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A Unified Variational Framework for Quantum Excited States

T0 review · 2 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read Minimizing $\mathrm{Tr}(\mathbf{S}^{-1}\mathbf{H})$ over non-orthogonal states simultaneously returns the lowest $N_s$ energy levels of a Hamiltonian, with no orthogonality constraints.

desk verdict A mathematically correct but known variational principle, made useful by clean demonstrations across MPS, quantics tensor train, and VQE; the main gap is the overclaimed novelty and the heuristic robustness of the optimizer. read the letter →

arxiv 2504.21459 v1 pith:AIHVX43X submitted 2025-04-30 quant-ph cond-mat.str-el

Shi-Xin Zhang , Lei Wang This is my paper · ORCID
classification quant-phcond-mat.str-el
keywords quantumexcitedstatesvariationalprinciplenon-orthogonalgeneralizedeigenvalueproblemmatrixproductquanticstensortraineigensolverlow-energyspectrum
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

The paper proposes that $\mathrm{Tr}(\mathbf{S}^{-1}\mathbf{H})$, built from a set of deliberately non-orthogonal variational states, is a variational target whose minimization returns the lowest $N_s$ energy levels of a quantum Hamiltonian in a single optimization run. Because the loss equals the sum of the Ritz values of the subspace generalized eigenproblem, minimizing it pulls the whole spanned subspace down toward the lowest $N_s$ exact eigenstates, with the minimax principle guaranteeing these variational energies stay above the exact ones. The authors demonstrate the principle on a 1D spin chain with matrix product states, a Morse oscillator with quantics tensor trains, and a 2D Hubbard model with variational quantum circuits, recovering multiple low-lying levels accurately in each case. The appeal is that excited states, usually obtained by state-specific optimizations or penalty terms, come from one unconstrained loss that works for any variational wavefunction family.

What carries the argument

The load-bearing object is the loss $L=\mathrm{Tr}(\mathbf{S}^{-1}\mathbf{H})$ formed from the $N_s\times N_s$ overlap and Hamiltonian matrices of non-orthogonal variational states. When the states are linearly independent, $\mathbf{S}$ is Hermitian positive definite, so the trace is well defined and equals the sum of the generalized eigenvalues (Ritz values) of $\mathbf{H}\mathbf{c}=E\mathbf{S}\mathbf{c}$. The minimax, or Rayleigh-Ritz, principle supplies the variational guarantee that these Ritz values are upper bounds on the exact low-lying eigenvalues, so reducing their sum pushes the span of the states toward the exact low-energy subspace. Around this core, the paper's machinery includes tensor-network contractions and automatic differentiation for matrix product states and quantics tensor trains, measurement protocols for circuits on quantum hardware, and a final generalized diagonalization to read out orthonormal eigenstates.

What would settle it

On a small exactly solvable system, run many optimizations with an ansatz that provably contains the lowest $N_s$ eigenstates; if the minimum of $\mathrm{Tr}(\mathbf{S}^{-1}\mathbf{H})$ found over all restarts stays strictly above the sum of the exact $N_s$ eigenvalues, or if the computed states have nonzero energy variance, the claim that this loss locates the low-energy subspace is not holding in practice.

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Extended reading notes

Core claim

The central claim is that a set of $N_s$ independent, non-orthogonal, unnormalized variational states can be optimized simultaneously to approximate the lowest $N_s$ eigenstates of a Hamiltonian by minimizing $L=\mathrm{Tr}(\mathbf{S}^{-1}\mathbf{H})$, where $\mathbf{H}_{ij}=\langle\psi_i|H|\psi_j\rangle$ and $\mathbf{S}_{ij}=\langle\psi_i|\psi_j\rangle$. The loss is exactly the sum $E_1+\cdots+E_{N_s}$ of the Ritz values obtained from the generalized eigenvalue problem $\mathbf{H}\mathbf{c}=E\mathbf{S}\mathbf{c}$ in the variational subspace, and each Ritz value is an upper bound on the corresponding exact eigenvalue by the minimax principle. The optimization therefore drags the entire low-energy subspace downward instead of targeting one state at a time, and after convergence the final orthonormal eigenstates are obtained by one post-optimization diagonalization of the generalized eigenproblem. The paper's numerical demonstrations on spin chains, a Morse potential, and the Hubbard model are offered as evidence that the principle is accurate and transferable across ansatzes.

Load-bearing premise

The method's practical success rests on gradient-based optimization finding a good minimum of a non-convex loss while the overlap matrix stays invertible, a convergence and stability behavior the paper observes empirically but does not guarantee.

Editorial extensions

If this is right

  • One optimization run replaces $N_s$ separate state-specific calculations, because the inverse overlap couples all variational states into a single loss.
  • No penalty coefficients or orthogonality constraints need tuning; the tested runs kept the overlap matrix well conditioned without explicit regularization.
  • The same loss transfers across wavefunction families: matrix product states, quantics tensor trains, and parameterized quantum circuits all converge to accurate low-lying spectra.
  • On quantum hardware, the required $\mathbf{H}$ and $\mathbf{S}$ entries can be obtained with standard non-orthogonal measurement protocols, making simultaneous multi-state excited-state computation a single variational loop.
  • The framework is positioned to connect with neural quantum states through an equivalent variational-Monte-Carlo formulation, which the paper identifies as a promising direction.

