Special Issue: Commemorating the 110th Anniversary of TANG Au-chin's Birthday Calculation of the Green's function on near-term quantum computers via Cartan decomposition

Jie Liu; Jinlong Yang; Lingyun Wan

REVIEW 3 major objections 4 minor 57 references

Special Issue: Commemorating the 110th Anniversary of TANG Au-chin's Birthday Calculation of the Green's function on near-term quantum computers via Cartan decomposition

T0 review · 3 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Constant-depth circuits compute Green's functions for any time

desk verdict Genuine but incremental: analytic gradients for Cartan decomposition plus a Green's-function pipeline; the two-site Hubbard test is solid, but the TFIM results and the fixed-depth claim are unverified. read the letter →

arxiv 2509.09248 v1 pith:26YNDIIY submitted 2025-09-11 quant-ph

Lingyun Wan , Jie Liu , Jinlong Yang This is my paper · ORCID

classification quant-ph

keywords Green'sfunctionCartandecompositionHamiltoniansimulationfixed-depthquantumcircuitsspectralFermi-Hubbardmodeltransverse-fieldIsingnear-termcomputers

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 introduces a quantum algorithm that computes real-time Green's functions using quantum circuits whose depth does not grow with simulation time. The key move is a Cartan decomposition of the Hamiltonian's Lie algebra: the time-evolution operator e^{-iHt} is rewritten as K e^{-i h t} K†, where K is a fixed unitary and h is a commuting sum of Pauli terms, so the time-dependent part factorizes into a product of single-Pauli rotations. With that factorization, the retarded Green's function is assembled from Hadamard-test overlaps of the fixed circuit acting on simple states. The authors demonstrate the construction on the two-site Fermi-Hubbard model and on transverse-field Ising chains, reproducing exact spectral functions and capturing a metal-to-Mott-insulator crossover. A sympathetic reader would care because near-term quantum devices tolerate only shallow circuits, and this construction makes long-time spectral calculations compatible with that constraint.

What carries the argument

The Cartan decomposition of the Hamiltonian's Lie algebra, a split g=k⊕m satisfying [k,k]⊂k, [m,m]⊂k, [k,m]⊂m, together with the KHK theorem: for H∈m there exists a fixed unitary K∈e^k and a Cartan subalgebra element h∈h such that H=KhK†. This turns time evolution into U(t)=K e^{-i h t} K†, and because h is Abelian, e^{-i h t} is a product of commuting Pauli rotations, giving fixed-depth circuits for arbitrary t. The paper finds K by minimizing the Killing form f(K)=Tr(K v K† H) over factorized unitaries K=∏ e^{i θ_i k_i}, with analytical gradients derived for efficient BFGS optimization; the gradients come from the factorized form, allowing each exponential to act on v and H as a Pauli rota

What would settle it

Run the Cartan-BFGS procedure on a transverse-field Ising chain with six or more sites, compute the residual norm ‖K†HK−h‖ after optimization, and compare the resulting U(t) against exact diagonalization for a long time t. If the residual is not at machine precision, or if the Green's function deviates from the exact result beyond rounding, the fixed-depth claim fails for that Hamiltonian.

Watch

Extended reading notes

Core claim

The central claim is that for a Hamiltonian H whose Pauli-string Lie algebra admits a Cartan decomposition with H in the noncompact part m, the real-time evolution operator is exactly U(t)=K_0 e^{-i h t} K_0†, with K_0 time-independent and h a Cartan subalgebra element. Because h is Abelian, e^{-i h t} is a product of commuting rotations around single Pauli strings, so its circuit depth is constant in t. The retarded Green's function is then obtained from ground-state overlaps of the form ⟨Ψ| c_a K_0 e^{-i h t} K_0† c_b† |Ψ⟩, evaluated by Hadamard tests, plus the conjugate term. Numerical experiments on the two-site Fermi-Hubbard model show the fixed-depth circuits reproduce the exact Green'

Load-bearing premise

The algorithm's correctness rests on the classical optimizer always reaching a global minimum of f(K)=Tr(KvK†H) so that K†HK exactly lands in the Cartan subalgebra; the paper demonstrates this on its benchmark Hamiltonians but provides no guarantee, and if the optimizer stalls the time-evolution operator is not e^{-iHt} at all.

