# Error reduction of quantum algorithms

## TL;DR

This paper introduces a novel quantum error reduction technique called Amplitude Separation, which improves the efficiency of quantum algorithms in determining properties of inputs, especially when error probabilities are high.

## Contribution

The paper proposes the Amplitude Separation method that combines amplitude amplification and estimation to reduce quantum algorithm error with fewer executions than existing methods.

## Key findings

- Achieves $O(1/(\sqrt{1-
ho} - \sqrt{
ho}))$ executions, better than previous approaches for high errors.
- Provides a quadratic speedup in the Multiple-Weight Decision Problem over classical and existing quantum algorithms.
- Demonstrates improved error reduction efficiency in quantum algorithms for property testing.

## Abstract

We give a technique to reduce the error probability of quantum algorithms that determine whether its input has a specified property of interest. The standard process of reducing this error is statistical processing of the results of multiple independent executions of an algorithm. Denoting by $\rho$ an upper bound of this probability (wlog., assume $\rho \le \frac{1}{2}$), classical techniques require $O(\frac{\rho}{[(1-\rho) - \rho]^2})$ executions to reduce the error to a negligible constant. We investigated when and how quantum algorithmic techniques like amplitude amplification and estimation may reduce the number of executions. On one hand, the former idea does not directly benefit algorithms that can err on both yes and no answers and the number of executions in the latter approach is $O(\frac{1}{(1-\rho) - \rho})$. We propose a novel approach named as {\em Amplitude Separation} that combines both these approaches and achieves $O(\frac{1}{\sqrt{1-\rho} - \sqrt{\rho}})$ executions that betters existing approaches when the errors are high. In the Multiple-Weight Decision Problem, the input is an $n$-bit Boolean function $f()$ given as a black-box and the objective is to determine the number of $x$ for which $f(x)=1$, denoted as $wt(f)$, given some possible values $\{w_1, \ldots, w_k\}$ for $wt(f)$. When our technique is applied to this problem, we obtain the correct answer, maybe with a negligible error, using $O(\log_2 k \sqrt{2^n})$ calls to $f()$ that shows a quadratic speedup over classical approaches and currently known quantum algorithms.