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The Guided Local Hamiltonian Problem for Stoquastic Hamiltonians

Gabriel Waite

TL;DR

This work extends the Guided Local Hamiltonian framework to stoquastic Hamiltonians, proving (promise) $BPP$-hardness even for $2$-local and square-lattice instances by mapping $BPP$ circuits to stoquastic Hamiltonians using a Feynman-Kitaev construction and a semi-classical encoded subset guiding state. It further shows $BQP$-hardness for guided pinned stoquastic models, via gadget-embedding techniques that preserve the guiding state's overlap with the ground state. The paper also develops a formal complexity-theoretic context (BPPq, StoqP) and discusses a multi-chotomy and duality conjecture that situates guided versus unguided Hamiltonians within a spectrum between $P$ and $BQP$, highlighting potential intermediate complexity classes. Overall, the results illuminate classical-quantum boundaries in stoquastic settings and raise open questions about guiding-state discovery and the exact complexity class of guided stoquastic Hamiltonians.

Abstract

We show that the Guided Local Hamiltonian problem for stoquastic Hamiltonians is (promise) BPP-hard. The Guided Local Hamiltonian problem is a variant of the Local Hamiltonian problem that incorporates an additional input known as a guiding state, which is promised to overlap with the ground state. For a range of local Hamiltonian families, this problem is (promise) BQP-hard, though for stoquastic Hamiltonians, the complexity was previously unknown. Our results are achieved 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. Subsequent analysis shows the BPP-hardness is not dependent on the locality, i.e., the result holds for 2-local stoquastic Hamiltonians. Additional arguments further the BPP-hardness to Hamiltonians restricted to a square lattice. We also find for stoquastic Hamiltonians with a fixed local constraint on a subset of the system qubits, the Guided Local Hamiltonian problem is BQP-hard.

The Guided Local Hamiltonian Problem for Stoquastic Hamiltonians

TL;DR

This work extends the Guided Local Hamiltonian framework to stoquastic Hamiltonians, proving (promise) -hardness even for -local and square-lattice instances by mapping circuits to stoquastic Hamiltonians using a Feynman-Kitaev construction and a semi-classical encoded subset guiding state. It further shows -hardness for guided pinned stoquastic models, via gadget-embedding techniques that preserve the guiding state's overlap with the ground state. The paper also develops a formal complexity-theoretic context (BPPq, StoqP) and discusses a multi-chotomy and duality conjecture that situates guided versus unguided Hamiltonians within a spectrum between and , highlighting potential intermediate complexity classes. Overall, the results illuminate classical-quantum boundaries in stoquastic settings and raise open questions about guiding-state discovery and the exact complexity class of guided stoquastic Hamiltonians.

Abstract

We show that the Guided Local Hamiltonian problem for stoquastic Hamiltonians is (promise) BPP-hard. The Guided Local Hamiltonian problem is a variant of the Local Hamiltonian problem that incorporates an additional input known as a guiding state, which is promised to overlap with the ground state. For a range of local Hamiltonian families, this problem is (promise) BQP-hard, though for stoquastic Hamiltonians, the complexity was previously unknown. Our results are achieved 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. Subsequent analysis shows the BPP-hardness is not dependent on the locality, i.e., the result holds for 2-local stoquastic Hamiltonians. Additional arguments further the BPP-hardness to Hamiltonians restricted to a square lattice. We also find for stoquastic Hamiltonians with a fixed local constraint on a subset of the system qubits, the Guided Local Hamiltonian problem is BQP-hard.

Paper Structure

This paper contains 32 sections, 24 theorems, 50 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Outline and Contributions.
  3. Preliminaries
  4. Subset states
  5. Complexity classes
  6. Simulator Hamiltonians
  7. Problem statement
  8. Quantum BPP Circuits
  9. Stoquastic Arthur
  10. The action of classically reversible gates
  11. Classical Hardness
  12. Pre-idling.
  13. Existence of a guiding state.
  14. Perturbed Hamiltonian.
  15. State overlap.
  16. ...and 17 more sections

Key Result

Proposition 1

A tensor product of a polynomial number of semi-classical encoded subset states is a semi-classical encoded subset state.

Figures (1)

  • Figure 1: A diagram of the complexity classes discussed in this work. The conjectured class is denoted by X (in violet).

Theorems & Definitions (44)

  • Definition 1: Subset State
  • Definition 2: Semi-Classical Subset State
  • Definition 3: Semi-Classical Encoded Subset States
  • Proposition 1
  • Proposition 2: CKFH+23
  • Definition 4: BPP
  • Definition 5: BQP
  • Proposition 3
  • proof
  • Definition 6: The Guided Local Hamiltonian Problem
  • ...and 34 more