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arXiv:2206.08449

Adaptive Algorithm for Quantum Amplitude Estimation

Yunpeng Zhao, Haiyan Wang, Kuai Xu, Yue Wang, Ji Zhu, Feng Wang

Published Jun 16, 2022• 2 citations
arXivHTMLPDF

Abstract

Quantum amplitude estimation is a key sub-routine of a number of quantum algorithms with various applications. We propose an adaptive algorithm for interval estimation of amplitudes. The quantum part of the algorithm is based only on Grover's algorithm. The key ingredient is the introduction of an adjustment factor, which adjusts the amplitude of good states such that the amplitude after the adjustment, and the original amplitude, can be estimated without ambiguity in the subsequent step. We show with numerical studies that the proposed algorithm uses a similar number of quantum queries to achieve the same level of precision $ε$ compared to state-of-the-art algorithms, but the classical part, i.e., the non-quantum part, has substantially lower computational complexity. We rigorously prove that the number of oracle queries achieves $O(1/ε)$, i.e., a quadratic speedup over classical Monte Carlo sampling, and the computational complexity of the classical part achieves $O(\log(1/ε))$, both up to a double-logarithmic factor.

DOI
10.48550/arxiv.2206.08449

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Quantum amplitude estimation is a key sub-routine of a number of quantum algorithms with various applications. We propose an adaptive algorithm for interval estimation of amplitudes. The quantum part of the algorithm is based only on Grover's algorithm. The key ingredient is the introduction of an adjustment factor, which adjusts the amplitude of good states such that the amplitude after the adjustment, and the original amplitude, can be estimated without ambiguity in the subsequent step. We show with numerical studies that the proposed algorithm uses a similar number of quantum queries to achieve the same level of precision εεε compared to state-of-the-art algorithms, but the classical part, i.e., the non-quantum part, has substantially lower computational complexity. We rigorously prove that the number of oracle queries achieves O(1/ε)O(1/ε)O(1/ε), i.e., a quadratic speedup over classical Monte Carlo sampling, and the computational complexity of the classical part achieves O(log⁡(1/ε))O(\log(1/ε))O(log(1/ε)), both up to a double-logarithmic factor.
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.
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.
In Chs. 2 and 3 we described several quantum algorithms that are exponentially faster than classical algorithms for the same problem, under some measure of complexity. Unfortunately, the algorithms of Ch. 2 apply to rather contrived problems that do not have known direct applications. We now describe a quantum algorithm that gives only a polynomial—more specifically, quadratic—speedup with respect to classical algorithms, but it applies to a large class of problems and finds many useful applications, both directly and as an algorithmic design tool. The algorithm is known as Grover’s search [89], and its generalization is known as amplitude amplification [43]. Amplitude amplification is widely used as part of many quantum algorithms. It also serves as the basis for amplitude estimation, a way of estimating probabilities quadratically faster than with classical Monte Carlo, and for quantum minimum finding. All these topics are discussed in this chapter.
Abstract We introduce the Real Quantum Amplitude Estimation (RQAE) algorithm, an extension of Quantum Amplitude Estimation (QAE) which is sensitive to the sign of the amplitude. RQAE is an iterative algorithm which offers explicit control over the amplification policy through an adjustable parameter. We provide a rigorous analysis of the RQAE performance and prove that it achieves a quadratic speedup, modulo logarithmic corrections, with respect to unamplified sampling. Besides, we corroborate the theoretical analysis with a set of numerical experiments.
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.