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Toward Separating QMA from QCMA with a Classical Oracle

Mark Zhandry

TL;DR

This work offers a new approach towards proving a classical oracle separation for QCMA, a class of languages that can be decided by an efficient quantum verifier given a quantum witness, and shows that this approach is sound assuming a natural statistical conjecture which may have other applications to quantum query complexity lower bounds.

Abstract

QMA is the class of languages that can be decided by an efficient quantum verifier given a quantum witness, whereas QCMA is the class of such languages where the efficient quantum verifier only is given a classical witness. A challenging fundamental goal in quantum query complexity is to find a classical oracle separation for these classes. In this work, we offer a new approach towards proving such a separation that is qualitatively different than prior work, and show that our approach is sound assuming a natural statistical conjecture which may have other applications to quantum query complexity lower bounds.

Toward Separating QMA from QCMA with a Classical Oracle

TL;DR

This work offers a new approach towards proving a classical oracle separation for QCMA, a class of languages that can be decided by an efficient quantum verifier given a quantum witness, and shows that this approach is sound assuming a natural statistical conjecture which may have other applications to quantum query complexity lower bounds.

Abstract

QMA is the class of languages that can be decided by an efficient quantum verifier given a quantum witness, whereas QCMA is the class of such languages where the efficient quantum verifier only is given a classical witness. A challenging fundamental goal in quantum query complexity is to find a classical oracle separation for these classes. In this work, we offer a new approach towards proving such a separation that is qualitatively different than prior work, and show that our approach is sound assuming a natural statistical conjecture which may have other applications to quantum query complexity lower bounds.

Paper Structure

This paper contains 25 sections, 21 theorems, 27 equations.

Table of Contents

  1. Introduction
  2. Our Work.
  3. Our Separating Oracles
  4. ${\sf QCMA}$ hardness
  5. $V$ that do not query $U$.
  6. $k$-wise independent $U$.
  7. Our $U$ are "close" to $k$-wise independent.
  8. Is this "close" enough?
  9. Preliminaries and notation
  10. Complex Normal Distribution.
  11. Quantum Computation.
  12. Defining QMA and QCMA relative to Oracles
  13. Some Useful Quantum Lemmas
  14. Our Oracle Separation
  15. Our Statistical Conjecture
  16. ...and 10 more sections

Key Result

Theorem 2.3

There exists a classical oracle ${\mathcal{O}}$ such that ${\sf QCMA}^{\mathcal{O}}\neq{\sf QMA}^{\mathcal{O}}$ if and only if ${\sf OI\text{-}QCMA}\neq{\sf OI\text{-}QMA}$.

Theorems & Definitions (40)

  • Remark 1.1
  • Remark 1.2
  • Definition 2.1: Oracle-Aided QMA/QCMA
  • Definition 2.2: Oracle-Input QMA/QCMA
  • Theorem 2.3
  • Lemma 2.4: BBBV97 Theorem 3.1+3.3
  • Lemma 2.5: C:Zhandry12 Theorem 3.1
  • Lemma 2.6: HamMag23 Theorem 5.5
  • Definition 3.1: Substitution Distance
  • Conjecture 3.2
  • ...and 30 more