Reading between the lines

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

  • Because the loss weights every targeted level equally, a weighted variant $\mathrm{Tr}(\mathbf{W}\mathbf{S}^{-1}\mathbf{H})$ could rebalance accuracy between the lowest and highest states in the window; this is a testable extension the paper does not discuss.
  • The reliance on $\mathbf{S}^{-1}$ suggests that near-degenerate spectra or larger $N_s$ will make the overlap matrix increasingly ill-conditioned, so explicit regularization or periodic subspace reorganization may be needed outside the tested regimes.
  • Building the spectrum incrementally by adding one variational state at a time and warm-starting from the converged subspace would turn the method into an iterative spectrum-construction algorithm, an extension not explored here.
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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

2 major / 5 minor

Summary. The manuscript proposes minimizing the loss L = Tr(S^{-1}H) over a set of N_s non-orthogonal, unnormalized variational states in order to simultaneously approximate the lowest N_s energy levels of a quantum Hamiltonian without explicit orthogonality constraints or penalty terms. The authors prove, via the minimax principle, that the Ritz values of the spanned subspace upper-bound the exact eigenvalues, so minimizing the sum of Ritz values targets the low-energy subspace. The method is demonstrated on three problems: a 16-site Heisenberg spin chain with periodic matrix product states, a Morse oscillator spectrum with quantics tensor trains, and a 2x3 Hubbard model with hardware-efficient variational quantum circuits, the last benchmarked against subspace VQE. The Supplementary Material contains the minimax proof, a Cholesky reduction of the generalized eigenproblem, a discussion of relations to prior subspace methods, and additional numerical results and implementation details.

Significance. The mathematical core is correct and standard: for any linearly independent set of states, L equals the sum of the Ritz values of the spanned subspace, and the minimax argument in SM I establishes the upper-bound property. If the optimization can be made reliable, the loss is an attractive ansatz-agnostic objective that avoids penalty terms and sequential orthogonality constraints, with natural compatibility with automatic differentiation. The numerical demonstrations span three different ansatz families and include checks against exact diagonalization, which is a genuine strength. The main unaddressed issues are the gap between the unconditional wording of the central claim and the actual optimizer behavior, and the limited statistical reporting of the quantum-circuit comparison. These are fixable within the scope of the manuscript.

major comments (2)
  1. [Method (Eq. 3) and Discussions] The variational statement is mathematically correct for any nonsingular S, but the advertised behavior ('the optimization drives the variational states to span the subspace corresponding to the lowest possible sum') is a statement about the global minimizer, not about L-BFGS on a non-convex loss whose feasible set is open. Near the singular boundary of S the generalized eigenvalue problem becomes ill-conditioned, and the authors themselves note that convergence to the global minimum is not guaranteed and that S requires careful monitoring, with SM VI reporting condition numbers up to O(10^4) and occasional training instability. I ask that the main text state explicitly that the central claim holds when the optimization remains inside the nonsingular region and reaches a sufficiently good global minimum, and that the authors report the smallest eigenvalue or condition number of S along the optimization trajectory for each application. If no regularization is used, a short explanation of why the reported L-BFGS runs avoid the singular boundary would make the practical claim concrete.
  2. [Hubbard results (Fig. 4) and SM VI] The comparison with subspace VQE reports only the best of 10 trials in the energy-error insets and displays raw loss curves without aggregate statistics, so the sentence 'the variance of the loss across different trials is also smaller in our case' is not quantitatively supported. Similarly, SM VI mentions that the Morse condition number O(10^4) 'correlated with occasional training instability' but does not state how often or how severely the training failed. Please report the success rate and the mean/median and spread of converged losses and energy errors over all independent initializations for each application, and for the Hubbard benchmark show the full trial distribution for both methods. This is necessary to support the robustness and advantage claims.
minor comments (5)
  1. [References] Reference [3] contains stray annotative text after the bibliographic entry ('1010.1992The work give the rebirth ...'); this should be removed.
  2. [References] Reference [72] lists 'G. K. lic Lic Chan'; the author name should be corrected to G. K.-L. Chan.
  3. [Fig. 4 caption] In the caption of Fig. 4, 'hardware effcient ansatz' should read 'hardware-efficient ansatz'.
  4. [Fig. 2(a) and Fig. 3 captions] The phrase 'the loss is shifted by the sum of exact eigenvalues' is ambiguous about the sign of the shift; specify that the plotted quantity is L minus the sum of the exact eigenvalues.
  5. [Spin-chain results] The statement 'All 2NsNchi^2 trainable parameters' would be clearer as 2 N_s N chi^2, with the power of chi indicated explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the variational principle is self-contained, benchmarks are external, and self-citations are implementation-level only.

