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BQP and the Polynomial Hierarchy

Scott Aaronson

TL;DR

The paper investigates whether quantum computation (BQP) lies outside the polynomial hierarchy (PH), delivering the first formal relativized evidence via oracle separations built around Fourier Fishing and Fourier Checking. It develops quantum algorithms that solve these problems with very few queries, while showing classical AC$^0$ and related models require superpolynomial resources, thereby connecting quantum advantages to deep circuit lower bounds. Central to the approach is the Generalized Linial-Nisan Conjecture (GLN), a near-extension of Braverman’s Linial-Nisan result, which would imply strong separations like BQP ∉ PH under plausible conjectures about almost $k$-wise independence and low-fat polynomial approximations. The work also introduces the Low-Fat Sandwich Conjecture as a concrete route to GLN, highlighting a rich interplay between quantum complexity, pseudorandomness, and circuit complexity with potential broader implications for understanding the capabilities of quantum computation beyond PH.

Abstract

The relationship between BQP and PH has been an open problem since the earliest days of quantum computing. We present evidence that quantum computers can solve problems outside the entire polynomial hierarchy, by relating this question to topics in circuit complexity, pseudorandomness, and Fourier analysis. First, we show that there exists an oracle relation problem (i.e., a problem with many valid outputs) that is solvable in BQP, but not in PH. This also yields a non-oracle relation problem that is solvable in quantum logarithmic time, but not in AC0. Second, we show that an oracle decision problem separating BQP from PH would follow from the Generalized Linial-Nisan Conjecture, which we formulate here and which is likely of independent interest. The original Linial-Nisan Conjecture (about pseudorandomness against constant-depth circuits) was recently proved by Braverman, after being open for twenty years.

BQP and the Polynomial Hierarchy

TL;DR

The paper investigates whether quantum computation (BQP) lies outside the polynomial hierarchy (PH), delivering the first formal relativized evidence via oracle separations built around Fourier Fishing and Fourier Checking. It develops quantum algorithms that solve these problems with very few queries, while showing classical AC and related models require superpolynomial resources, thereby connecting quantum advantages to deep circuit lower bounds. Central to the approach is the Generalized Linial-Nisan Conjecture (GLN), a near-extension of Braverman’s Linial-Nisan result, which would imply strong separations like BQP ∉ PH under plausible conjectures about almost -wise independence and low-fat polynomial approximations. The work also introduces the Low-Fat Sandwich Conjecture as a concrete route to GLN, highlighting a rich interplay between quantum complexity, pseudorandomness, and circuit complexity with potential broader implications for understanding the capabilities of quantum computation beyond PH.

Abstract

The relationship between BQP and PH has been an open problem since the earliest days of quantum computing. We present evidence that quantum computers can solve problems outside the entire polynomial hierarchy, by relating this question to topics in circuit complexity, pseudorandomness, and Fourier analysis. First, we show that there exists an oracle relation problem (i.e., a problem with many valid outputs) that is solvable in BQP, but not in PH. This also yields a non-oracle relation problem that is solvable in quantum logarithmic time, but not in AC0. Second, we show that an oracle decision problem separating BQP from PH would follow from the Generalized Linial-Nisan Conjecture, which we formulate here and which is likely of independent interest. The original Linial-Nisan Conjecture (about pseudorandomness against constant-depth circuits) was recently proved by Braverman, after being open for twenty years.

Paper Structure

This paper contains 22 sections, 23 theorems, 107 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Motivation
  3. Our Results
  4. In Defense of Oracles
  5. Preliminaries
  6. Problems
  7. Complexity Classes
  8. Quantum Algorithms
  9. Quantum Algorithm for Fourier Fishing
  10. Quantum Algorithm for Fourier Checking
  11. The Classical Complexity of Fourier Fishing
  12. Constant-Depth Circuit Lower Bounds
  13. Secretly Biased Fourier Coefficients
  14. Putting It All Together
  15. The Classical Complexity of Fourier Checking
  16. ...and 7 more sections

Key Result

Theorem 1

There exists an oracle $A$ relative to which $\mathsf{FBQP}^{A}\not \subset \mathsf{FBPP}^{\mathsf{PH}^{A}}$, where $\mathsf{FBQP}$ and $\mathsf{FBPP}$ are the relation versions of $\mathsf{BQP}$ and $\mathsf{BPP}$ respectively.Confusingly, the $\mathsf{F}$ stands for “ function”; we are simply foll

Figures (1)

  • Figure 1: The Fourier coefficients of a random Boolean function follow a Gaussian distribution, with mean $0$ and variance $1$. However, larger Fourier coefficients are more likely to be observed by the quantum algorithm.

Theorems & Definitions (30)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Lemma 7
  • Lemma 8
  • Theorem 9
  • Corollary 10
  • ...and 20 more