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arxiv: 2606.23967 · v1 · pith:JWIXBCVZnew · submitted 2026-06-22 · 🪐 quant-ph

Polynomial-time exact diagonalization via sparse guided eigenwalks

Zachary E. Chin , Mario Motta , Javier Robledo Moreno , Antonio Mezzacapo , Isaac L. Chuang , William Kirby This is my paper

Pith reviewed 2026-06-26 07:45 UTC · model grok-4.3

classification 🪐 quant-ph
keywords sparse eigenvectorseigenwalkexact diagonalizationsupport localizationgraph searchHermitian matricessparse principal component analysisquantum ground states
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The pith

For any sparse eigenvector there exists another with the same eigenvalue whose support has diameter linear only in the sparsity.

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

The paper introduces the eigenwalk problem of finding the support of a sparse eigenvector by walking the graph of nonzero matrix entries. It proves that any sparse eigenvector has a counterpart with identical eigenvalue whose support lies inside a ball whose radius grows linearly with the number of nonzeros and does not depend on the dimension of the space. When one vertex inside that support is already known, the entire support can therefore be recovered by a bounded-depth search, turning exact diagonalization of certain exponentially large but sparse Hamiltonians into a polynomial-time classical procedure. The same guarantee applies to non-extremal eigenvectors provided they are strictly sparser than all eigenvectors belonging to other eigenvalues. The localization principle requires no assumptions about degeneracy, locality, or spectral gaps and extends directly to sparse principal-component analysis.

Core claim

For every sparse eigenvector of a Hermitian matrix there exists (possibly different) sparse eigenvector with the same eigenvalue whose support is contained in a subgraph whose diameter scales linearly with the sparsity and is independent of the total number of vertices. Consequently, if a 2^n-dimensional poly(n)-sparse Hamiltonian possesses an O(1)-sparse extremal eigenvector and a single support vertex is supplied, an exact eigenvector of the same eigenvalue can be recovered in poly(n) time; the same holds for a non-extremal eigenvector that is sparser than every eigenvector with a different eigenvalue.

What carries the argument

The support-localization theorem for sparse eigenvectors on the graph induced by nonzero entries, which guarantees a representative eigenvector whose support diameter is linear in its sparsity.

If this is right

  • Exact eigenvector recovery becomes polynomial-time when the Hamiltonian is poly(n)-sparse, the target eigenvector is O(1)-sparse and extremal, and one support index is known.
  • The same polynomial-time guarantee holds for non-extremal eigenvectors that are strictly sparser than all eigenvectors of different eigenvalues.
  • No assumptions on degeneracy, locality, spectral width, or gap are required for the polynomial-time result.
  • The identical localization principle applies to sparse principal component analysis without further modification.

Where Pith is reading between the lines

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

  • If a reliable heuristic for guessing an initial support vertex can be supplied, the method yields a practical classical algorithm for finding sparse ground states of large sparse Hamiltonians.
  • The same bounded-diameter search can be used as a subroutine inside hybrid quantum-classical algorithms that only need to certify or refine a candidate support.
  • Because the diameter bound is independent of dimension, the approach scales to any matrix whose sparsity graph has bounded degree, including adjacency matrices of regular graphs of arbitrary size.

Load-bearing premise

At least one vertex inside the desired support must already be known or guessed.

What would settle it

A concrete Hermitian matrix whose every eigenvector with a given sparsity level has support diameter strictly larger than linear in that sparsity.

Figures

Figures reproduced from arXiv: 2606.23967 by Antonio Mezzacapo, Isaac L. Chuang, Javier Robledo Moreno, Mario Motta, William Kirby, Zachary E. Chin.

Figure 1
Figure 1. Figure 1: The eigenwalk problem for a Hermitian matrix [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Three possible graphs, each with a subset [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Two sufficient conditions for eigenvector degeneracy when the support is partitioned into subsets [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Example graphs saturating diameter bound for irreducible support components. Let the subset [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Illustration of the polynomial-time algorithm for the guided sparse eigenwalk problem (Theorem [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Proof sketch of Theorem 4 (approximately sparse support localization). (a) The extremal eigenvector |ψ⟩ with eigenvalue λ is approximately supported on the vertex subset A. Within the induced subgraph of A in G, the guiding vertex v belongs to a connected component whose vertex set is S; the remaining vertices in A belong to S ′ . Vertices outside A carry the leakage amplitude γ, with |γ| 2 ≤ ϵ. The locali… view at source ↗
Figure 7
Figure 7. Figure 7: More graph examples, each with a subset S of shaded vertices. For the graph in (a), there exists a graph-consistent Hermitian matrix H admitting a nondegenerate non-extremal eigenvector whose support is S. For the graph in (b), there exists a graph-consistent Hermitian matrix H admitting a nondegenerate extremal eigenvector whose support is S. The graph in (c) admits a graph-consistent Hermitian matrix H w… view at source ↗
read the original abstract

