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Iterative Harrow-Hassidim-Lloyd quantum algorithm for solving resonances with eigenvector continuation

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Pith's one-line read The paper claims that solving the fixed-point equation $C(E,1)\phi=\phi$ with the iterative HHL algorithm yields nuclear resonance energies, demonstrated on the $\alpha$-$\alpha$ G-wave resonance at $11.8211-1.8107i$ MeV in agreement with…

desk verdict A correct fixed-point rewrite of the complex-scaled eigenvalue problem with a new EC+IHHL combination, but the demonstration is curated and the convergence claim outruns the evidence; proof-of-principle, conditional. read the letter →

arxiv 2506.20929 v1 pith:E7PEVJHK submitted 2025-06-26 quant-ph nucl-th

Hantao Zhang , Dong Bai , Zhongzhou Ren This is my paper
classification quant-phnucl-th
keywords quantumcomputingiterativeHHLalgorithmeigenvectorcontinuationcomplexscalingmethodnuclearresonancenon-Hermitianeigenvalueproblemalpha-alphascatteringfixed-pointiteration
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Resonances are complex eigenvalues of a non-Hermitian Hamiltonian, and quantum algorithms for such problems are largely missing. This paper claims that one can turn the complex-scaled Schrodinger equation into the fixed-point equation $C(E,1)\phi=\phi$ and solve it by iterating the Harrow-Hassidim-Lloyd (HHL) quantum linear-systems algorithm, so that the converged pair $(\phi,E)$ is an eigenpair of the complex-scaled Hamiltonian. Combined with eigenvector continuation to reduce the Hilbert space, the scheme is tested on the $\alpha$-$\alpha$ G-wave resonance, converging to $11.8211-1.8107i$ MeV in a few iterations, close to the R-matrix value $11.8079-1.8085i$ MeV. The interest is that it gives a concrete route to non-Hermitian eigenvalue problems on a quantum computer.

What carries the argument

The load-bearing object is the matrix $C(E,\beta)$ defined by $C_{ij}(E,\beta)=(\phi_i|e^{-2i\theta}T+V_N(re^{i\theta})+e^{-i\theta}V_C-(E-\beta)|\phi_j)/\beta$, with $\beta=1$ so the Schrodinger equation becomes the fixed-point condition $C(E,1)\phi=\phi$. The algorithm starts from a trial wavefunction $\phi$ and its expectation energy $E$, embeds the non-Hermitian $C$ into the larger Hermitian matrix $A$, solves $Ax=b$ with HHL, reads off a new wavefunction $\phi^*$, and recomputes $E^*$ until the change falls below $\varepsilon$; the c-product replaces the usual inner product because the complex-scaled states are not Hermitian-conjugate. Eigenvector continuation provides the small orthonormal basis: bound-state wavefunctions at several coupling parameters are orthogonalized and used to build the complex-scaled matrix, which cuts the Hilbert-space dimension and hence the qubit count. A projection step makes each newly found eigenvector orthogonal to previously found ones, allowing successive eigenvalues to be extracted.

What would settle it

Run the same iterative HHL scheme starting from a random initial wavefunction on the $8\times8$ complex-scaled Hamiltonian $H_{\theta=20^\circ}$ of the appendix and check whether the sequence of energies converges to $11.8079-1.8110i$ MeV, converges to a different eigenpair, or fails to converge; alternatively, compute the spectral radius of the iteration map at the resonance energy to test whether it is a contractive fixed point.

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Extended reading notes

Core claim

The central claim is that the fixed-point iteration $C(E,1)\phi=\phi$, with $C(E,1)=H_\theta-(E-1)I$ acting in the c-product, converges to the eigenpairs of the complex-scaled Hamiltonian $H_\theta$. Each iteration solves a linear system $Ax=b$ by HHL, where $A$ is the Hermitian block embedding $A=\begin{pmatrix}0&C\\C^\dagger&0\end{pmatrix}$, and the updated energy is the c-expectation value of $H_\theta$ with the new wavefunction. Numerically, six iterations at tolerance $\varepsilon=10^{-4}$ MeV produce the first eigenpair and five iterations produce the second, which is the resonance at $11.8211-1.8107i$ MeV, in agreement with direct diagonalization ($11.8079-1.8110i$ MeV) and with the R-matrix result ($11.8079-1.8085i$ MeV). The authors take this agreement as evidence that the iterative HHL algorithm reliably computes eigenvalues of non-Hermitian matrices.

