arxiv: 2509.25829 · v2 · submitted 2025-09-30 · 🪐 quant-ph · cs.CC

The Guided Local Hamiltonian Problem for Stoquastic Hamiltonians

Gabriel Waite This is my paper

Pith reviewed 2026-05-18 13:10 UTC · model grok-4.3

classification 🪐 quant-ph cs.CC

keywords guided local Hamiltonianstoquastic HamiltoniansBPP-hardnessground-state energy estimationquantum complexitysemi-classical statesreduction from circuits

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The pith

Estimating the ground-state energy of stoquastic Hamiltonians is promise BPP-hard even with a guiding state that overlaps the true ground state.

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

The paper shows that the Guided Local Hamiltonian problem stays hard for classical probabilistic algorithms when the Hamiltonian is restricted to be stoquastic. It reaches this conclusion by building a reduction from BPP circuits that produces both a 6-local stoquastic Hamiltonian and a special guiding state whose overlap with the ground state is guaranteed. The hardness result survives when the interactions are lowered to two-body terms and when the qubits are arranged on a square lattice. A reader cares because stoquastic Hamiltonians lack the sign problem that usually blocks classical simulation, so one might have hoped a guiding state would make the problem tractable; the result says that hope does not hold in general.

Core claim

We show that the Guided Local Hamiltonian problem for stoquastic Hamiltonians is (promise) BPP-hard. The proof proceeds by reducing from quantum-inspired BPP circuits to 6-local stoquastic Hamiltonians whose ground states are guided by semi-classical encoded subset states that maintain sufficient overlap. The same BPP-hardness carries over to 2-local stoquastic Hamiltonians and to Hamiltonians placed on a square lattice. For stoquastic Hamiltonians that carry an additional fixed local constraint on a subset of qubits the problem becomes BQP-hard. Under constant promise gap, constant overlap, and constant spectral gap, when the guiding state is preparable by a constant-depth geometrically лок

What carries the argument

Semi-classical encoded subset states that act as guiding states with a promised overlap to the ground state of the constructed stoquastic Hamiltonian while preserving the original BPP circuit hardness.

If this is right

The BPP-hardness continues to hold when the Hamiltonian is restricted to two-body interactions.
The hardness result extends to stoquastic Hamiltonians whose qubits lie on a square lattice.
Adding a fixed local constraint on a subset of the qubits upgrades the problem to BQP-hard.
When the promise gap, overlap, and spectral gap are all constant and the guiding state is prepared by a constant-depth geometrically local circuit, a deterministic classical approximation algorithm exists.

Where Pith is reading between the lines

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

Guiding states of this form appear insufficient to render general stoquastic systems classically tractable.
The boundary between BPP and BQP for guided ground-state problems may be narrower than expected inside sign-problem-free families.
Similar reductions could be attempted with guiding states prepared by other classical means to test how much classical side information is needed to cross the hardness threshold.

Load-bearing premise

The semi-classical encoded subset states can be built so that they keep enough overlap with the ground state of the stoquastic Hamiltonian without erasing the BPP-hardness inherited from the circuit problem.

What would settle it

A classical polynomial-time algorithm that approximates the ground-state energy of 2-local stoquastic Hamiltonians to within the promised constant gap, given a guiding state with constant overlap, would falsify the BPP-hardness claim.

Figures

Figures reproduced from arXiv: 2509.25829 by Gabriel Waite.

read the original abstract

We show that the Guided Local Hamiltonian problem for stoquastic Hamiltonians is (promise) BPP-hard. The Guided Local Hamiltonian problem extends the Local Hamiltonian problem by incorporating an additional input known as a guiding state, which is promised to overlap with the ground state. For a range of local Hamiltonian families, prior work shows this problem is (promise) BQP-hard, though for stoquastic Hamiltonians, the complexity was previously unknown. We obtain our results by first reducing from quantum-inspired BPP circuits to 6-local stoquastic Hamiltonians. We prove particular classes of quantum states, known as semi-classical encoded subset states, can guide the estimation of the ground-state energy. Our analysis shows that this BPP-hardness does not depend on locality, i.e., the result holds for 2-local stoquastic Hamiltonians. Additional arguments extend this BPP-hardness to Hamiltonians restricted to a square lattice. We further show that for stoquastic Hamiltonians with a fixed local constraint on a subset of the system qubits, the Guided Local Hamiltonian problem is BQP-hard. In addition to these hardness results, we present a deterministic classical approximation algorithm for the problem under the conditions of constant promise gap, constant overlap, and constant spectral gap, when the guiding state is preparable in constant depth by a geometrically local circuit.

