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A Cautionary Note on Quantum Oracles

Avantika Agarwal, Srijita Kundu

TL;DR

The paper demonstrates that quantum-relativized oracle techniques can yield separations between QMA and polyQCPH that do not carry over to classical oracle models, and similarly for distributional oracles, urging caution in applying non-standard oracle frameworks to quantum-classical separations. It extends AK07’s quantum-oracle separation to the broader quantum-classical polynomial hierarchy and shows QMA ⊆ PQP under quantum oracles, while constructing a language that lies in QMA^U but not in polyQCPH^U. Parallel results in distributional-oracle settings show QMA ⊆ PSPACE under distributional oracles, but also reveal L ∈ QMA^O while L ∉ polyQCPH^O with high probability, reinforcing that distributional models can diverge from classical intuition. Collectively, these results highlight the delicate boundary between quantum and classical resources in relativized models and stress the need for caution when using quantum and distributional oracle constructions to derive separations.

Abstract

In recent years, the quantum oracle model introduced by Aaronson and Kuperberg (2007) has found a lot of use in showing oracle separations between complexity classes and cryptographic primitives. It is generally assumed that proof techniques that do not relativize with respect to quantum oracles will also not relativize with respect to classical oracles. In this note, we show that this is not the case: specifically, we show that there is a quantum oracle problem that is contained in the class QMA, but not in a class we call polyQCPH. The class polyQCPH is equal to PSPACE with respect to classical oracles, and it is a well-known result that QMA is contained in PSPACE (also with respect to classical oracles). We also show that the same separation holds relative to a distributional oracle, which is a model introduced by Natarajan and Nirkhe (2024). We believe our findings show the need for some caution when using these non-standard oracle models, particularly when showing separations between quantum and classical resources.

A Cautionary Note on Quantum Oracles

TL;DR

The paper demonstrates that quantum-relativized oracle techniques can yield separations between QMA and polyQCPH that do not carry over to classical oracle models, and similarly for distributional oracles, urging caution in applying non-standard oracle frameworks to quantum-classical separations. It extends AK07’s quantum-oracle separation to the broader quantum-classical polynomial hierarchy and shows QMA ⊆ PQP under quantum oracles, while constructing a language that lies in QMA^U but not in polyQCPH^U. Parallel results in distributional-oracle settings show QMA ⊆ PSPACE under distributional oracles, but also reveal L ∈ QMA^O while L ∉ polyQCPH^O with high probability, reinforcing that distributional models can diverge from classical intuition. Collectively, these results highlight the delicate boundary between quantum and classical resources in relativized models and stress the need for caution when using quantum and distributional oracle constructions to derive separations.

Abstract

In recent years, the quantum oracle model introduced by Aaronson and Kuperberg (2007) has found a lot of use in showing oracle separations between complexity classes and cryptographic primitives. It is generally assumed that proof techniques that do not relativize with respect to quantum oracles will also not relativize with respect to classical oracles. In this note, we show that this is not the case: specifically, we show that there is a quantum oracle problem that is contained in the class QMA, but not in a class we call polyQCPH. The class polyQCPH is equal to PSPACE with respect to classical oracles, and it is a well-known result that QMA is contained in PSPACE (also with respect to classical oracles). We also show that the same separation holds relative to a distributional oracle, which is a model introduced by Natarajan and Nirkhe (2024). We believe our findings show the need for some caution when using these non-standard oracle models, particularly when showing separations between quantum and classical resources.
Paper Structure (20 sections, 11 theorems, 32 equations, 1 figure)

This paper contains 20 sections, 11 theorems, 32 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Our observations
  3. Quantum oracles.
  4. Distributional oracles.
  5. Definitions and Preliminaries
  6. Complexity Classes
  7. Other Preliminaries
  8. Classical Oracles
  9. Quantum Oracles
  10. QMA is contained in PQP
  11. $\mathsf{TQBF}\xspace$ completeness fails for quantum oracles.
  12. Oracle Separation between QMA and polyQCPH
  13. AK07 Quantum oracle problem.
  14. Discussion
  15. Distributional Oracles
  16. ...and 5 more sections

Key Result

lemma 2.13

Given a $p$-uniform probability measure $\sigma$ over $N$-dimensional pure states, and a density matrix $\rho$,

Figures (1)

  • Figure 1: Complexity class containments in the unrelatized setting, and with respect to classical, quantum and distributional oracles. All of the containments shown here were known previously in the unrelativized setting; we verify that most of the containments hold in the quantum and distributional oracle settings, except$\mathsf{QMA}\xspace \subseteq \mathsf{polyQCPH}\xspace$, for which we show a separation.

Theorems & Definitions (31)

  • definition 2.1: True Quantified Boolean Formula ($\mathsf{TQBF}\xspace$)
  • definition 2.2: Quantum Turing Machine Wat09
  • definition 2.3: $\mathsf{BQPSPACE}\xspace$ Wat09
  • definition 2.4: $\mathsf{PQP}\xspace$
  • definition 2.5: $\mathsf{QMA}\xspace$
  • definition 2.6: $\mathsf{QC\Sigma}_i$
  • definition 2.7: $\mathsf{QC\Pi}_i$
  • definition 2.8: Quantum-Classical Polynomial Hierarchy ($\mathsf{QCPH}\xspace$)
  • definition 2.9: $\mathsf{polyQC\Sigma}_i$
  • definition 2.10: $\mathsf{polyQC\Pi}_i$
  • ...and 21 more