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arxiv: 2509.25821 · v2 · submitted 2025-09-30 · 🪐 quant-ph · cs.CC

On the Complexity of the Succinct State Local Hamiltonian Problem

Gabriel Waite , Karl Lin This is my paper

Pith reviewed 2026-05-18 13:15 UTC · model grok-4.3

classification 🪐 quant-ph cs.CC
keywords succinct quantum stateslocal Hamiltonian problemMA-completenesspromise problemsFeynman-Kitaev constructionqubit Hamiltoniansquantum complexitycircuit-to-Hamiltonian reduction
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The pith

The succinct-state 2-local Hamiltonian problem on qubits is promise-MA-complete.

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 problem of approximating the ground-state energy of a 2-local qubit Hamiltonian becomes promise-MA-complete when the ground state is promised to have a succinct classical description. The argument rests on a precise arithmetic characterization of which quantum states count as succinct, combined with a technical reduction that takes the standard Feynman-Kitaev circuit Hamiltonian and lowers its locality from six to two while leaving the number of particles unchanged. A reader should care because the result draws a sharp boundary: below a certain locality threshold the succinct version stays inside MA, while higher-locality versions sit at higher complexity classes.

Core claim

The Succinct State 2-Local Hamiltonian problem for qubit Hamiltonians is promise-MA-complete. The proof proceeds by first giving a systematic arithmetic description of succinct quantum states using operations over particular number fields, then applying a refined circuit-to-Hamiltonian reduction that achieves two-local interactions without raising the dimension of the particles.

What carries the argument

Refined Feynman-Kitaev reduction that lowers circuit-Hamiltonian locality from 6 to 2 for qubits while preserving succinct ground-state descriptions.

If this is right

  • A complexity phase transition appears when locality is varied while keeping the succinct-state promise fixed.
  • The set of known MA-complete instances is enlarged to include 2-local qubit Hamiltonians.
  • Succinctness is shown to be preserved under a locality-lowering circuit-to-Hamiltonian construction.
  • The boundary between efficiently describable and efficiently verifiable quantum systems is tightened for low-locality cases.

Where Pith is reading between the lines

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

  • Similar arithmetic characterizations might extend to other promise classes such as QMA or PP.
  • The same reduction technique could be tested on Hamiltonians with fixed interaction graphs to see whether MA-completeness survives.
  • If the number-field arithmetic can be made effective, one could search for concrete small instances that witness the MA-hardness.

Load-bearing premise

Succinct quantum states admit an arithmetic characterization over chosen number fields that remains intact after the locality-reduction step.

What would settle it

An explicit 2-local qubit Hamiltonian whose promised succinct ground state cannot be verified by any MA verifier would falsify the completeness claim.

Figures

Figures reproduced from arXiv: 2509.25821 by Gabriel Waite, Karl Lin.

Figure 1
Figure 1. Figure 1: FIG. 1. A hierarchy of succinct states. Not to scale. [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. A flow diagram of the complexity of the [PITH_FULL_IMAGE:figures/full_fig_p023_2.png] view at source ↗
read the original abstract

We study the computational complexity of the Local Hamiltonian problem under the promise that its ground state is succinctly represented. We show that the Succinct State 2-Local Hamiltonian problem, for qubit Hamiltonians, is (promise) MA-complete. The approach combines a systematic characterisation of succinct quantum states, defined through arithmetic over specific number fields, with a refined reduction that lowers the locality of Feynman-Kitaev circuit-Hamiltonians from 6 to 2, without increasing particle dimension. This reveals a complexity phase transition, parameterised by locality, and extends the scope of previously known MA-complete problem instances. Our results further clarify how succinctness behaves under circuit-based constructions, and progresses toward a better understanding of the boundary between efficiently describable and efficiently verifiable quantum systems.

