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Separating Quantum and Classical Advice with Good Codes

John Bostanci, Andrew Huang, Vinod Vaikuntanathan

TL;DR

The techniques yield the first unconditional classical oracle separation between the class of languages that can be decided with quantum advice and the class of languages that can be decided with classical advice.

Abstract

We show an unconditional classical oracle separation between the class of languages that can be verified using a quantum proof ($\mathsf{QMA}$) and the class of languages that can be verified with a classical proof ($\mathsf{QCMA}$). Compared to the recent work of Bostanci, Haferkamp, Nirkhe, and Zhandry (STOC 2026), our proof is conceptually and technically simpler, and readily extends to other oracle separations. In particular, our techniques yield the first unconditional classical oracle separation between the class of languages that can be decided with quantum advice ($\mathsf{BQP}/\mathsf{qpoly}$) and the class of languages that can be decided with classical advice ($\mathsf{BQP}/\mathsf{poly}$), improving on the quantum oracle separation of Aaronson and Kuperberg (CCC 2007) and the classically-accessible classical oracle separation of Li, Liu, Pelecanos and Yamakawa (ITCS 2024). Our oracles are based on the code intersection problem introduced by Yamakawa and Zhandry (FOCS 2022), combined with codes that have extremely good list-recovery properties.

Separating Quantum and Classical Advice with Good Codes

TL;DR

The techniques yield the first unconditional classical oracle separation between the class of languages that can be decided with quantum advice and the class of languages that can be decided with classical advice.

Abstract

We show an unconditional classical oracle separation between the class of languages that can be verified using a quantum proof () and the class of languages that can be verified with a classical proof (). Compared to the recent work of Bostanci, Haferkamp, Nirkhe, and Zhandry (STOC 2026), our proof is conceptually and technically simpler, and readily extends to other oracle separations. In particular, our techniques yield the first unconditional classical oracle separation between the class of languages that can be decided with quantum advice () and the class of languages that can be decided with classical advice (), improving on the quantum oracle separation of Aaronson and Kuperberg (CCC 2007) and the classically-accessible classical oracle separation of Li, Liu, Pelecanos and Yamakawa (ITCS 2024). Our oracles are based on the code intersection problem introduced by Yamakawa and Zhandry (FOCS 2022), combined with codes that have extremely good list-recovery properties.
Paper Structure (35 sections, 33 theorems, 106 equations, 2 figures)

This paper contains 35 sections, 33 theorems, 106 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Our Work
  3. Technical Overview
  4. The Yamakawa-Zhandry Algorithm.
  5. Lifting the Separation.
  6. An Entropic Viewpoint.
  7. List Recovery and Code Expansion.
  8. A Final Complication.
  9. A General Recipe for a Classical Oracle Separation.
  10. Moving to the Advice Setting: $\BQP/\mathrm{qpoly}$ vs. $\BQP/\mathrm{poly}$.
  11. Some Concluding Remarks.
  12. Discussion and Open Questions
  13. Structured versus randomness in classical oracle separations
  14. QMA-completeness of a decoding problem
  15. Simplifications to the separation
  16. ...and 20 more sections

Key Result

Theorem 1.1

There exists a classical oracle $\mathcal{O}$ such that $\QMA^{\mathcal{O}} \not\subseteq \QCMA^{\mathcal{O}}$.

Figures (2)

  • Figure 1: Biased Yamakawa-Zhandry Algorithm $\mathsf{BiasedYZ}_{\mathtt{Dec}_{C^{\perp}}}(\ket{\mathsf{adv}_H}, \mathbf{b})$
  • Figure 2: The Hash Value Guesser, given a successful $\QCMA$ verifier for the code intersection subset size problem.

Theorems & Definitions (89)

  • Theorem 1.1: Informal
  • Theorem 1.2: Informal
  • Definition 1: Modified from BHNZ25
  • Definition 2: BHNZ25
  • Definition 3: Oracle $\QCMA$
  • Definition 4: Oracle $\QMA$
  • Definition 5: Oracle $\BQP/\mathrm{poly}$
  • Definition 6: Oracle $\BQP/\mathrm{qpoly}$
  • Definition 7: Query mass
  • Theorem 4.1: Hybrid method BBBV97
  • ...and 79 more