Bounds for Small-Error and Zero-Error Quantum Algorithms

TL;DR

This paper analyzes quantum algorithms with small or zero error, establishing bounds, separations from classical algorithms, and implications for quantum complexity and evasiveness conjectures.

Contribution

It provides tight bounds on quantum search trade-offs, nearly optimal quantum-classical separations for zero-error functions, and challenges to classical conjectures in quantum settings.

Findings

Tight bounds on quantum search query trade-offs.

Nearly optimal quantum-classical separations for zero-error monotone functions.

Quantum separations in graph property evasiveness challenge classical conjectures.

Abstract

We present a number of results related to quantum algorithms with small error probability and quantum algorithms that are zero-error. First, we give a tight analysis of the trade-offs between the number of queries of quantum search algorithms, their error probability, the size of the search space, and the number of solutions in this space. Using this, we deduce new lower and upper bounds for quantum versions of amplification problems. Next, we establish nearly optimal quantum-classical separations for the query complexity of monotone functions in the zero-error model (where our quantum zero-error model is defined so as to be robust when the quantum gates are noisy). Also, we present a communication complexity problem related to a total function for which there is a quantum-classical communication complexity gap in the zero-error model. Finally, we prove separations for monotone graph properties in the zero-error and other error models which imply that the evasiveness conjecture for such properties does not hold for quantum computers.