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On the Complexity of the Succinct State Local Hamiltonian Problem

Gabriel Waite, Karl Lin

TL;DR

The paper studies the Local Hamiltonian problem under the promise that the ground state is succinctly representable and proves that the Succinct State $3$-Local Hamiltonian problem is promise $MA$-complete. It formalizes four natural families of succinct states with exact binary encodings, analyzes how they behave under common quantum gates and circuit compositions, and develops locality-reduction techniques to push MA-hardness to $3$-locality. A core technical contribution is showing that history states from MA circuits can be realized as succinct subset-like states, enabling a robust MA-hardness reduction that survives complex-to-real and real-to-stoquastic transformations. The results illuminate a boundary between quantum and classical verification when the witness is structurally constrained, providing insight into the verification of ground-state energies and guiding future work on robustness to encoding precision and geometric Hamiltonians.

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 3-Local Hamiltonian problem is (promise) MA-complete. Our proof proceeds by systematically characterising succinct quantum states and modifying the original MA-hardness reduction. In particular, we show that a broader class of succinct states suffices to capture the hardness of the problem, extending and strengthening prior results to classes of Hamiltonians with lower locality.

On the Complexity of the Succinct State Local Hamiltonian Problem

TL;DR

The paper studies the Local Hamiltonian problem under the promise that the ground state is succinctly representable and proves that the Succinct State -Local Hamiltonian problem is promise -complete. It formalizes four natural families of succinct states with exact binary encodings, analyzes how they behave under common quantum gates and circuit compositions, and develops locality-reduction techniques to push MA-hardness to -locality. A core technical contribution is showing that history states from MA circuits can be realized as succinct subset-like states, enabling a robust MA-hardness reduction that survives complex-to-real and real-to-stoquastic transformations. The results illuminate a boundary between quantum and classical verification when the witness is structurally constrained, providing insight into the verification of ground-state energies and guiding future work on robustness to encoding precision and geometric Hamiltonians.

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 3-Local Hamiltonian problem is (promise) MA-complete. Our proof proceeds by systematically characterising succinct quantum states and modifying the original MA-hardness reduction. In particular, we show that a broader class of succinct states suffices to capture the hardness of the problem, extending and strengthening prior results to classes of Hamiltonians with lower locality.

Paper Structure

This paper contains 35 sections, 58 theorems, 120 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Related Prior Work.
  3. Preliminaries and Technical Summary
  4. Binary Representations
  5. Binary Number Classes
  6. Computational Complexity
  7. Technical Summary
  8. Succinct States
  9. Properties of Subset States
  10. Operations with Subset States
  11. Operations with Hybrid Subset States
  12. Properties of General Succinct States
  13. Multi-Alphabet Query Access
  14. The Succinct State Local Hamiltonian problem
  15. Class Containment
  16. ...and 20 more sections

Key Result

Lemma 1

Consider two $\mathbb{S}$-succinct states $(\lvert \psi \rangle,\mathbb{S}_{p(n)},\mathcal{Q}_{\psi})$ and $(\lvert \phi \rangle, \mathbb{S}_{q(m)},\mathcal{Q}_{\phi})$, where $p(n)$ and $q(m)$ are polynomial functions on $n$ and $m$ respectively. Then the tensor product $\lvert \psi \rangle\lvert \

Figures (2)

  • Figure 1: A hierarchy of succinct states. Not to scale.
  • Figure 2: A flow diagram of the complexity of the Succinct State Local Hamiltonian problem. Arrows (loosely) represent modifications/reductions. We note that bold (solid) arrows indicate the flow of ideas akin to a reduction. The dashed arrows represent the combination of results needed to establish new complexity classifications. Smaller boxes with a grey background represent results from prior work, while orange boxes denote results from this work. The larger three boxes represent groupings of specific complexity results; namely MA- hardness and MA containment. Also note that some arrows have been omitted to improve readability.

Theorems & Definitions (111)

  • Definition 1: Semi-Classical Verification Circuit
  • Definition 2: MAq BDOT06
  • Lemma 1
  • Lemma 1
  • Lemma 1
  • Remark 1
  • Lemma 1
  • Lemma 1
  • Claim 1
  • Proposition 1
  • ...and 101 more