full rationale

The paper's central derivation is the minimax/Rayleigh-Ritz upper-bound argument in Supplemental Section I: for any Ns-dimensional variational subspace, the Ritz values E_alpha from Hc=ESc satisfy E_alpha >= E_exact_alpha, and minimizing L = Tr(S^{-1}H) = sum_alpha E_alpha optimizes the subspace itself. The loss is not constructed from the target eigenvalues; the exact spectra are used only as post-hoc benchmarks (exact diagonalization for the spin chain and Hubbard model, analytic Morse formula for the oscillator). Nothing in the training loop uses the reported energies as inputs. The paper's self-citations (Refs. 59-61 for TensorCircuit-NG and differentiable tensor networks, and contextual prior VMC/VQE work) concern software, gradient computation, and related methods; they do not supply the load-bearing mathematical fact, which is proved in the supplement rather than imported by citation. The acknowledged limitations -- non-convex landscape, no global convergence guarantee, condition number of S, best-of-10 trials reported for the VQE benchmark -- are optimization robustness caveats stated in the Discussions and Supplement VI, not circular dependencies. Hence no prediction reduces by construction to a fitted parameter or to a self-citation chain.

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

The ledger reveals no invented physical entities and no data-fitted constants. The method rests on standard functional analysis (minimax theorem) and on numerical hyperparameters chosen by hand. The main non-standard assumptions are the practical ones: optimizer convergence and S non-singularity.

free parameters (5)
  • number of variational states N_s = 32, 16, 16
    Chosen by hand; determines how many excited states are targeted.
  • MPS bond dimension chi = 16 or 24 (spin chain); 128 (Morse TT)
    Controls ansatz expressiveness; larger values improve accuracy.
  • VQE ansatz depth D = 5
    Hardware-efficient circuit depth; chosen by hand.
  • initial parameter scale sigma = 1.0 (spin chain), 0.02 (Morse), 0.01 (Hubbard)
    Initialization standard deviation; chosen per application for stable training.
  • Morse grid bits N_d = 16 or 18
    Grid resolution for real-space discretization; chosen to balance accuracy and cost.
assumptions (5)
  • standard math Minimax (Courant-Fischer) principle for Ritz values
    Used in Sec. I of SM to prove each Ritz value is an upper bound on the exact eigenvalue; standard theorem for self-adjoint operators.
  • domain assumption Hamiltonian is self-adjoint and bounded below with discrete low-energy spectrum
    Required for the minimax theorem and for low-lying eigenstates to exist; satisfied by the finite-size and 1D models considered.
  • domain assumption Variational states remain linearly independent (S positive definite) throughout optimization
    The loss is undefined when S is singular; the paper relies on the optimizer avoiding state collapse, with condition numbers reported as manageable.
  • domain assumption Ansatz expressiveness is sufficient to represent low-energy subspace
    The paper states results are valid 'if the ansatz is sufficiently expressive'; holds for chosen bond dimensions/depths in the small demonstrations.
  • standard math Jordan-Wigner mapping for Hubbard model
    Used to map the fermionic Hubbard Hamiltonian to Pauli strings on 12 qubits; standard and exact.

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Pith. "Pith review of A Unified Variational Framework for Quantum Excited States." pith.science (2026). https://pith.science/paper/AIHVX43X

@misc{pith2026250421459,
  author       = {Pith},
  title        = {Pith review of: A Unified Variational Framework for Quantum Excited States},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AIHVX43X}},
  note         = {Machine review of arXiv:2504.21459}
}
abstract

Determining quantum excited states is crucial across physics and chemistry but presents significant challenges for variational methods, primarily due to the need to enforce orthogonality to lower-energy states, often requiring state-specific optimization, penalty terms, or specialized ansatz constructions. We introduce a novel variational principle that overcomes these limitations, enabling the \textit{simultaneous} determination of multiple low-energy excited states. The principle is based on minimizing the trace of the inverse overlap matrix multiplied by the Hamiltonian matrix, $\mathrm{Tr}(\mathbf{S}^{-1}\mathbf{H})$, constructed from a set of \textit{non-orthogonal} variational states $\{|\psi_i\rangle\}$. Here, $\mathbf{H}_{ij} = \langle\psi_i | H | \psi_j\rangle$ and $\mathbf{S}_{ij} = \langle\psi_i | \psi_j\rangle$ are the elements of the Hamiltonian and overlap matrices, respectively. This approach variationally optimizes the entire low-energy subspace spanned by $\{|\psi_i\rangle\}$ without explicit orthogonality constraints or penalty functions. We demonstrate the power and generality of this method across diverse physical systems and variational ansatzes: calculating the low-energy spectrum of 1D Heisenberg spin chains using matrix product states, finding vibrational spectrum of Morse potential using quantics tensor trains for real-space wavefunctions, and determining excited states for 2D fermionic Hubbard model with variational quantum circuits. In all applications, the method accurately and simultaneously obtains multiple lowest-lying energy levels and their corresponding states, showcasing its potential as a unified and flexible framework for calculating excited states on both classical and quantum computational platforms.

Figures

Figures reproduced from arXiv: 2504.21459 by the authors.

Figure 1
Figure 1. FIG. 1. End-to-end variational optimization with the loss [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. (a) Convergence of the loss function [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Training dynamics for the Morse potential spectrum [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Training dynamics for the VQE calculation of the [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]

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