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Quantum Lower Bounds by Polynomials

Robert Beals, Harry Buhrman, Richard Cleve, Michele Mosca, Ronald de Wolf

TL;DR

This work establishes that in the quantum black-box model, exponential speedups for partial problems do not extend to total functions: any total function computable by a T-query quantum algorithm with bounded error can be computed exactly by a classical algorithm with only O(T^6) queries (improved to O(T^4) for monotone and O(T^2) for symmetric functions). The authors develop a quantum polynomial method, showing that T queries induce degree-2T polynomials constraining acceptance probabilities, and use symmetrization and degree results (Paturi, Minsky–Papert) to derive tight bounds for symmetric functions and specific functions like OR, AND, PARITY, and MAJORITY. They further relate quantum and classical decision-tree complexities, obtaining D(f) ∈ O(Q2(f)^6) and improved exact bounds D(f) ≤ 32 ~deg(f)^4, along with a detailed analysis via block sensitivity. The results give precise, sometimes optimal, quantum query bounds and illuminate the limits of quantum speedups in the oracle model, with broad implications for understanding quantum advantage and the polynomial method in complexity theory.

Abstract

We examine the number T of queries that a quantum network requires to compute several Boolean functions on {0,1}^N in the black-box model. We show that, in the black-box model, the exponential quantum speed-up obtained for partial functions (i.e. problems involving a promise on the input) by Deutsch and Jozsa and by Simon cannot be obtained for any total function: if a quantum algorithm computes some total Boolean function f with bounded-error using T black-box queries then there is a classical deterministic algorithm that computes f exactly with O(T^6) queries. We also give asymptotically tight characterizations of T for all symmetric f in the exact, zero-error, and bounded-error settings. Finally, we give new precise bounds for AND, OR, and PARITY. Our results are a quantum extension of the so-called polynomial method, which has been successfully applied in classical complexity theory, and also a quantum extension of results by Nisan about a polynomial relationship between randomized and deterministic decision tree complexity.

Quantum Lower Bounds by Polynomials

TL;DR

This work establishes that in the quantum black-box model, exponential speedups for partial problems do not extend to total functions: any total function computable by a T-query quantum algorithm with bounded error can be computed exactly by a classical algorithm with only O(T^6) queries (improved to O(T^4) for monotone and O(T^2) for symmetric functions). The authors develop a quantum polynomial method, showing that T queries induce degree-2T polynomials constraining acceptance probabilities, and use symmetrization and degree results (Paturi, Minsky–Papert) to derive tight bounds for symmetric functions and specific functions like OR, AND, PARITY, and MAJORITY. They further relate quantum and classical decision-tree complexities, obtaining D(f) ∈ O(Q2(f)^6) and improved exact bounds D(f) ≤ 32 ~deg(f)^4, along with a detailed analysis via block sensitivity. The results give precise, sometimes optimal, quantum query bounds and illuminate the limits of quantum speedups in the oracle model, with broad implications for understanding quantum advantage and the polynomial method in complexity theory.

Abstract

We examine the number T of queries that a quantum network requires to compute several Boolean functions on {0,1}^N in the black-box model. We show that, in the black-box model, the exponential quantum speed-up obtained for partial functions (i.e. problems involving a promise on the input) by Deutsch and Jozsa and by Simon cannot be obtained for any total function: if a quantum algorithm computes some total Boolean function f with bounded-error using T black-box queries then there is a classical deterministic algorithm that computes f exactly with O(T^6) queries. We also give asymptotically tight characterizations of T for all symmetric f in the exact, zero-error, and bounded-error settings. Finally, we give new precise bounds for AND, OR, and PARITY. Our results are a quantum extension of the so-called polynomial method, which has been successfully applied in classical complexity theory, and also a quantum extension of results by Nisan about a polynomial relationship between randomized and deterministic decision tree complexity.

Paper Structure

This paper contains 11 sections, 23 theorems, 9 equations.

Table of Contents

  1. Introduction
  2. Summary of results
  3. Preliminaries
  4. Boolean functions and polynomials
  5. The framework of quantum networks
  6. General lower bounds on the number of queries
  7. Bounds for error-free computation
  8. Bounds for computation with bounded-error
  9. Lower bounds in terms of block sensitivity
  10. Polynomial relation between classical and quantum complexity
  11. Some particular functions

Key Result

Theorem 3.1

If $f$ is a Boolean function that depends on $N$ variables, then $deg(f)\geq \log N-O(\log\log N)$.

Theorems & Definitions (25)

  • Theorem 3.1: Nisan, Szegedy
  • Lemma 3.2: Minsky, Papert
  • Theorem 3.3: Paturi
  • Lemma 4.1
  • Lemma 4.2
  • Theorem 4.3
  • Corollary 4.4
  • Theorem 4.5
  • Theorem 4.6: Von zur Gathen, Roche
  • Corollary 4.7
  • ...and 15 more