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A Note on the Complexity of the Spectral Gap Problem

Justin Yirka

Abstract

The problem of estimating the spectral gap of a local Hamiltonian is known to be contained in the class $P^{QMA[log]}$: polynomial time with access to a logarithmic number of QMA queries. The problem was shown to be hard for $P^{UQMA[log]}$, a weaker class, under Turing reductions by Gharibian and Yirka [arXiv:1606.05626]. I give a brief proof that the Spectral Gap problem is QMA-hard under a many-one (Karp) reduction. Consequently, the problem is $P^{QMA[log]}$-complete under truth-table reductions. It remains open to characterize the complexity of the Spectral Gap problem under many-one reductions. I conjecture that the problem belongs to a strict subclass of $P^{QMA[log]}$.

A Note on the Complexity of the Spectral Gap Problem

Abstract

The problem of estimating the spectral gap of a local Hamiltonian is known to be contained in the class : polynomial time with access to a logarithmic number of QMA queries. The problem was shown to be hard for , a weaker class, under Turing reductions by Gharibian and Yirka [arXiv:1606.05626]. I give a brief proof that the Spectral Gap problem is QMA-hard under a many-one (Karp) reduction. Consequently, the problem is -complete under truth-table reductions. It remains open to characterize the complexity of the Spectral Gap problem under many-one reductions. I conjecture that the problem belongs to a strict subclass of .

Paper Structure

This paper contains 3 sections, 5 theorems, 2 equations.

Table of Contents

  1. Results

Key Result

Lemma 3

Spectral Gap is contained in $\mathsf{P}\xspace^{\mathsf{QMA}\xspace[\log]}$.

Theorems & Definitions (11)

  • Definition 1: Spectral Gap
  • Definition 2
  • Lemma 3: Ambainis14-hardThanQMA
  • proof : Proof sketch
  • Lemma 4
  • proof
  • Corollary 5
  • Theorem 6
  • proof
  • Theorem 7
  • ...and 1 more