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

1 major / 3 minor

Summary. The paper claims that the Succinct State 2-Local Hamiltonian problem for qubit Hamiltonians is promise MA-complete. It establishes this via a systematic characterization of succinct quantum states defined through arithmetic over specific number fields, combined with a refined reduction that lowers the locality of Feynman-Kitaev circuit Hamiltonians from 6 to 2 without increasing particle dimension, thereby revealing a locality-parameterized complexity phase transition.

Significance. If the central claims hold, the result extends the known MA-complete instances in quantum complexity theory and clarifies how succinctness is preserved under circuit-to-Hamiltonian mappings. The number-field arithmetic characterization and the locality-reduction technique constitute concrete technical progress toward delineating the boundary between efficiently describable and efficiently verifiable quantum systems.

major comments (1)
  1. [Section 4] Section 4 (Reduction from 6-local to 2-local): the proof that the refined mapping preserves both the succinct representation of the ground state and the promise gap must explicitly track the scaling of the energy gap with the number of ancilla particles introduced; without quantitative bounds, it is unclear whether the MA promise remains intact after the locality reduction.
minor comments (3)
  1. [Abstract] Abstract and Section 2: the phrase 'specific number fields' should be replaced by an explicit list or definition of the fields used in the succinct-state characterization.
  2. [Section 3] Figure 1 (if present) or the diagram in Section 3: labels on the reduction arrows should indicate which properties (succinctness, locality, gap) are preserved at each step.
  3. [References] References: add citations to prior work on succinct representations of quantum states and on MA-completeness of local Hamiltonian variants for context.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough review and constructive feedback. We appreciate the positive assessment and the recommendation for minor revision. We address the single major comment below and will incorporate the requested clarification.

read point-by-point responses

  1. Referee: [Section 4] Section 4 (Reduction from 6-local to 2-local): the proof that the refined mapping preserves both the succinct representation of the ground state and the promise gap must explicitly track the scaling of the energy gap with the number of ancilla particles introduced; without quantitative bounds, it is unclear whether the MA promise remains intact after the locality reduction.

    Authors: We agree that an explicit derivation of the gap scaling will improve clarity. The locality-reduction construction in Section 4 introduces a number of ancilla qubits linear in the size of the original circuit (specifically, O(s) ancillas for a circuit of size s). Because the underlying Feynman-Kitaev Hamiltonian already encodes an inverse-polynomial promise gap and the additional penalty terms for the ancillas are normalized by a constant factor independent of system size, the overall gap of the final 2-local Hamiltonian remains inverse polynomial in the total number of particles N. This scaling is sufficient to preserve the MA promise gap. In the revised manuscript we will add a short lemma (or dedicated paragraph) in Section 4 that states the precise dependence of the gap on the ancilla count and confirms that the MA promise is unchanged. We view this as a presentational improvement rather than a substantive gap in the argument. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper derives promise MA-completeness for the Succinct State 2-Local Hamiltonian problem via an explicit reduction from the standard Feynman-Kitaev circuit Hamiltonian (refined to 2-local qubits) combined with an independent arithmetic characterization of succinct states over number fields. No step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation; the argument preserves succinctness and promise gap through standard complexity-preserving transformations that are externally verifiable. This is the normal case of an independent reduction-based proof.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on standard definitions of the local Hamiltonian problem, the MA complexity class, and the Feynman-Kitaev construction; a new characterisation of succinct states is introduced but no free parameters or invented physical entities appear.

axioms (1)
  • standard math Standard definitions and closure properties of the MA complexity class and of k-local Hamiltonians on qubits.
    Invoked as the target class and problem family for the completeness proof.

pith-pipeline@v0.9.0 · 5648 in / 1143 out tokens · 52801 ms · 2026-05-18T13:15:53.280847+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Unentangled stoquastic Merlin-Arthur proof systems: the power of unentanglement without destructive interference

    quant-ph 2026-04 unverdicted novelty 7.0

    StoqMA(2) contains NP with Õ(√n)-qubit proofs and completeness error 2^{-polylog(n)}, is contained in EXP, and satisfies StoqMA(k)=StoqMA(2) for k≥2 when completeness error is negligible.

Reference graph

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