Computing quantum ground states is generically difficult, but additional structure can sometimes allow diagonalization to be recast as a more feasible problem. For example, when the desired ground state is sparse in a given basis, diagonalization can be facilitated via graph search. We make this reformulation precise by introducing the eigenwalk problem, which seeks the support of a sparse eigenvector of a Hermitian matrix by exploring the graph induced by its nonzero entries. However, it is not obvious whether the relevant support vertices must always be efficiently reachable by a search on the graph. To resolve this question, we prove that for every sparse eigenvector, there exists a (possibly different) sparse eigenvector with the same eigenvalue whose support is tightly localized in the graph, with diameter scaling only linearly in the sparsity and independently of the total number of vertices. As a consequence, if a $2^n$-dimensional, ${\rm poly}(n)$-sparse Hamiltonian has an $\mathcal{O}(1)$-sparse extremal eigenvector and one support element is known, then an exact eigenvector with the same eigenvalue can be computed classically in ${\rm poly}(n)$ time. The same conclusion follows when the $\mathcal{O}(1)$-sparse eigenvector is non-extremal, provided that it is sparser than every eigenvector with a different eigenvalue. These results hold with no assumptions on the degeneracy, locality, spectral width, or spectral gap of the Hamiltonian, and the underlying support-localization principle also extends to problems beyond exact diagonalization, such as sparse principal component analysis.

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 / 1 minor

Summary. The paper introduces the eigenwalk problem of recovering the support of a sparse eigenvector of a Hermitian matrix by search on the graph of its nonzero entries. It proves a localization theorem stating that for every sparse eigenvector there exists (possibly different) sparse eigenvector with identical eigenvalue whose support has graph diameter linear in the sparsity level and independent of dimension. As a consequence, exact diagonalization of a 2^n-dimensional poly(n)-sparse Hamiltonian possessing an O(1)-sparse extremal eigenvector (or a non-extremal eigenvector sparser than all others) can be performed in poly(n) time provided one support vertex is already known. The result requires no assumptions on degeneracy, locality, spectral gap or width, and is claimed to extend to sparse PCA.

Significance. If the localization theorem is correct, the work supplies a parameter-free, assumption-light route to exact classical diagonalization for a nontrivial class of sparse high-dimensional Hermitian operators once a single support vertex is supplied. The absence of fitted parameters, self-referential definitions, or reductions to quantities defined inside the paper is a clear strength. The result is of theoretical interest for structured quantum many-body problems and for sparse principal-component analysis, though its algorithmic utility remains conditional on the availability of a seed vertex.

major comments (2)
  1. The central localization theorem is presented as an existence result whose proof is not reproduced in the provided abstract or reader notes; without the explicit derivation steps (presumably in §3 or §4) it is impossible to verify whether the diameter bound is obtained without hidden assumptions on the support or on the matrix entries.
  2. [Abstract] Abstract, final paragraph: the poly(n)-time claim is correctly qualified by the proviso that one support element must already be known. No mechanism is supplied for locating such a seed in the worst case, so the algorithmic consequence remains conditional exactly as stated; this qualification is load-bearing for any claim of unconditional polynomial-time diagonalization.
minor comments (1)
  1. Notation for the sparsity graph and the precise definition of 'eigenwalk' should be introduced with a small illustrative example early in the text to aid readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful review and for acknowledging the potential significance of the localization theorem. We respond point-by-point to the major comments below.

read point-by-point responses

  1. Referee: The central localization theorem is presented as an existence result whose proof is not reproduced in the provided abstract or reader notes; without the explicit derivation steps (presumably in §3 or §4) it is impossible to verify whether the diameter bound is obtained without hidden assumptions on the support or on the matrix entries.

    Authors: The complete proof of the localization theorem, including all explicit derivation steps, appears in Sections 3 and 4 of the full manuscript. These sections derive the linear-diameter bound directly from the eigenvector sparsity and the graph induced by nonzero matrix entries, without additional assumptions on support structure or matrix values. revision: no

  2. Referee: Abstract, final paragraph: the poly(n)-time claim is correctly qualified by the proviso that one support element must already be known. No mechanism is supplied for locating such a seed in the worst case, so the algorithmic consequence remains conditional exactly as stated; this qualification is load-bearing for any claim of unconditional polynomial-time diagonalization.

    Authors: We agree that the result is conditional on the availability of one support vertex and that the abstract states this limitation accurately. The manuscript neither claims nor supplies a general method for locating such a seed, so the algorithmic consequence is correctly presented as conditional. revision: no

Circularity Check

0 steps flagged

No circularity: mathematical existence proof is self-contained

full rationale

The paper presents a graph-theoretic existence theorem establishing that any sparse eigenvector has a counterpart with the same eigenvalue whose support has diameter linear in the sparsity (independent of dimension). This is proved directly from the definition of the induced graph on nonzero matrix entries and standard linear-algebra facts about eigenspaces; no parameters are fitted, no quantities are defined in terms of the result itself, and no load-bearing steps reduce to self-citations or prior ansatzes by the same authors. The poly-time algorithm is explicitly conditional on an external seed vertex, which is stated as an assumption rather than derived internally. The derivation chain therefore contains no reductions of the enumerated circular kinds.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on standard linear-algebra and graph-theoretic facts with no free parameters or new postulated entities.

axioms (1)
  • standard math Properties of Hermitian matrices and the graph induced by their nonzero entries
    Invoked to define the eigenwalk graph and to reason about eigenvector support.

pith-pipeline@v0.9.1-grok · 5816 in / 1168 out tokens · 31122 ms · 2026-06-26T07:45:31.441876+00:00 · methodology

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