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arXiv:2208.14612 · Society for Industrial and Applied Mathematics eBooks

Modified Iterative Quantum Amplitude Estimation is Asymptotically Optimal

Shion Fukuzawa, Christopher Ho, Sandy Irani, Jasen Zion

Published Jan 1, 2023• 6 citations
arXivHTMLPDF

Abstract

In this work, we provide the first QFT-free algorithm for Quantum Amplitude Estimation (QAE) that is asymptotically optimal while maintaining the leading numerical performance. QAE algorithms appear as a subroutine in many applications for quantum computers. The optimal query complexity achievable by a quantum algorithm for QAE is log queries, providing a speedup of a factor of 1/ε over any other classical algorithm for the same problem. The original algorithm for QAE utilizes the quantum Fourier transform (QFT) which is expected to be a challenge for near-term quantum hardware. To solve this problem, there has been interest in designing a QAE algorithm that avoids using QFT. Recently, the iterative QAE algorithm (IQAE) [1] was introduced with a near-optimal query complexity and small constant factors. In this work, we combine ideas from the preceding line of work to introduce a QFT-free QAE algorithm that maintains the asymptotically optimal query complexity while retaining small constant factors. We supplement our analysis with numerical experiments comparing our performance with IQAE where we find that our modifications retain the high performance, and in some cases even improve the numerical results.

DOI
10.1137/1.9781611977561.ch12

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In this work, we provide the first QFT-free algorithm for Quantum Amplitude Estimation (QAE) that is asymptotically optimal while maintaining the leading numerical performance. QAE algorithms appear as a subroutine in many applications for quantum computers. The optimal query complexity achievable by a quantum algorithm for QAE is log queries, providing a speedup of a factor of 1/ε over any other classical algorithm for the same problem. The original algorithm for QAE utilizes the quantum Fourier transform (QFT) which is expected to be a challenge for near-term quantum hardware. To solve this problem, there has been interest in designing a QAE algorithm that avoids using QFT. Recently, the iterative QAE algorithm (IQAE) [1] was introduced with a near-optimal query complexity and small constant factors. In this work, we combine ideas from the preceding line of work to introduce a QFT-free QAE algorithm that maintains the asymptotically optimal query complexity while retaining small constant factors. We supplement our analysis with numerical experiments comparing our performance with IQAE where we find that our modifications retain the high performance, and in some cases even improve the numerical results.
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.
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.
Abstract This paper focuses on the quantum amplitude estimation algorithm, which is a core subroutine in quantum computation for various applications. The conventional approach for amplitude estimation is to use the phase estimation algorithm, which consists of many controlled amplification operations followed by a quantum Fourier transform. However, the whole procedure is hard to implement with current and near-term quantum computers. In this paper, we propose a quantum amplitude estimation algorithm without the use of expensive controlled operations; the key idea is to utilize the maximum likelihood estimation based on the combined measurement data produced from quantum circuits with different numbers of amplitude amplification operations. Numerical simulations we conducted demonstrate that our algorithm asymptotically achieves nearly the optimal quantum speedup with a reasonable circuit length.
We demonstrate that the problem of amplitude estimation, a core subroutine used in many quantum algorithms, can be mapped directly to a problem in signal processing called direction of arrival (DOA) estimation. The DOA task is to determine the direction of arrival of an incoming wave with the fewest possible measurements. The connection between amplitude estimation and DOA allows us to make use of the vast amount of signal processing algorithms to post-process the measurements of the Grover iterator at predefined depths. Using an off-the-shelf DOA algorithm called ESPRIT together with a compressed-sensing based sampling approach, we create a phase-estimation free, parallel quantum amplitude estimation (QAE) algorithm with a worst-case sequential query complexity of ∼4.3/ε\sim 4.3/\varepsilon∼4.3/ε and a parallel query complexity of ∼0.26/ε\sim 0.26/\varepsilon∼0.26/ε at 95% confidence. This performance is statistically equivalent and a 16×16\times16× improvement over Rall and Fuller [Quantum 7, 937 (2023)], for sequential and parallel query complexity respectively, which to our knowledge is the best published result for amplitude estimation. The approach presented here provides a simple, robust, parallel method to performing QAE, with many possible avenues for improvement borrowing ideas from the wealth of literature in classical signal processing.
Monte Carlo integration is a widely used numerical method for approximating integrals, which is often computationally expensive. In recent years, quantum computing has shown promise for speeding up Monte Carlo integration, and several quantum algorithms have been proposed to achieve this goal. In this paper, we present an application of Quantum Machine Learning (QML) and Grover's amplification to build a new quantum algorithm for estimating Monte Carlo integrals. Our method, which we call Quantum Fourier Iterative Amplitude Estimation (QFIAE), decomposes the target function into its Fourier series using a Parametrized Quantum Circuit (PQC), specifically a Quantum Neural Network (QNN), and then inte- grates each trigonometric component using Iterative Quantum Amplitude Estimation (IQAE). This approach builds on Fourier Quantum Monte Carlo Integration (FQMCI) method, which also decomposes the target function into its Fourier series, but QFIAE avoids the need for numerical integration of Fourier coefficients. This approach reduces the computational load while maintaining the quadratic speedup achieved by IQAE. To evaluate the performance of QFIAE, we apply it to a test function corresponding to a particle physics scattering process and compare its accuracy with other quantum integration methods and the analytic result. Our results show that QFIAE achieves comparable accuracy while being suitable for execution on real hardware. We also demonstrate how the accuracy of QFIAE improves by increasing the number of terms in the Fourier series.
In this paper, we introduce an efficient algorithm for the quantum amplitude estimation task which is tailored for near-term quantum computers. The quantum amplitude estimation is an important problem which has various applications in fields such as quantum chemistry, machine learning, and finance. Because the well-known algorithm for the quantum amplitude estimation using the phase estimation does not work in near-term quantum computers, alternative approaches have been proposed in recent literature. Some of them provide a proof of the upper bound which almost achieves the Heisenberg scaling. However, the constant factor is large and thus the bound is loose. Our contribution in this paper is to provide the algorithm such that the upper bound of query complexity almost achieves the Heisenberg scaling and the constant factor is small.