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.

Desk Editor's Note private letter to a colleague

The paper shows guided stoquastic local Hamiltonian is BPP-hard down to 2-local lattice cases via a direct reduction that was left open before.

read the letter

The main takeaway is that the guided local Hamiltonian problem stays BPP-hard when restricted to stoquastic Hamiltonians, and the hardness carries through to 2-local interactions on a square lattice. They close the gap left open in earlier work by reducing from standard BPP circuits to 6-local stoquastic Hamiltonians first, then showing the same hardness for the 2-local and lattice versions. The guiding states are semi-classical encoded subset states that keep enough overlap with the ground state to preserve the promise gap. They also add a deterministic classical approximation algorithm that works when the promise gap, overlap, and spectral gap are all constant and the guiding state comes from a constant-depth local circuit. This last part is a useful complement because it spells out conditions where classical methods can still succeed on these sign-problem-free models. The reduction approach is straightforward and builds cleanly on prior results without circularity. The lattice extension adds some physical relevance. The potential weak point is whether the overlap lower bound survives the gadgets used to drop from 6-local to 2-local. The stress-test note flags exactly this risk: extra terms could push the ground-state support away from the subset-state subspace and collapse the promise. The paper claims the result is independent of locality, which implies they have bounds that hold, but a referee would still want to check the explicit overlap estimates after each reduction step to confirm nothing slips below inverse-polynomial. This is aimed at complexity theorists working on quantum Hamiltonians and stoquastic systems in quantum chemistry or lattice models. Someone who follows guided problems or computational limits for sign-problem-free cases will get direct value from the new hardness statement and the approximation conditions. It deserves a serious referee because the claim is substantive, the reduction is explicit, and the details are checkable. I would send it out for review.

Referee Report

1 major / 2 minor

Summary. The manuscript proves that the Guided Local Hamiltonian problem for stoquastic Hamiltonians is promise-BPP-hard. The central argument reduces from quantum-inspired BPP circuit problems to the task of estimating the ground-state energy of a 6-local stoquastic Hamiltonian, using semi-classical encoded subset states as guiding states that are promised to have non-negligible overlap with the true ground state. The authors then extend the reduction to 2-local stoquastic Hamiltonians and to Hamiltonians supported on a square lattice while preserving the BPP-hardness. A separate argument shows BQP-hardness when an additional fixed local constraint is imposed on a subset of qubits. The paper also supplies a deterministic classical algorithm that approximates the ground-state energy to constant additive error when the promise gap, guiding-state overlap, and spectral gap are all constant and the guiding state is preparable by a constant-depth geometrically local circuit.

Significance. If the reductions and overlap guarantees are correct, the result supplies the first explicit complexity classification for the guided local Hamiltonian problem in the stoquastic setting, placing it in promise-BPP rather than promise-BQP. The claimed independence from locality and the extension to square-lattice geometry strengthen the statement. The classical approximation algorithm is a constructive positive result that applies under the stated constant-parameter regime. The explicit construction from BPP circuits and the use of semi-classical states are verifiable strengths.

major comments (1)

[2-local reduction and overlap analysis] The 2-local reduction (following the 6-local construction): the central claim requires an explicit inverse-polynomial lower bound on the overlap between the semi-classical encoded subset state and the ground state of the final 2-local stoquastic Hamiltonian. The manuscript must show that the additional gadget terms do not shift the ground-state support outside the relevant subspace by more than an inverse-polynomial amount; otherwise the promise gap for guided energy estimation collapses and BPP-hardness fails to transfer. Please supply the concrete overlap calculation or lemma that survives the locality reduction.

minor comments (2)

[Introduction] The phrase 'quantum-inspired BPP circuits' appears in the abstract and introduction without a self-contained definition or pointer to the precise circuit model used; a short paragraph or reference would improve readability.
[Preliminaries] Notation for the semi-classical encoded subset states should be introduced once with a clear definition of the encoding map before it is used in the overlap arguments.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough review and for identifying this key point in the 2-local reduction. We address the comment directly below and will incorporate the requested clarification in the revised manuscript.

read point-by-point responses

Referee: [2-local reduction and overlap analysis] The 2-local reduction (following the 6-local construction): the central claim requires an explicit inverse-polynomial lower bound on the overlap between the semi-classical encoded subset state and the ground state of the final 2-local stoquastic Hamiltonian. The manuscript must show that the additional gadget terms do not shift the ground-state support outside the relevant subspace by more than an inverse-polynomial amount; otherwise the promise gap for guided energy estimation collapses and BPP-hardness fails to transfer. Please supply the concrete overlap calculation or lemma that survives the locality reduction.