Editorial extensions

If this is right

For any Hamiltonian admitting the Cartan decomposition, the quantum circuit needed to evolve a state to arbitrary time has constant depth, eliminating Trotter error accumulation and variational drift over long times.
The spectral function A_k(ω) is recovered by Fourier transforming the real-time Green's function, so the method gives direct access to Hubbard bands and quasiparticle features without summing over excited states.
The construction can be reused for any initial state: once K_0 and h are fixed classically, each Green's function entry is just a Hadamard-test overlap on the same short circuit.
The demonstrated crossover from a correlated metal at U=3 to a Mott insulator at U=6 in the two-site Fermi-Hubbard model shows the fixed-depth circuits preserve enough dynamical information to distinguish phases.
The method separates the expensive classical search for K_0 from the quantum sampling, which is the natural division for near-term hardware.

Reading between the lines

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

Beyond the paper's examples, the same fixed-depth identity applies to any retarded or advanced correlation function of the form ⟨Ψ| O_1 e^{-iHt} O_2 |Ψ⟩, since the time-evolution operator is the only part that needs U(t).
If the classical optimization of K can be made locality-aware, the approach could move past the exponential scaling the authors note for the Lie algebra dimension; a natural test is a one-dimensional chain with local interactions where a block-structured K may suffice.
The method's advantage is likely to be most visible on devices where two-qubit gate errors dominate: once K is compiled, the same circuit is reused for every time step, so calibration overhead is amortized over the whole spectral function.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Two things to know. First, this is a real but incremental step: it ports the Cartan-decomposition fixed-depth simulator from refs. [43,44] to the Green's function setting, with the genuinely new piece being the analytic gradient for the factorized K (Eqs. 14-18). Second, the two-site Hubbard demonstration is legitimate—the exact curves in Figs. 1-3 match, and I checked that Eq. (23), Eq. (37), and the Cartan h basis in Eq. (39) are correct. The puzzle is the TFIM section: no exact comparison is shown, so the central claim that U(t)=K e^{-iht}K† holds for those systems is unsupported.

The new contributions are the factorized gradient formulas and the explicit time-domain pipeline for the retarded Green's function (Eqs. 23-26). If the Cartan decomposition can be made exact, the fixed-depth property is nice, and the analytic gradient is reusable in other Cartan-based circuits. The conclusion is also refreshingly honest about the exponential classical overhead and the 10-qubit ceiling—that limitation belongs in the abstract.

The weak spot is the optimization. The whole construction rests on BFGS finding a K such that K† H K lies in h. For the two-site Hubbard model, the exact benchmarks indirectly validate it. For the TFIM, there is no residual measure ||K†HK-h|| and no exact spectral function to compare against; the optimization might have converged to a local minimum that does not actually put H into h. The paper even claims, without proof, that the product form in Eq. (9) is equivalent to the exponential sum in Eq. (8); for non-commuting generators that is not generally true, so the parameterization may not be expressive enough. That is a load-bearing gap for the 'fixed-depth for arbitrarily long times' headline. The interpretation of Mott crossover in a 2-site cluster and quantum criticality in a 6-site chain also outruns the systems, and the typos in Eqs. (38) and (42) are minor but avoidable.

This paper deserves a serious referee because the core equations are checkable and the gradient formula is reusable. The referee should ask for a residual report and at least one exact TFIM benchmark, and a statement about the coverage of the product ansatz. With those additions, the contribution is publishable, if modest.

Referee Report

3 major / 4 minor

Summary. The paper presents a hybrid quantum–classical algorithm for computing retarded Green's functions on near-term quantum hardware. The key idea is to use a Cartan decomposition of the Hamiltonian Lie algebra: after a classical optimization step, the authors seek a unitary K and an Abelian subalgebra element h such that H = K h K†. Time evolution then takes the fixed-depth form U(t) = K e^{-i h t} K†, independent of t. The Green's function is assembled from Hadamard-test overlaps of time-evolved excited states, and spectral functions are obtained by Fourier transformation. The method is demonstrated on the two-site Fermi–Hubbard model with exact benchmarks in Figs. 1–3, and on transverse-field Ising chains of two, four, and six sites in Fig. 4. The conclusion explicitly acknowledges that the dimension of K scales exponentially and limits the method to small systems.

