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The Prospect of Quantum Amplitude Estimation Based on the Development of the Quantum Volume

Carmen Rosa Hoek, Jins de Jong

Published Oct 31, 2023• 0 citations
Publisher

Abstract

Quantum amplitude estimation is a numerical integration technique and a candidate for the first operationally used quantum algorithm providing a quantum advantage in the NISQ era. Its main metric is the generated Fisher information per second, as it describes the accuracy of the numerical integration. The development of this metric can be coupled to the quantum volume of the used quantum processor with a phenomenological noise model. Since the increase of the quantum volume seems to follow an exponential pattern, insight in the integration power of quantum amplitude estimation for the upcoming years can be obtained. This shows that quantum amplitude estimation as a numerical integration technique will not provide an advantage for functions without significant improvements to the connectivity, reliability and speed of quantum processors. Beyond that, this method may be used the estimate the development of many more NISQ era algorithms.

DOI
10.20944/preprints202310.2077.v1

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Quantum amplitude estimation is a numerical integration technique and a candidate for the first operationally used quantum algorithm providing a quantum advantage in the NISQ era. Its main metric is the generated Fisher information per second, as it describes the accuracy of the numerical integration. The development of this metric can be coupled to the quantum volume of the used quantum processor with a phenomenological noise model. Since the increase of the quantum volume seems to follow an exponential pattern, insight in the integration power of quantum amplitude estimation for the upcoming years can be obtained. This shows that quantum amplitude estimation as a numerical integration technique will not provide an advantage for functions without significant improvements to the connectivity, reliability and speed of quantum processors. Beyond that, this method may be used the estimate the development of many more NISQ era algorithms.
The Quantum Amplitude Estimation (QAE) algorithm is a major quantum algorithm designed to achieve a quadratic speed-up. Until fault-tolerant quantum computing is achieved, being competitive over classical Monte Carlo (MC) remains elusive. Alternative methods have been developed so as to require fewer resources while maintaining an advantageous theoretical scaling. We compared the standard QAE algorithm with two Noisy Intermediate-Scale Quantum (NISQ)-friendly versions of QAE on a numerical integration task, with the Monte Carlo technique of Metropolis–Hastings as a classical benchmark. The algorithms were evaluated in terms of the estimation error as a function of the number of samples, computational time, and length of the quantum circuits required by the solutions, respectively. The effectiveness of the two QAE alternatives was tested on an 11-qubit trapped-ion quantum computer in order to verify which solution can first provide a speed-up in the integral estimation problems. We concluded that an alternative approach is preferable with respect to employing the phase estimation routine. Indeed, the Maximum Likelihood estimation guaranteed the best trade-off between the length of the quantum circuits and the precision in the integral estimation, as well as greater resistance to noise.
The approximation of an integral with numerical integration methods is a relevant problem in many fields, from finance to chemistry. Monte Carlo methods are a family of methods widely used for numerical integration but which are not always applicable to industrial tasks. Instead, there are quantum computing methods such as Quantum Amplitude Estimation able to offer a speed-up compared to the Monte Carlo methods for numerical integration which promise a significant impact, especially in finance. The Quantum Amplitude Estimation algorithm is a fundamental quantum algorithm with the potential to achieve a quadratic speed-up. Unfortunately, being competitive over classical Monte Carlo methods would require hardware performances currently not available. Therefore, alternative methods have been developed in order to exploit fewer resources while maintaining an advantageous theoretical scaling. We present a comparison between the standard QAE algorithm and two NISQ-friendly versions of the QAE on a numerical integration task. The Monte Carlo technique of Metropolis-Hastings instead is used as a classical benchmark. The scaling of the different algorithms are evaluated in terms of estimation error as a function of the number of samples, computational time, and length of the quantum circuits required by the quantum solutions. The conclusion highlights that an alternative approach is preferable with respect to employng the phase estimation routine. Indeed, the maximum likelihood estimation guarantees the best trade-off between the length of the quantum circuits and the precision in the integral estimation.
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Quantum advantage requires overcoming noise-induced degradation of quantum systems. Conventional methods for reducing noise such as error mitigation face scalability issues in deep circuits. Specifically, noise hampers the extraction of amplitude and observable information from quantum systems. In this work, we present a novel algorithm that enhances the estimation of amplitude and observable under noise. Remarkably, our algorithm exhibits robustness against noise that varies across different depths of the quantum circuits. We assess the accuracy of amplitude and observable using numerical analysis and theoretically analyze the impact of gate-dependent noise on the results. This algorithm is a potential candidate for noise-resilient approaches that have high computational accuracy.
Quantum Amplitude Estimation (QAE) -- a technique by which the amplitude of a given quantum state can be estimated with quadratically fewer queries than by standard sampling -- is a key sub-routine in several important quantum algorithms, including Grover search and Quantum Monte-Carlo methods. An obstacle to implementing QAE in near-term noisy intermediate-scale quantum (NISQ) devices has been the need to perform Quantum Phase Estimation (QPE) -- a costly procedure -- as a sub-routine. This impediment was lifted with various QPE-free methods of QAE, wherein Grover queries of varying depths / powers (often according to a "schedule") are followed immediately by measurements and classical post-processing techniques like maximum likelihood estimation (MLE). Existing analyses as to the optimality of various query schedules in these QPE-free QAE schemes have hitherto assumed noise-free systems. In this work, we analyse QPE-free QAE under common noise models that may afflict NISQ devices and report on the optimality of various query schedules in the noisy regime. We demonstrate that, given an accurate noise characterization of one's system, one must choose a schedule that balances the trade-off between the greater ideal performance achieved by higher-depth circuits, and the correspondingly greater accumulation of noise-induced error.
Quantum Amplitude Estimation (QAE) is a fundamental component of many quantum computing algorithms and applications. The practical implementation of QAE workflows on current Noisy Intermediate-Scale Quantum (NISQ) platforms requires accurate probability amplitude representation and reliable measurement recovery. Understanding the behavior of individual computational stages is therefore essential for improving the overall performance of practical quantum estimation procedures. This study presents a component-wise validation of the amplitude encoding process within a controlled quantum simulation environment. The experimental implementation was developed using the Qiskit framework, where classical probability values were encoded into single-qubit quantum states through parameterized rotation operations and subsequently recovered from measurement statistics. A systematic evaluation was performed across the probability interval from 0.1 to 0.9 using 8192 measurement shots for each experiment. The experimental results demonstrate that the recovered measurement probabilities closely match the corresponding target values throughout the investigated domain. Statistical evaluation produced a Mean Absolute Error (MAE) of 0.003521, a Mean Squared Error (MSE) of 1.67 × 10⁻⁵, and a Root Mean Squared Error (RMSE) of 0.004087, indicating high numerical consistency under ideal simulation conditions. The findings suggest that the amplitude encoding stage accurately represents classical probability values and provides a reliable baseline for studying more advanced quantum estimation workflows. The component-wise validation methodology adopted in this work offers a practical framework for investigating individual stages of quantum algorithms and supports future research on hybrid quantum-classical processing strategies for NISQ-era quantum computing.