A general statistical approach to quantum algorithms in a circuit model based on the expectation and standard deviation of each gate separately

TL;DR

This paper introduces a unified statistical framework for quantum algorithms using expectation and standard deviation of gates, enabling a fundamental basis transformation based on a new identity.

Contribution

It proposes a novel statistical approach to describe quantum algorithms, extending the Aharonov-Vaidman identity for basis switching.

Findings

Provides a general method to analyze quantum algorithms statistically.

Enables basis transformation in quantum states using a new identity.

Offers insights into the structure of quantum algorithms through statistical measures.

Abstract

Recently, there has been a growing literature exploring the generalization of quantum algorithms, such that different quantum algorithms are special examples of a more fundamental structure. In this short paper, we provide a general approach to describe quantum algorithms as a quantum state with amplitudes that are constructed from the expected value and standard deviation of each quantum gate or a sub-sequence of gates in the algorithm. The proposed statistical-based description relies on the celebrated Aharonov-Vaidman identity. We present a more fundamental identity that, unlike the previous one, allows us to switch the basis of the states into a desired form.