Load-bearing premise

The iteration $C(E,1)\phi=\phi$ is assumed to converge to the sought eigenpair, with no proof of contraction or divergence control, and the demonstration relies on two hand-picked initial wavefunctions (given in the appendix) that place the resonance as the second solved eigenvalue.

Editorial extensions

If this is right

  • Five iterations of the IHHL algorithm locate the alpha-alpha G-wave resonance at $11.8211-1.8107i$ MeV, within about $0.02$ MeV of the R-matrix value.
  • With the projection method, all eight eigenvalues of the complex-scaled Hamiltonian are obtained in ascending order of real part, matching direct diagonalization.
  • Eigenvector continuation compresses the problem to a subspace spanned by a handful of bound-state training vectors, reducing the qubit count needed for the quantum solve.
  • Because the non-Hermitian operator is embedded in a Hermitian block matrix, the same iterative HHL route applies to other non-Hermitian eigenvalue problems beyond nuclear resonances.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural stress test the authors did not run is to vary the initial wavefunction systematically; the admitted sensitivity of the eigenvalue ordering to that choice suggests the iteration may need deflation or a contraction guarantee before it can be a black-box resonance solver.
  • The fixed-point formulation could be adapted to compute scattering phase shifts in the complex domain, an extension the paper points toward.
  • The quantum advantage of HHL for this problem is not yet demonstrated end to end: the 1-bit precision used in the simulation and the iteration count would need a resource estimate on actual hardware.
  • Combining the IHHL iteration with classical subspace methods, such as using several parallel initial vectors, might turn the observed fast local convergence into a robust global eigensolver for non-Hermitian matrices.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The manuscript proposes an iterative Harrow-Hassidim-Lloyd (IHHL) algorithm combined with eigenvector continuation and complex scaling to compute resonant eigenvalues of non-Hermitian Hamiltonians. The central construction is the fixed-point equation C(E,1)φ = φ, Eq. (7), with C(E,1) = H_θ - (E-1)I in the EC basis; the non-Hermitian matrix C is embedded in a larger Hermitian matrix A (Eq. (9)), and HHL is used to solve the resulting linear system iteratively while updating the complex energy from a c-product. The method is demonstrated on the α-α G-wave resonance, yielding 11.8211 - 1.8107i MeV from IHHL versus 11.8079 - 1.8110i MeV by direct diagonalization and 11.8079 - 1.8085i MeV from the R-matrix method.

Significance. The algebraic fixed-point identity behind the algorithm is correct, and the idea of combining eigenvector continuation with complex scaling to reduce the dimension of a quantum eigensolver is a sensible and potentially useful direction. The paper is also transparent about the dependence of eigenvalue ordering on the initial wave function, which is a genuine limitation that the authors do not hide. However, the numerical demonstration is a single 8x8 example with hand-picked initial states, the claimed 10^-4 MeV stopping accuracy is not reflected in the reported table, and no convergence or target-selection analysis is provided. If these gaps are addressed, the method could serve as a useful proof-of-principle; as it stands, the manuscript's general claims are stronger than the evidence presented.

