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REVIEW 3 major objections 4 minor 1 cited by

Classical optimization algorithms for diagonalizing quantum Hamiltonians

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read A classical optimization algorithm can diagonalize a Hamiltonian without ever getting stuck in a spurious minimum.

desk verdict A clean landscape theorem is buried under an invalid headline example: Example 1's D contains Y Paulis, so it is not diagonal by the paper's own Eq. (2), and the claimed exponential-Lie-algebra family is unproven. read the letter →

arxiv 2506.17883 v1 pith:RRT5FLWZ submitted 2025-06-22 quant-ph cs.NAmath.NAmath.OC

Taehee Ko , Sangkook Choi , Hyowon Park , Xiantao Li This is my paper · ORCID
classification quant-phcs.NAmath.NAmath.OC
keywords HamiltoniandiagonalizationPaulibasiscostlandscapeglobalminimafast-forwardingLiealgebragradientdescentquantumsimulation
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

This paper aims to establish that Hamiltonian diagonalization, the key primitive for long-time quantum simulation, can be recast as a classical optimization problem with a benign landscape: every nonzero stationary point of the cost function is a global minimum and yields a valid diagonalization. The cost function is built from Pauli-string coefficients, penalizing off-diagonal terms and enforcing unitarity through an orthogonality constraint. If true, this provides a route to polynomial-time diagonalization for a broader class of fast-forwardable Hamiltonians, including cases whose Lie algebras are exponentially large. It also supplies a posteriori error bounds that convert optimization error into diagonalization accuracy, plus a randomized-coordinate variant with cheaper per-iteration cost.

What carries the argument

The central object is a parameterized Pauli-basis ansatz $K(r,\theta)=\sum_{j=1}^d r_j e^{i\theta_j} P_j$, whose coefficients $r_j e^{i\theta_j}$ are exactly the Pauli-basis coefficients of $K$ in polar form. The cost $F$ combines $f(r,\theta)$, the squared off-diagonal Pauli coefficients of $K^\dagger H K$, with an orthogonality penalty built from coefficients $\phi_P$ of $K^\dagger K$. The identity that carries the argument is $\sum_j r_j \partial F/\partial r_j = 4F(r,\theta)$, which, together with the explicit gradient formula, forces $F=0$ at any nonzero stationary point; the orthogonality part then forces $K^\dagger K = \|r\|^2 I$, making the stationary $K$ unitary up to scaling.

What would settle it

A concrete falsifier is a numerical scan on a small Hamiltonian, such as the four-qubit XXZ model, with a fixed Pauli ansatz known to contain the exact diagonalizing unitary: run Algorithm 1 from many random initial points and check whether any nonzero stationary point with $F>0$ is reached, since the theorem predicts none. A complementary test would construct a Hamiltonian whose diagonalizing unitary provably lies outside every polynomial-size Pauli ansatz and show the algorithm's output is not a diagonalization, exposing the practical scope limitation.

Watch

Extended reading notes

Core claim

The central claim is Theorem 3: any nonzero stationary point $(r_c,\theta_c)$ of the total cost $F(r,\theta)=\sum_{P\in G_1} \mathrm{tr}(K(r,\theta)^\dagger H K(r,\theta) P)^2 + \sum_{P\in G_2} \phi_P(r,\theta)^2$ is a global minimum, and $K(r_c,\theta_c)=\sum_{j=1}^d r_j e^{i\theta_j} P_j$ is unitary up to scaling, giving $H = \|r\|^{-4} K h K^\dagger$ with $h=K^\dagger H K$ diagonal. Hence every nontrivial fixed point of gradient descent with normalization is a genuine diagonalization. The paper also proves convergence under a uniform Kurdyka-Lojasiewicz condition and derives an a posteriori bound that maps small $F$ to closeness of $H$ to $\widetilde{H}=K h_0 K^\dagger$ and of the corresponding eigenspace projectors.

Load-bearing premise

The algorithm must be handed a Pauli-string ansatz $K(r,\theta)$ whose span contains a unitary that diagonalizes $H$, and in every numerical experiment that ansatz and its initialization are taken from the known eigendecomposition of the target Hamiltonian; the paper gives no procedure to discover such an ansatz from $H$ alone.

Editorial extensions

If this is right

  • Any nontrivial stationary point of the optimization is a correct diagonalization, so gradient descent with normalization cannot get stuck at spurious minima; the only failure mode is convergence to $r=0$, which the normalization step is designed to avoid.
  • The convergence theorem gives sublinear or linear rates under a uniform KL condition, with iteration count $O\left(\epsilon^{-\max\{\alpha-1,0\}} \log(1/\epsilon)\right)$, translating numerical optimization effort into a diagonalization guarantee.
  • The a posteriori error bound means a user can certify approximate diagonalization and eigenspace accuracy directly from the final cost function value.
  • For Hamiltonians whose diagonalizing unitary is sparse in the Pauli basis, the per-iteration cost is polynomial: $O(d^4 M^2)$ deterministic and $O(d^2 M)$ for the randomized-coordinate variant.
  • The constructed family of Hamiltonians shows that quantum-diagonalizable Hamiltonians strictly contain those with polynomially sized Lie algebras, so exponential Lie-algebra dimension alone does not preclude efficient diagonalization.

