# Optimal one-shot quantum algorithm for EQUALITY and AND

## TL;DR

This paper presents an optimal one-shot quantum algorithm for computing the Boolean functions EQUALITY and AND, achieving minimal error probabilities in the quantum black box model.

## Contribution

It introduces a quantum algorithm that minimizes the error probability for EQUALITY and AND functions with a single query, improving understanding of quantum query complexity.

## Key findings

- Lowest error probability for AND_n is 1/2 - n/(n^2+1)
- Lowest error probability for EQUALITY_{n+1} is 1/2 - n/(n^2+1)
- Optimal one-shot quantum algorithms can achieve these error bounds

## Abstract

We study the computation complexity of Boolean functions in the quantum black box model. In this model our task is to compute a function $f:\{0,1\}\to\{0,1\}$ on an input $x\in\{0,1\}^n$ that can be accessed by querying the black box. Quantum algorithms are inherently probabilistic; we are interested in the lowest possible probability that the algorithm outputs incorrect answer (the error probability) for a fixed number of queries. We show that the lowest possible error probability for $AND_n$ and $EQUALITY_{n+1}$ is $1/2-n/(n^2+1)$.