Quantum Lower Bounds by Polynomials

TL;DR

This paper establishes bounds on quantum query complexity for total Boolean functions, showing limitations on quantum speed-up and providing tight characterizations for symmetric functions and specific cases like AND, OR, and PARITY.

Contribution

It extends the polynomial method to quantum computing, providing new bounds and characterizations for quantum query complexity of Boolean functions.

Findings

Quantum speed-up for total functions is limited, unlike partial functions.

Tight bounds are provided for symmetric functions in various error settings.

New bounds are established for AND, OR, and PARITY functions.

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.