Reading between the lines

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

  • A natural testable extension is a bootstrap that grows the Pauli ansatz from the terms of $H$ until the stationary-point condition $F=0$ is met, since the paper gives no autonomous procedure for discovering the ansatz.
  • The construction of Example 1 suggests a general recipe for generating hard-to-diagonalize-looking Hamiltonians: conjugate a diagonal Pauli-sparse $D$ by products of a few anticommuting Pauli rotations, which may inflate the dynamical Lie algebra to $su(2^n)$ while keeping the eigendecomposition Pauli-sparse.
  • The continuation observation that optimal parameters for one Hamiltonian initialize well for nearby parameters points toward a practical parameter-sweep workflow for tunable or time-dependent models, though the paper demonstrates this only numerically.
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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 / 4 minor

Summary. The manuscript presents classical optimization algorithms for Hamiltonian diagonalization. Given a Hamiltonian H expressed as a sum of M Pauli strings, the authors parameterize a candidate unitary K(r,θ) = Σ_j r_j e^{iθ_j} P_j and define a cost F(r,θ) consisting of the squared off-diagonal Pauli coefficients of K†HK plus a penalty enforcing K†K ∝ I. They prove (Theorems 2 and 3) that every nonzero stationary point of F is a global minimum and corresponds to a KHK decomposition of H, and they give a posteriori error bounds linking approximate cost minimization to approximate diagonalization. A deterministic gradient-descent-with-normalization algorithm is analyzed under a uniform KL condition (Assumption 1), and a randomized-coordinate variant is proposed with reduced per-iteration cost. The paper also claims a new family of quantum-diagonalizable Hamiltonians with exponentially large Lie algebras but polynomially sparse Pauli decompositions (Examples 1 and 2). Numerical experiments on random sparse Hamiltonians, the XXZ model, and a Hubbard model are reported.

Significance. If all claims were valid, the paper would make a useful contribution: a classically implementable optimization landscape with no spurious stationary points, polynomial per-iteration cost for Hamiltonians with sparse Pauli diagonalizations, and error certificates would complement existing Lie-algebraic and variational diagonalization methods. The algebraic core—the stationary-point argument in Theorems 2 and 3 and the error bound in Lemma 2—appears sound and is the strongest part of the paper. The code is publicly available and the numerical experiments are clearly described. However, the claimed new family of quantum-diagonalizable Hamiltonians is not established as presented, and the polynomial-efficiency claim presupposes a known ansatz containing the diagonalizing unitary; these issues affect the paper's central advertised contributions. The convergence theorem also has a regime where the proof does not support the stated rate.

major comments (3)
  1. [§IV, Example 1 (Eq. (35)) and Definition 2] The matrix D = I + Σ_{j=1}^n d_j Y_j is not diagonal in the computational basis under the paper's own classification in Eq. (2), which places every Pauli string containing an X or Y factor in the off-diagonal part. Consequently H = U D U† is not an eigen-decomposition and U is not a diagonalizing unitary, so the example does not satisfy Definition 2. This is not a typo: Appendix C uses the Y_j factors in D to obtain X_1, Z_1, Z_1Z_2 and the commutators that generate the set (C1); replacing Y_j by Z_j would break the Lie-algebra argument. The claimed new family of quantum-diagonalizable Hamiltonians with exponentially large Lie algebras, and the discussion in Section VI that this family strictly contains the PLA Hamiltonians, are therefore unsupported.
  2. [§II–§V, Eq. (4) and Algorithm 1] The optimization problem is only defined relative to a user-supplied Pauli ansatz {P_j}, and the correctness and efficiency claims assume that this ansatz spans a diagonalizing unitary of H with d = poly(n). No procedure is given to construct such an ansatz from H alone; in the Section V experiments the initial K is obtained from an eigensolver and then expanded in Pauli strings, so the ansatz and initialization are derived from the known eigendecomposition. The abstract's claim of polynomial-time efficiency for diagonalizing unknown H is therefore not established; the paper demonstrates polynomial per-iteration cost for a subproblem whose input includes an oracle-like ansatz.
  3. [Theorem 4 and Appendix A] The convergence-rate derivation is not valid for the claimed range α ∈ [0,1]. In Eqs. (A9)–(A10), the factor 1 − μa F(x_t)^{α−1} is used; when α < 1 and F(x_t) is small, F^{α−1} = 1/F^{1−α} is large and the factor can become negative, so the inequality ε ≤ (1 − μa ε^{α−1})^T F(x_0) used to extract T is meaningless. The stated bound T = O(ε^{−max{α−1,0}} log(1/ε)), which sets the exponent to zero for α ∈ [0,1], does not follow from the proof; a separate argument is needed for the linear-convergence claim in that regime.
minor comments (4)
  1. [Algorithms 1 and 2] The text contains the typo 'learing rate schedule'; it should read 'learning rate schedule'.
  2. [§II, Eq. (3) and definition of G1] As written, G1 is defined through K(r,θ) but K depends on the optimization parameters; the definition should clarify that G1 is the fixed set of all off-diagonal Pauli strings that can appear in the symbolic expansion of K†HK, with zero coefficients allowed at particular parameter values.
  3. [§III.D] The phrase 'quadratically reduces the per-iteration cost from O(d^4 M^2) to O(d^2 M)' is imprecise, since the ratio is O(d^2 M); it would be clearer to say the reduction factor is O(d^2 M).
  4. [Abstract and §III.D] The randomized-coordinate algorithm is presented as a practical contribution, but Section III.D states that no convergence guarantee is available for it ('deriving a lower bound of the norm of r_{t+1} and a convergence guarantee is still an open issue'); the abstract and conclusion should state this limitation explicitly rather than presenting the variant as fully supported.

