switching fromH/λtoU=e −iH/λ∆t we have introduced a new parameter ∆t. The benefit is that ∆tcan be tuned to cancel out adverse effects ofλ, however, the choice of ∆tgreatly affects the performance of the method in the presence of finite sampling precision. Before considering the effects of finite precision we first show restrictions ...

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  65. [67]

    We see from Eq

    Estimation precision As we have seen from Appendix B 2, the quantum Arnoldi iteration reduces to the estimation of⟨χ 0|U l|χ0⟩for l= 0,1, ..., r. We see from Eq. A27, and from simulated data, that the main effect of finite estimation precision is to introduce a bias proportional to the precision. Tighta prioribounds are difficult to derive in the presence...

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  66. [68]

    To handle this, a broadening parameter (iγ) is added to the frequency and the Green’s function on the real axis is defined in the limit thatγ→0 +

    Zero-temperature Green’s functions At zero temperature these expectation values are evaluated with respect to the ground state G+ pq(t) = Θ(t)⟨ψ 0|U †(t)apU(t)a † q |ψ0⟩and (C2a) G− pq(t) = Θ(−t)⟨ψ 0|U †(t)a† pU(t)a q |ψ0⟩.(C2b) The Green’s functions are expressed in the frequency domain through a Fourier transform, however, they will contain poles at the...

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  67. [69]

    The main difficulty lies in expressing the Green’s function in the frequency domain when the initial state is not an eigenstate ofH, as is the case forT= 0

    Thermal Green’s functions It might seem more complicated to apply the Arnoldi method at nonzero temperature. The main difficulty lies in expressing the Green’s function in the frequency domain when the initial state is not an eigenstate ofH, as is the case forT= 0. In obtaining Equation C5b we used the fact that⟨ψ 0|e iHt =⟨ψ 0|e iE0t in order to commute ...

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  68. [70]

    The spectral function For smallγ, the imaginary component ofG(ω) describes the probability distribution of single-particle excitations. This is called the spectral function Apq(ω, γ) =−π −1Im(Gpq(ω+iγ)),(C23) It will be useful to consider the effects ofγwhen estimating the spectral function, therefore we explicitly account for it in our definition. Append...

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    [54] and [55]

    The matrix inversion polynomial We now show the polynomial used for matrix inversion following Refs. [54] and [55]. Like all methods for matrix inversion the complexity depend on the matrix’s condition numberκ. Hereκis aneffectivecondition number for the block-encoded matrix given as the ratio of the subnormalization factor to the eigenvalue of smallest a...

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  70. [72]

    The Hadamard test is a well-known protocol to estimate the expectation values of unitaries matrices

    The Hadamard test for block-encoded matrices Using the matrix inversion polynomial we use QSVT to transform a block-encoding ofω+iγ±(H−E 0) into a block-encoding of [ω+iγ±(H−E 0)]−1, what remains is to estimate the expectation value of this matrix with respect to thea p |ψ0⟩ora † p |ψ0⟩states. The Hadamard test is a well-known protocol to estimate the exp...

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    However, one of the main drawbacks of the ROQAM is the lack of tight a prioribounds under finite-precision estimation

    Ana posterioribound for the ROQAM The motivation for the ROQAM for Green’s function estimation is to avoid needing to employ a seperate instance of QSVT or QLSA for every frequencyω. However, one of the main drawbacks of the ROQAM is the lack of tight a prioribounds under finite-precision estimation. Ana posterioribound can be established if we are allowe...

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  72. [74]

    two-site DMFT

    The single-impurity Anderson model The SIAM is a simple model of a strongly correlated system, describing a localized quantum impurity coupled to a bath of free fermions. The SIAM Hamiltonian givenn bath spin orbitals is defined as H=H imp +H bath +H int Himp = X σ (εimp −µ)a † σaσ +U a † ↓a↓a† ↑a↑ Hbath = nbathX σ,j=1 (εj −µ)c † jσ cjσ Hint = nbathX σ,j=...

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  73. [75]

    noiseless

    Emulating the ROQAM To estimateG + σ (ω) at zero temperature we performed the Arnoldi iteration on the starting state|χ 0⟩=a σ |ψ0⟩ using the generating matrixU=e −iH∆t. We used a Python program to calculate the expectation values⟨χ 0|U l |χ0⟩. To emulate the effects of finite sampling, independently random numbers with mean zero and varianceδ 2 l were ad...

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  74. [76]

    Behavior under finite-precision errors We also studied the behavior of the ROQAM in the presence of finite-precision estimation. To mimic estimating the expectation values⟨χ 0|U l|χ0⟩to precisionδ l, we add independent random variables with standard deviationδ l to the real and imaginary components of⟨χ 0|U l|χ0⟩. The main effect of finite precision will ...

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  75. [77]

    These resource estimates do not include the cost of state preparation or the cost to estimate E0, which is a input to the problem

    Resource estimation We estimate theTgates required to estimate the SIAMG(ω) to 1% mean relative error on the real and imaginary axes at zero temperature. These resource estimates do not include the cost of state preparation or the cost to estimate E0, which is a input to the problem. As such, these resource estimates should not be seen as precise estimate...

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