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Exponential Quantum Speed-ups are Generic

Fernando G. S. L. Brandao, Michal Horodecki

TL;DR

The paper investigates which quantum circuits yield exponential speed-ups in query complexity by introducing U-Circuit Checking, a Fourier-Checking–style oracle problem parameterized by a dispersiveness measure C(U). It shows that almost any sufficiently long quantum circuit provides an exponential quantum-versus-postselected-classical query separation, provided C(U) is large; this is demonstrated for both quantum Fourier transforms over finite groups and almost all elements of certain approximate unitary 3-designs, and extends to random circuits via a rigorous 3-design proof. The authors develop a spectral-gap approach to bound convergence of random circuits to a 3-design, and they analyze both lower and upper bounds on classical query complexity, including independent-query and approximately-sparse-unitary regimes. Collectively, the results suggest that typical polynomial-size quantum circuits can deliver exponential speed-ups in the query model, with broad implications for identifying robust quantum advantages beyond QFT-based techniques.

Abstract

A central problem in quantum computation is to understand which quantum circuits are useful for exponential speed-ups over classical computation. We address this question in the setting of query complexity and show that for almost any sufficiently long quantum circuit one can construct a black-box problem which is solved by the circuit with a constant number of quantum queries, but which requires exponentially many classical queries, even if the classical machine has the ability to postselect. We prove the result in two steps. In the first, we show that almost any element of an approximate unitary 3-design is useful to solve a certain black-box problem efficiently. The problem is based on a recent oracle construction of Aaronson and gives an exponential separation between quantum and classical bounded-error with postselection query complexities. In the second step, which may be of independent interest, we prove that linear-sized random quantum circuits give an approximate unitary 3-design. The key ingredient in the proof is a technique from quantum many-body theory to lower bound the spectral gap of local quantum Hamiltonians.

Exponential Quantum Speed-ups are Generic

TL;DR

The paper investigates which quantum circuits yield exponential speed-ups in query complexity by introducing U-Circuit Checking, a Fourier-Checking–style oracle problem parameterized by a dispersiveness measure C(U). It shows that almost any sufficiently long quantum circuit provides an exponential quantum-versus-postselected-classical query separation, provided C(U) is large; this is demonstrated for both quantum Fourier transforms over finite groups and almost all elements of certain approximate unitary 3-designs, and extends to random circuits via a rigorous 3-design proof. The authors develop a spectral-gap approach to bound convergence of random circuits to a 3-design, and they analyze both lower and upper bounds on classical query complexity, including independent-query and approximately-sparse-unitary regimes. Collectively, the results suggest that typical polynomial-size quantum circuits can deliver exponential speed-ups in the query model, with broad implications for identifying robust quantum advantages beyond QFT-based techniques.

Abstract

A central problem in quantum computation is to understand which quantum circuits are useful for exponential speed-ups over classical computation. We address this question in the setting of query complexity and show that for almost any sufficiently long quantum circuit one can construct a black-box problem which is solved by the circuit with a constant number of quantum queries, but which requires exponentially many classical queries, even if the classical machine has the ability to postselect. We prove the result in two steps. In the first, we show that almost any element of an approximate unitary 3-design is useful to solve a certain black-box problem efficiently. The problem is based on a recent oracle construction of Aaronson and gives an exponential separation between quantum and classical bounded-error with postselection query complexities. In the second step, which may be of independent interest, we prove that linear-sized random quantum circuits give an approximate unitary 3-design. The key ingredient in the proof is a technique from quantum many-body theory to lower bound the spectral gap of local quantum Hamiltonians.

Paper Structure

This paper contains 12 sections, 17 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. Our results
  3. Related Work
  4. Preliminaries
  5. The oracle problem and its quantum solution
  6. Dispersing Circuits
  7. Approximate Unitary Designs
  8. Families of Flat Unitaries
  9. Classical lower bounds
  10. Classical upper bounds
  11. Random Circuits are approximate $3$-designs
  12. Acknowledgments

Key Result

Theorem I

For any circuit $U$ acting on $n$ qubits with $C(U) = \Omega(n)$, the problem ${\sf{U}}$-$\textsc{Circuit Checking}$ shows an exponential separation of quantum and postselected classical query complexities.

Theorems & Definitions (22)

  • Theorem I
  • Theorem II
  • Theorem III
  • Theorem IV
  • Theorem V
  • Theorem VI
  • Proposition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • ...and 12 more