Circularity Check

2 steps flagged · score 7.0 of 10

Polynomial-time diagonalization is demonstrated only when the diagonalizing unitary is supplied as the ansatz and/or initialization; the numerical tests construct H from a known U,D or warm-start from an eigensolver. Example 1's non-diagonal D adds a separate correctness flaw.

  1. fitted input called prediction [Section V, Numerical Result, Fig. 1 setup]
    "First, we sample a pair of polynomial numbers of Pauli strings, say, ({P_{i_a}}_{i_a∈I_a}, {P_{i_b}}_{i_b∈I_b}) ... a diagonal matrix is constructed as D=∑_{i_a∈I_a} c_{i_a}P_{i_a}, and a unitary matrix is formulated as the product of Pauli rotations, say, U=∏_{i_b∈I_b} exp(i c_{i_b}P_{i_b}). Lastly, with this pair of unitary and diagonal, we construct a Hamiltonian that is represented by UDU†."

    The Hamiltonians used to test Algorithm 1 are created by choosing U and D first; the ansatz dimension d in K(r,θ) is then read off from the Pauli support of the chosen U, as in 'the number of parameters in K in (3) is 8,32,8,32'. Thus the algorithm is run inside a parameter manifold defined by a known diagonalizing unitary. The claimed polynomial efficiency for Definition 2 is conditional on this supplied decomposition; no procedure selects the ansatz from H alone.

  2. fitted input called prediction [Section V, Numerical Result, XXZ/Hubbard initialization]
    "we compute its eigen-decomposition using an eigensolver, obtain a unitary K, expand K in terms of Pauli strings as (3) and use the coefficients as the initial parameters for the optimization (6) with the Hamiltonian defined by the target value of Δ."

    Even when the target Hamiltonian is not constructed from the same U, the initial point is the Pauli expansion of an eigendecomposition computed by an eigensolver for a nearby Hamiltonian. The cost function is then minimized from a point already near a true diagonalization; the experiments therefore do not show that the algorithm can find the ansatz or the solution from H alone. The diagonalization is supplied, up to a parameter perturbation, before optimization begins.

full rationale

The algebraic landscape theorem (Theorem 3) is self-contained: the proof that a nonzero stationary point has F=0 follows from the Euler identity Σ_j r_j ∂F/∂r_j=4F and the sum-of-squares form of F, with no hidden fitted quantity. However, the paper's central algorithmic claim is circular in its setup. No procedure is given to select the Pauli ansatz K(r,θ)=Σ_j r_j e^{iθ_j}P_j (the integer d and the strings P_j) from H alone; Definition 2 assumes H=UDU† with polynomially sparse U,D, and the numerical demonstrations use that known decomposition. The Fig. 1 tests generate H by first sampling U and D, and the XXZ/Hubbard tests initialize from an eigensolver's decomposition of a nearby Hamiltonian. Thus the claimed polynomial-time diagonalization is not computed from H alone; the decomposition is an input by construction. This warrants a score of 7. Separately, Example 1 defines D=I+Σ_j d_jY_j, which contradicts Eq. (2)'s classification of Y-containing strings as off-diagonal; hence the claimed family of quantum-diagonalizable Hamiltonians with exponentially large Lie algebras is unsupported. That is a correctness flaw, not an additional circular step, and I do not count it in the score beyond noting it weakens the breadth claim. The self-citation [20] in Appendix B supports only a secondary perturbation bound and is not load-bearing for the main diagonalization claim, so it does not raise the score further.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The central claims rest on standard Pauli-basis math plus the domain assumption that H has polynomially many Pauli terms. The main extra weight is placed on a user-supplied Pauli ansatz that includes the diagonalizing unitary, on unproved Assumption 1, and on self-cited spectral bounds. The new Hamiltonian family in Example 1 is internally inconsistent because D is not diagonal.