major comments (3)
  1. [Sec. 2.1, Eqs. (7)-(9), Fig. 1] The fixed-point identity C(E,1)φ = φ is algebraically correct, but the actual iteration used in the HHL loop is not analyzed. From the description of solving A x = b, the map is effectively φ_{n+1} = C(E_n,1)^{-1} φ_n (or an equivalent linear solve), yet no contractivity, spectral-radius, or basin-of-attraction analysis is provided. Because H_θ is non-Hermitian and the c-product is bilinear, standard Hermitian inverse-iteration arguments do not apply, and complex weights can cancel, so convergence to the desired resonance is not guaranteed for generic initial states. The claim in Sec. 3 that "using iterative HHL algorithm the converged eigenenergies will be obtained after a few iterations" is therefore supported only by the single curated example. The authors should add a convergence analysis or, failing that, a numerical basin study over random initial vectors and training intervals.
  2. [Sec. 3, Table 1] The stated stopping tolerance ε = 10^-4 MeV is inconsistent with the reported results. The IHHL resonance (11.8211 - 1.8107i MeV) differs from direct diagonalization (11.8079 - 1.8110i MeV) by 0.0132 MeV in the real part, and other states differ by up to about 0.03 MeV (e.g., 97.9506 vs 97.9800). If ε refers to the change between successive iterations rather than the error relative to the exact eigenenergy, that definition should be stated and the actual residuals reported; if the discrepancy is due to the 1-bit HHL precision, the text should say so and quantify how this limits the achievable accuracy. As written, the claim that the results "reach the specified precision ε = 10^-4 MeV" is not supported by Table 1.
  3. [Sec. 3 after Fig. 4; Appendix A, Eqs. (A.2)-(A.3)] The resonance is selected by construction rather than found by a generic criterion. The paper states that the order of eigenvalue solution changes with different initial wave functions, and the initial vectors in Appendix A are chosen so that the resonance appears as the second solved eigenvalue. No procedure is given for identifying which converged complex eigenvalue is the physical resonance when the answer is not already known from R-matrix or direct diagonalization, and only one complex-scaling angle θ = 20° is tested. A demonstration of robustness across initial vectors, training intervals, and scaling angles, or an explicit criterion for resonance identification, is needed to support the claimed general algorithm.
minor comments (6)
  1. [Sec. 2.1] The sentence about separating real and imaginary parts of complex wave functions for HHL is under-specified; it should state whether two separate HHL runs are made and how the real and imaginary parts of x are recombined to form φ*.
  2. [Sec. 3] The definition of "1-bit precision" and its relation to "4 additional quantum bits" is unclear; please specify whether precision refers to the number of phase-estimation bits or to a decimal digit count.
  3. [Sec. 1 and Sec. 4] The paper claims HHL offers exponential speedup, but no end-to-end complexity accounting is given for the iterative loop, including the cost of estimating the new energy E* from the output state; a resource estimate (qubits, circuit depth, number of measurements) would strengthen the quantum-advantage claim.
  4. [Appendix A] The matrices and vectors are typeset in a way that is hard to read (spaces within numbers, e.g., "0 .0808"); please typeset them properly and consider making the data available in machine-readable form.
  5. [Throughout] There are several typographical issues: "o ffers" in the abstract, "As a important topic" in Sec. 1, and "int he" in Appendix A should be corrected.
  6. [References] Reference [102] is a bare URL without version or access date; please provide a full citation for the pyQPanda package.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the IHHL fixed-point iteration genuinely solves the complex-scaled eigenproblem; the observed limitations are convergence and selection concerns, not circular reductions.

full rationale

The derivation chain is self-contained rather than circular. The fixed-point map C(E,1) phi = phi in Eqs. (7)-(9) is algebraically equivalent to the Schrodinger equation H_theta phi = E phi: with beta=1, C(E,1)=H_theta-(E-1)I, so C phi = phi iff H_theta phi = E phi. The IHHL loop is therefore a genuine fixed-point iteration over the same non-Hermitian matrix H_theta, not a quantity fitted to the R-matrix value; the benchmark energy 11.8079-1.8085i MeV enters only as an external comparison, not as an input or constraint. The EC training vectors, training interval [1.45,1.75], theta=20 degrees, and initial vectors are chosen with knowledge of the bound spectrum and of where the resonance appears (Appendix A, Sec. 3), which weakens the generality of the demonstration, and the paper explicitly notes that the order of solved eigenvalues can change with different initial vectors and that no stabilization over theta is performed. These are correctness and completeness limitations, not circular reductions: none of the final numbers (11.8211-1.8107i MeV IHHL vs 11.8079-1.8110i diagonalization) is forced by construction, and no load-bearing result is imported solely from the authors' previous work.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard complex-scaling theory and eigenvector continuation assumptions, plus two ad hoc assumptions specific to this paper: convergence of the fixed-point iteration and sufficiency of 1-bit HHL precision. The free parameters are choices of training interval, rotation angle, initial vectors, tolerance, and beta; none of them are fitted to the target resonance energy, but the initial vectors and single-angle choice are informed by the known bound and resonance structure.