Significance. The central algebraic reduction is valuable and largely correct: Eq. (23) reduces the retarded Green's function to two time-evolution overlaps, Eq. (37) is the correct Jordan–Wigner two-site Hubbard Hamiltonian, and the basis h in Eq. (39) is pairwise commuting, as required for a Cartan subalgebra. For the Hubbard model, the exact comparisons in Figs. 1–3 give credible evidence that the Cartan decomposition succeeded for the parameters used. If the fixed-depth identity Eq. (19) can be reliably achieved, the algorithm offers a genuine advantage over Trotter-based simulation for small NISQ-era systems, and the analytical gradients in Eqs. (14)–(18) are a useful technical contribution. The paper is less convincing for the transverse-field Ising model, where no exact benchmark or optimization residual is provided, and the physical discussion of quantum criticality in Fig. 4 rests on unvalidated curves.

major comments (3)

[One-dimensional Spin Chains / Fig. 4] The TFIM demonstration does not validate the central claim. Unlike Figs. 1–3, Fig. 4 contains only Cartan-derived spectral functions: no exact (diagonalization or Trotter) reference curves, no time-domain overlap fidelity, and no residual measure ||K†HK − h|| or BFGS success statistics are reported. Since Eq. (19) is exact only if the optimization of Eq. (6) actually finds K satisfying Eq. (7), the TFIM results cannot be distinguished from an optimization failure. Please add exact benchmarks for at least one system size and report residual norms or success statistics for all Cartan decompositions used in the paper.
[Method, Eqs. (6)–(9)] The statement after Eq. (8) that the exponential form K = e^{Σ ia_i k_i} and the product form K = ∏ e^{ia_i k_i} are 'equivalent' is not justified, and it is load-bearing. A local minimum of f(K) over the fixed-order product submanifold Eq. (9) is only a constrained stationary point; the tangent space of that submanifold is not obviously all of k at every point, so the critical-point condition that would imply K†HK ∈ h need not hold. BFGS has no convergence guarantee for this nonconvex problem. The paper should either prove the equivalence for the relevant Lie algebras or report residual norms and multiple-random-restart success rates; otherwise the identity U(t) = K e^{-iht} K† is an assumption rather than a derived result.
[Abstract / Conclusion] The abstract claims 'an efficient algorithm... requires only fixed-depth quantum circuits for arbitrarily long time simulations.' The conclusion correctly states that the dimension of K scales exponentially and limits the method to roughly 10 qubits. This is a significant qualification: the fixed-depth property holds only after a successful classical optimization whose cost is exponential and whose success is, for TFIM, unverified. Please state this limitation prominently in the abstract or introduction, and separate the quantum circuit depth from the classical preprocessing cost in the claims.

minor comments (4)

[Eq. (38)] The last basis element of k is written as Z0Y1Z2X0, which repeats Z0 and is inconsistent with the pattern of the other elements; it should almost certainly be Z0Y1Z2X3.
[Eq. (24)] The Jordan–Wigner expression is written as Q_a^† ⊗ Z_{a−1} ⊗ ... ⊗ Z_1; the ordering of the tensor factors should be specified explicitly to avoid ambiguity about which qubit is the least significant.
[One-dimensional Spin Chains] The notation k is used both for momentum labels in the Hubbard section and for the system-size index in Eq. (45), where 'different k means different sizes.' This is confusing; use N or n for the chain length.
[Throughout] There are several typographical and grammatical issues: 'homomrophism', 'Cartan decomposition has been emerged', and 'We use [PiPj to represent' should be corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Cartan decomposition is an exact Hamiltonian identity, and the Green's-function predictions are benchmarked against independent exact results.

full rationale

The paper's central construction is not circular. The parameters θ in K are obtained by minimizing f(K)=Tr(KvK†H) (Eq. 6), a classical optimization whose target is the exact identity H=K0 h K0† (Eq. 7). This is a Hamiltonian decomposition, not a fit to Green's-function data; the Green's function is then computed from the resulting U(t)=K0 e^{-iht} K0† (Eq. 19). No quantity entering the Cartan decomposition is defined in terms of the target Green's function or spectral function, so the prediction is not forced by construction. The numerical benchmarks in Figs. 1–3 compare against independently computed exact results, providing external validation. The only self-citation (ref. [12]) appears in a broad list of prior quantum algorithms and is not load-bearing; no uniqueness claim or ansatz is imported from the authors' own prior work. Concerns about BFGS convergence or the absence of exact TFIM comparisons are correctness/robustness issues, not circularity.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