free parameters (3)
  • Ansatz Pauli set {P_j} for K = d = 8, 32, 8, 32 in Fig. 1; full 4^n basis for XXZ/Hubbard
    The algorithm needs a user-supplied set of Pauli strings; in all experiments this set is chosen from the known eigendecomposition of H. Polynomial efficiency is conditional on this free choice.
  • Initial coefficients (r0, theta0) = coefficients of eigendecomposition unitary from classical eigensolver
    Section V initializes with the exact answer. This hand-supplied input is not derived by the algorithm.
  • KL constants delta_f, mu, alpha and learning-rate schedule a_t = not reported
    The convergence theorem depends on these constants and on a_t; none are specified or bounded in the paper, so the rate guarantees cannot be instantiated.
assumptions (6)
  • standard math The Pauli strings form an orthogonal basis for 2^n x 2^n matrices under the Hilbert-Schmidt inner product; every Hermitian A has expansion A = Σ a_j P_j with a_j = 2^{-n} tr(A P_j).
    Used in Eq. (2) and throughout to define f and F.
  • domain assumption The target Hamiltonian is expressed as a linear combination of polynomially many Pauli strings, M = O(poly(n)).
    Intro and Eq. (1); typical in quantum simulation but restricts scope to local/sparse Hamiltonians.
  • ad hoc to paper A Pauli ansatz set {P_j} that contains the support of a diagonalizing unitary of H is known in advance.
    Eq. (4); without this, d can be 4^n and the polynomial complexity claim fails. The paper supplies this by construction or eigensolver in all demonstrations.
  • ad hoc to paper Assumption 1: there exist delta_f > 0, mu > 0, alpha in [0,2) such that ||nabla F||^2 >= 4 mu F^alpha whenever F < delta_f.
    Invoked in Theorem 4; only 'numerically validated', not proved. It is a uniform strengthening of KL that may fail for a given Hamiltonian or initialization.
  • standard math The generating set of [31] (Z1, Z2, X1, X2, Z1Z2 and X2Y3...Zj, Z2Y3...Xj) generates all Pauli strings in the dynamical Lie algebra sense.
    Used in Appendix C to prove g(H) = su(2^n); relies on prior literature [31].
  • domain assumption The spectral projector bound invoked from [20, Lemma 5] is correct.
    Used in the proof of Theorem 5; it is a self-citation, and the lemma is not reproduced in this paper.

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Cite this review

Pith. "Pith review of Classical optimization algorithms for diagonalizing quantum Hamiltonians." pith.science (2026). https://pith.science/paper/RRT5FLWZ

@misc{pith2026250617883,
  author       = {Pith},
  title        = {Pith review of: Classical optimization algorithms for diagonalizing quantum Hamiltonians},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RRT5FLWZ}},
  note         = {Machine review of arXiv:2506.17883}
}
read the original abstract

Diagonalizing a Hamiltonian, which is essential for simulating its long-time dynamics, is a key primitive in quantum computing and has been proven to yield a quantum advantage for several specific families of Hamiltonians. Yet, despite its importance, only a handful of diagonalization algorithms exist, and correspondingly few families of fast-forwardable Hamiltonians have been identified. This paper introduces classical optimization algorithms for Hamiltonian diagonalization by formulating a cost function that penalizes off-diagonal terms and enforces unitarity via an orthogonality constraint, both expressed in the Pauli operator basis. We pinpoint a class of Hamiltonians that highlights severe drawbacks of existing methods, including exponential per-iteration cost, exponential circuit depth, or convergence to spurious optima. Our approach overcomes these shortcomings, achieving polynomial-time efficiency while provably avoiding suboptimal points. As a result, we broaden the known realm of fast-forwardable systems, showing that quantum-diagonalizable Hamiltonians extend to cases generated by exponentially large Lie algebras. On the practical side, we also present a randomized-coordinate variant that achieves a more efficient per-iteration cost than the deterministic counterpart. We demonstrate the effectiveness of these algorithms through explicit examples and numerical experiments.

Figures

Figures reproduced from arXiv: 2506.17883 by the authors.

Figure 1
Figure 1. FIG. 1: Optimization results (Top) and the corresponding values of [PITH_FULL_IMAGE:figures/full_fig_p014_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Optimization results for XXZ model and Hubbard model with four different choices of [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: The diagram illustrates examples of quantum diagonalizable Hamiltonians: this work [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗

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