free parameters (5)
  • Training parameter interval [1.45, 1.75]
    Eight training points chosen so the system is bound; no resonance information used, but the interval is hand-picked and no sensitivity analysis is given.
  • Complex scaling angle theta = 20 degrees
    Single angle satisfying 2*theta > arctan(|Im(E)/Re(E)|) for the known resonance; the standard stabilization scan over theta is not performed.
  • Initial wave functions phi0^(1st), phi0^(2nd)
    Hand-picked vectors (Appendix A) chosen so that the resonance appears as the second solved eigenvalue; the authors acknowledge the order changes with different initial functions.
  • Convergence tolerance epsilon = 1e-4 MeV
    Stopping criterion; the reported deviations from direct diagonalization (~0.01 MeV) exceed this tolerance, so the claim is inconsistent with the data.
  • beta in C(E,beta) = 1
    Set to 1 for simplicity; the denominator beta shifts the fixed-point equation without affecting the final eigenpair in exact arithmetic.
assumptions (5)
  • domain assumption Complex scaling (ABC theorem) turns resonances into L2 eigenstates of a non-Hermitian Hamiltonian
    Invoked in Sec. 2.1 to justify using eigenvector continuation on bound-state-like wavefunctions.
  • domain assumption The c-product is the correct inner product for complex-scaled wavefunctions
    Used in Eqs. (8), (12), and (14) without derivation or discussion of its domain of validity.
  • domain assumption Training eigenvectors from bound states span a subspace containing the target resonance
    Eigenvector continuation is assumed faithful for the alpha-alpha G-wave resonance; no error bound or validation of the 8-dimensional subspace is given.
  • ad hoc to paper The fixed-point iteration C(E,1) phi = phi converges to the desired eigenpair
    No convergence proof or contractivity analysis is provided; the paper only shows numerical convergence for this one example (Sec. 3).
  • ad hoc to paper HHL with 1-bit precision provides sufficient accuracy for the iteration
    The pyqpanda simulation is run at 1-bit precision without error analysis, yet the paper claims epsilon=1e-4 MeV convergence.

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Cite this review

Pith. "Pith review of Iterative Harrow-Hassidim-Lloyd quantum algorithm for solving resonances with eigenvector continuation." pith.science (2026). https://pith.science/paper/E7PEVJHK

@misc{pith2026250620929,
  author       = {Pith},
  title        = {Pith review of: Iterative Harrow-Hassidim-Lloyd quantum algorithm for solving resonances with eigenvector continuation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/E7PEVJHK}},
  note         = {Machine review of arXiv:2506.20929}
}
abstract

We propose a novel quantum algorithm for solving nuclear resonances, which is based on the iterative Harrow-Hassidim-Lloyd algorithm and eigenvector continuation with complex scaling. To validate this approach, we compute the resonant states of $\alpha-\alpha$ system and achieve results in good agreement with traditional methods. Our study offers a new perspective on calculating eigenvalues of non-Hermitian operators and lays some groundwork for further exploration of nuclear resonances using quantum computing.

Figures

Figures reproduced from arXiv: 2506.20929 by the authors.

Figure 1
Figure 1. Schematic diagram of iterative resonant state solution based on the HHL algorithm. Input part: Given an initial trial wave function [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Schematic of eigenvector continuation with complex scaling. The [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 4
Figure 4. The second eigenvalue obtained using the iterative HHL algorithm [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figures from the paper (1 more)
Figure 3
Figure 3. Figure 3: The first eigenvalue obtained using the iterative HHL algorithm. After [PITH_FULL_IMAGE:figures/full_fig_p004_3.png]

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Forward citations

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Reference graph

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