Authors: We agree that an explicit inverse-polynomial overlap bound is necessary to rigorously transfer the BPP-hardness through the locality reduction. The 6-local construction (Section 3) establishes the base overlap guarantee for semi-classical encoded subset states. The 2-local reduction (Section 4) replaces each 6-local term with a standard perturbative 2-local gadget that preserves stoquasticity. These gadgets are chosen so that their action is diagonal in the computational basis outside the encoded subspace and introduce only an inverse-polynomial perturbation within the low-energy subspace. In the revised manuscript we will add a dedicated lemma (new Lemma 4.6) that combines the gadget spectral gap (which is polynomial by construction) with the original overlap promise to show that the final ground-state overlap remains at least 1/poly(n). The proof proceeds by bounding the ground-state projection onto the orthogonal complement of the encoded subspace using standard perturbation theory for stoquastic Hamiltonians; the error term is controlled by the ratio of the gadget strength to the gap and is therefore inverse-polynomial. This addition does not change the main theorem statements but makes the overlap preservation fully explicit. revision: yes

Circularity Check

0 steps flagged

Explicit reduction from BPP circuits to guided stoquastic LH establishes hardness without circular definitions or self-referential steps

full rationale

The derivation proceeds by an explicit, constructive reduction from quantum-inspired BPP circuit instances to 6-local stoquastic Hamiltonians, followed by an analytic proof that semi-classical encoded subset states have non-negligible overlap with the ground state. The same construction is then shown to preserve the promise gap and overlap bound under reduction to 2-local and square-lattice geometries. No quantity in the target problem (energy estimate, guiding-state overlap, or promise gap) is defined in terms of itself or obtained by fitting a parameter to the output; the overlap lower bound is derived directly from the circuit-to-Hamiltonian mapping rather than assumed or renamed. No load-bearing self-citation or uniqueness theorem imported from prior author work appears in the argument. The central claim therefore remains independent of its own outputs and is self-contained against the external BPP hardness assumption.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proof rests on standard definitions of promise-BPP and stoquastic Hamiltonians together with the existence of the semi-classical guiding states used in the reduction; no numerical free parameters or new physical entities are introduced.

axioms (2)

standard math Standard promise-BPP and promise-gap definitions from complexity theory
The hardness statement is phrased in these terms.
domain assumption Existence and overlap properties of semi-classical encoded subset states for the constructed Hamiltonians
Invoked to ensure the guiding state satisfies the promise in the reduction.

pith-pipeline@v0.9.0 · 5761 in / 1412 out tokens · 75351 ms · 2026-05-18T13:10:31.436305+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

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IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

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Relation between the paper passage and the cited Recognition theorem.

We show that the Guided Local Hamiltonian problem for stoquastic Hamiltonians is (promise) BPP-hard... by first reducing from quantum-inspired BPP circuits to 6-local stoquastic Hamiltonians... semi-classical encoded subset states
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Apply the Schrieffer-Wolff transformation... perturbation gadgets... overlap |⟨ξ|ζ⟩|² ≥ (√δ − ε)²

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Forward citations

Cited by 1 Pith paper

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Reference graph

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As the quantum circuit can be simulated by a classical probabilistic circuit with random coin flips instead of |+p⟩, the same success probability can be achieved classically. Therefore, there exists a classical probabilistic circuit that outputs 1 with probability at least 2 3 forx∈Lyes. Case 2:Similarly, if x∈Lno, the quantum circuit M|x|outputs 1 with p...

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b.BPP ⊆BPPq.Now, let L = (Lyes,L no) be a promise problem inBPP

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Then:    ( √ B−A)2≤|⟨a|c⟩|2≤( √ B+A) 2 ifA≤ √ B, 0≤|⟨a|c⟩|2≤( √ B+A) 2 ifA> √ B

The quantum circuit, which simulates the classical circuit, also outputs 1 with probability at most 1 3, as the quantum ancillae provide the same type of randomness and the gates are reversible.■ Lemma 3.Let|a⟩,|b⟩,|c⟩∈(C2)⊗nbe normalised states, such that|||a⟩−|b⟩||≤Aand|⟨b|c⟩|2≥B. Then:    ( √ B−A)2≤|⟨a|c⟩|2≤( √ B+A) 2 ifA≤ √ B, 0≤|⟨a|c⟩|2≤( √ B+A) 2...

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