Iterative quantum amplitude estimation

Dmitry Grinko,Christa Zoufal,Stefan Woerner,Julien Gacon

doi:10.1038/s41534-021-00379-1

Dmitry Grinko, Christa Zoufal + Show 2 more

Open Access

https://doi.org/10.1038/s41534-021-00379-1

Copy DOI

Abstract

We introduce a variant of Quantum Amplitude Estimation (QAE), called Iterative QAE (IQAE), which does not rely on Quantum Phase Estimation (QPE) but is only based on Grover’s Algorithm, which reduces the required number of qubits and gates. We provide a rigorous analysis of IQAE and prove that it achieves a quadratic speedup up to a double-logarithmic factor compared to classical Monte Carlo simulation with provably small constant overhead. Furthermore, we show with an empirical study that our algorithm outperforms other known QAE variants without QPE, some even by orders of magnitude, i.e., our algorithm requires significantly fewer samples to achieve the same estimation accuracy and confidence level.

Highlights

Quantum Amplitude Estimation (QAE)[1] is a fundamental quantum algorithm with the potential to achieve a quadratic speedup for many applications that are classically solved through Monte Carlo (MC) simulation
While thepeffisffiffitffi imation error bound of classical MC simulation scales as Oð1= MÞ, where M denotes the number of samples, QAE achieves a scaling of Oð1=MÞ for M samples, indicating the aforementioned quadratic speedup
We introduced Iterative Quantum Amplitude Estimation, a new variant of QAE that realizes a quadratic speedup over classical MC simulation

Summary

INTRODUCTION

Quantum Amplitude Estimation (QAE)[1] is a fundamental quantum algorithm with the potential to achieve a quadratic speedup for many applications that are classically solved through Monte Carlo (MC) simulation. The canonical QAE follows the form of QPE: it uses m ancilla qubits—initialized in equal superposition—to represent the final result, it defines the number of quantum samples as M = 2m and applies geometrically increasing powers of Q controlled by the ancillas In the following, we will use the term QAE for the canonical QAE with the application of MLE to the y measurements to derive an improved estimate and confidence intervals based on the likelihood ratio. It utilizes a quantity Lmax—the maximum possible error, which could be returned on a given iteration using Nshots measurements It is calculated before the start of the algorithm for chosen ε, α and number of shots Nshots. It is derived by analogy with Supplementary Eq (39), where instead of Nmax(ε, α) one should use Nshots

Grinko et al 4

DISCUSSION

CODE AVAILABILITY