No new physical entities are postulated. The paper's "input" is the Hamiltonian together with the algebraic machinery of Cartan decomposition inherited from refs. [43,44]; the reader pays for the classical Lie-algebra computation and the nonconvex optimization rather than for any new physics.

free parameters (3)

eta (spectral broadening) = 0.2
Imaginary convergence factor in Eq. (27), chosen by hand for the Fourier transform; it sets the width of every reported spectral peak and is the only free knob in the extracted A_k(omega).
Cartan parameters theta_i of K = not reported (determined by BFGS)
Optimized classically by BFGS to satisfy Eq. (7). Their values are not fitted to Green's function data, but the fixed-depth circuit exists only if these parameters are found; the paper reports neither the values nor the residual of Eq. (7).
gamma (coefficients of generic v) = unspecified transcendental number
Introduced in the Method to make v "dense" in the Cartan subalgebra so that the minimization of Eq. (6) forces K†HK into h. Chosen by hand; standard trick from ref. [43].

assumptions (4)

standard math KHK theorem: for any m in m there exists K in e^k and h in h with m = K h K†.
Eq. (4), cited to refs. [44,47]. Standard result for compact semisimple Lie algebras; requires the Hamiltonian to lie in m.
domain assumption The Hamiltonian H belongs to m for the chosen Cartan decomposition g = k ⊕ m.
Stated in the Method ("Given a Cartan decomposition of g(H) such that H ∈ m"). Not explicitly verified in the text for either model, though the Hubbard numerics are consistent with it.
domain assumption BFGS minimization of the nonconvex f(K) reaches a global (not merely local) critical point with K†HK ∈ h.
Eqs. (6)-(7): the existence of K is guaranteed by the theorem, but its discovery by gradient descent is assumed. This is the most fragile classical step and is unquantified.
domain assumption The exact ground state is available (via ADAPT-VQE) and Hadamard tests give the required overlaps.
Eqs. (23)-(26): the Green's function is defined relative to the exact ground state; any ground-state error propagates directly into G(t).

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Cite this review

Pith. "Pith review of Special Issue: Commemorating the 110th Anniversary of TANG Au-chin's Birthday Calculation of the Green's function on near-term quantum computers via Cartan decomposition." pith.science (2026). https://pith.science/paper/26YNDIIY

@misc{pith2026250909248,
  author       = {Pith},
  title        = {Pith review of: Special Issue: Commemorating the 110th Anniversary of TANG Au-chin's Birthday Calculation of the Green's function on near-term quantum computers via Cartan decomposition},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/26YNDIIY}},
  note         = {Machine review of arXiv:2509.09248}
}

read the original abstract

Accurate computation of the Green's function is crucial for connecting experimental observations to the underlying quantum states. A major challenge in evaluating the Green's function in the time domain lies in the efficient simulation of quantum state evolution under a given Hamiltonian-a task that becomes exponentially complex for strongly correlated systems on classical computers. Quantum computing provides a promising pathway to overcome this barrier by enabling efficient simulation of quantum dynamics. However, for near-term quantum devices with limited coherence times and fidelity, the deep quantum circuits required to implement time-evolution operators present a significant challenge for practical applications. In this work, we introduce an efficient algorithm for computing Green's functions via Cartan decomposition, which requires only fixed-depth quantum circuits for arbitrarily long time simulations. Additionally, analytical gradients are formulated to accelerate the Cartan decomposition by leveraging a unitary transformation in a factorized form. The new algorithm is applied to simulate long-time Green's functions for the Fermi-Hubbard and transverse-field Ising models, extracting the spectral functions through Fourier transformation.

Figures

Figures reproduced from arXiv: 2509.09248 by the authors.

**Figure 2.** FIG. 2: Numerical simulation of the exact method and Cartan algorithm to compute the Green’s function in real [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** FIG. 3: Numerical simulation of the spectral function. We take [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: FIG. 4: Numerical simulation of the spectral function [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

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