Detailed Account of Complexity for Implementation of Some Gate-Based Quantum Algorithms

TL;DR

This paper provides a detailed analysis of the time and space complexity involved in implementing gate-based quantum algorithms, including state preparation and measurement, for solving linear systems and differential equations.

Contribution

It offers a comprehensive complexity assessment of full quantum algorithm implementations, highlighting steps often overlooked such as state preparation and readout.

Findings

Complexity of state preparation can dominate overall cost.

Measurement requirements significantly impact total resource estimates.

Implementation complexity varies with problem type and hardware constraints.

Abstract

In this work, we are interested in the detailed analysis of complexity aspects of both time and space that arises from the implementation of a quantum algorithm on a quantum based hardware. In particular, some steps of the implementation, as state preparation and readout processes, in most of the cases can surpass the complexity aspects of the algorithm itself. We present the complexity involved in the full implementation of quantum algorithms for solving linear systems of equations and linear system of differential equations, from state preparation to the number of measurements needed to obtain good statistics from the final states of the quantum system, in order to assess the overall complexity of the processes.