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arxiv: 2606.11455 · v1 · pith:YLY3QYTGnew · submitted 2026-06-09 · 🪐 quant-ph

Planted-Solution Pauli Hamiltonians as a Quantum Benchmarking Primitive

Amir Kalev , Itay Hen This is my paper

Pith reviewed 2026-06-27 13:01 UTC · model grok-4.3

classification 🪐 quant-ph
keywords Pauli Hamiltoniansplanted solutionsfrustration-free Hamiltoniansground-state energy estimationquantum benchmarkingblock-product statesconstraint satisfaction problems
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The pith

A construction produces Pauli Hamiltonians with exactly known ground-state energies by planting block-product states into sums of frustration-free local clauses.

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

The paper shows how to generate quantum Hamiltonians whose ground states and energies are known exactly in advance for use as benchmarks. It embeds a chosen block-product state so that it is the simultaneous ground state of multiple local clauses whose supports overlap yet remain frustration-free. The full Hamiltonian is then rewritten as a polynomial-size sum of Pauli operators. The same framework recovers classical planted constraint-satisfaction problems when restricted to diagonal terms, creating a direct route for classical hardness to appear in quantum instances. If successful, the method supplies reference problems against which ground-state estimation algorithms can be tested with certainty about the correct answer.

Core claim

The construction embeds a planted block-product state as the simultaneous ground state of a sum of frustration-free local clauses on overlapping supports, exposes the resulting model only as a polynomial-size linear combination of Pauli operators, and admits optional Clifford conjugation that preserves the spectrum. The framework subsumes classical planted constraint-satisfaction problems as a diagonal special case, providing a direct embedding channel through which classical hardness properties can be inherited.

What carries the argument

The sum of frustration-free local clauses on overlapping supports that share the planted block-product state as common ground state, rewritten as a polynomial number of Pauli terms.

If this is right

  • The resulting Hamiltonians serve as reference instances with known ground-state energies for benchmarking quantum ground-state estimation algorithms.
  • Classical planted constraint-satisfaction problems embed directly as the diagonal case, transferring known hardness properties into the quantum setting.
  • Optional Clifford conjugation generates new instances while leaving the spectrum unchanged.
  • All instances remain polynomial in the number of Pauli terms, keeping them tractable to write down.

Where Pith is reading between the lines

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

  • These instances could be used to measure how algorithm performance scales with the degree of classical hardness inherited from the planted problem.
  • The same planting technique might be adapted to produce reference states for other quantum simulation tasks beyond energy estimation.
  • Because the construction is explicit, one could generate families of instances with controlled locality or interaction range to isolate algorithmic bottlenecks.

Load-bearing premise

Frustration-free local clauses on overlapping supports can always be chosen so their sum has the planted block-product state as ground state and expands to only a polynomial number of Pauli operators.

What would settle it

An explicit construction of the clauses for some block-product state whose resulting Pauli Hamiltonian has an eigenstate of strictly lower energy than the planted state would falsify the central claim.

Figures

Figures reproduced from arXiv: 2606.11455 by Amir Kalev, Itay Hen.

Figure 1
Figure 1. Figure 1: FIG. 1. Construction pipeline for planted benchmark Hamiltonians. (a) Qubits are partitioned into disjoint blocks [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Fraction of instances with numerically degenerate [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Block-clause structure of the worked example. Nine [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
read the original abstract

We introduce a construction of Pauli Hamiltonians with exactly known ground-state energies, intended as reference instances for ground-state energy estimation algorithms. The construction embeds a planted block-product state as the simultaneous ground state of a sum of frustration-free local clauses on overlapping supports, exposes the resulting model only as a polynomial-size linear combination of Pauli operators, and admits optional Clifford conjugation that preserves the spectrum. The framework subsumes classical planted constraint-satisfaction problems as a diagonal special case, providing a direct embedding channel through which classical hardness properties can be inherited. Open-source software, certification keys, and example instances are made publicly available.

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

0 major / 1 minor

Summary. The manuscript introduces a construction of Pauli Hamiltonians with exactly known ground-state energies by embedding a planted block-product state as the simultaneous ground state of a sum of frustration-free local clauses defined on overlapping supports. The resulting Hamiltonian is exposed only as a polynomial-size linear combination of Pauli operators; the framework admits optional Clifford conjugation (which preserves the spectrum) and subsumes classical planted constraint-satisfaction problems as the diagonal special case. Open-source software, certification keys, and example instances are provided.

Significance. If the construction is valid, the work supplies a useful benchmarking primitive for ground-state energy estimation algorithms. The ability to plant known ground states while retaining a compact Pauli representation, together with the direct embedding channel from classical planted CSPs, allows controlled inheritance of hardness properties. Public release of software and instances further supports reproducibility and adoption as a standard test suite.

minor comments (1)
  1. The abstract states that the sum of frustration-free clauses yields the planted state as ground state but does not sketch the explicit construction of the clauses on overlapping supports; a one-sentence clarification in the abstract or introduction would improve immediate readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's construction directly defines frustration-free local clauses whose individual ground states are the planted block-product state by explicit design; the sum therefore has that state as ground state with energy equal to the sum of clause minima, and each constant-support clause expands to a fixed number of Pauli terms, yielding a polynomial-size Hamiltonian by elementary counting. Clifford conjugation is a unitary similarity transformation and therefore spectrum-preserving by definition. No parameter is fitted to data and then relabeled a prediction, no uniqueness theorem is imported from self-citation, and no ansatz is smuggled via prior work. The derivation is self-contained against the stated assumptions and does not reduce to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The construction rests on the domain assumption that suitable frustration-free clauses exist; no free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption Frustration-free local clauses on overlapping supports can share a common block-product ground state that remains the ground state of their sum.
    Core premise enabling the planted state to be the ground state of the full Hamiltonian.

pith-pipeline@v0.9.1-grok · 5620 in / 1239 out tokens · 34817 ms · 2026-06-27T13:01:02.092711+00:00 · methodology

discussion (0)

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Reference graph

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24 extracted references · 1 canonical work pages · 1 internal anchor

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    We drawK= 4 random subsets of blocks, each of size three, S1 ={1,2,4}, S 2 ={2,3,5}, S3 ={1,3,5}, S 4 ={3,4,5},(B2) which define the corresponding support regions Λk = [ i∈Sk Ai

    These block states define the planted global product state |Ψ⋆⟩= 5O i=1 |ψAi ⟩, with qubits ordered according to the block definitions above. We drawK= 4 random subsets of blocks, each of size three, S1 ={1,2,4}, S 2 ={2,3,5}, S3 ={1,3,5}, S 4 ={3,4,5},(B2) which define the corresponding support regions Λk = [ i∈Sk Ai. In this example, Λ 1 ={1,3,4,5,9}wit...