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arXiv:2010.11656 · New Journal of Physics

Modified Grover operator for quantum amplitude estimation

Shumpei Uno, Yohichi Suzuki, Keigo Hisanaga, Rudy Raymond, Tomoki Tanaka, Tamiya Onodera, Naoki Yamamoto

Published Aug 1, 2021• 19 citations
arXivHTMLPDF

Abstract

Abstract In this paper, we propose a quantum amplitude estimation method that uses a modified Grover operator and quadratically improves the estimation accuracy in the ideal case, as in the conventional one using the standard Grover operator. Under the depolarizing noise, the proposed method can outperform the conventional one in the sense that it can in principle achieve the ultimate estimation accuracy characterized by the quantum Fisher information in the limit of a large number of qubits, while the conventional one cannot achieve the same value of ultimate accuracy. In general this superiority requires a sophisticated adaptive measurement, but we numerically demonstrate that the proposed method can outperform the conventional one and approach to the ultimate accuracy, even with a simple non-adaptive measurement strategy.

DOI
10.1088/1367-2630/ac19da

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Abstract In this paper, we propose a quantum amplitude estimation method that uses a modified Grover operator and quadratically improves the estimation accuracy in the ideal case, as in the conventional one using the standard Grover operator. Under the depolarizing noise, the proposed method can outperform the conventional one in the sense that it can in principle achieve the ultimate estimation accuracy characterized by the quantum Fisher information in the limit of a large number of qubits, while the conventional one cannot achieve the same value of ultimate accuracy. In general this superiority requires a sophisticated adaptive measurement, but we numerically demonstrate that the proposed method can outperform the conventional one and approach to the ultimate accuracy, even with a simple non-adaptive measurement strategy.
Quantum-enhanced (i.e., higher performance by quantum effects than any classical methods) mean value estimation of observables is a fundamental task in various quantum technologies; in particular, it is an essential subroutine in quantum computing algorithms. Notably, the quantum estimation theory identifies the ultimate precision of such an estimator, which is referred to as the quantum Cramér-Rao (QCR) lower bound or equivalently the inverse of the quantum Fisher information. Because the estimation precision directly determines the performance of those quantum technological systems, it is highly demanded to develop a generic and practically implementable estimation method that achieves the QCR bound. Under imperfect conditions, however, such an ultimate and implementable estimator for quantum mean values has not been developed. In this paper, we propose a quantum-enhanced mean value estimation method in a depolarizing noisy environment that asymptotically achieves the QCR bound in the limit of a large number of qubits. To approach the QCR bound in a practical setting, the method adaptively optimizes the amplitude amplification and a specific measurement that can be implemented without any knowledge of state preparation. We provide a rigorous analysis for the statistical properties of the proposed adaptive estimator such as consistency and asymptotic normality. Furthermore, several numerical simulations are provided to demonstrate the effectiveness of the method, particularly showing that the estimator needs only a modest number of measurements to almost saturate the QCR bound.
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 Quantum amplitude estimation (QAE) is a pivotal quantum algorithm to estimate the squared amplitude a of the target basis state in a quantum state $|{\Phi}\rangle $ | Φ 〉 . Various improvements on the original quantum phase estimation-based QAE have been proposed for resource reduction. One of such improved versions is iterative quantum amplitude estimation (IQAE), which outputs an estimate â of a through the iterated rounds of the measurements on the quantum states like $G^{k}|{\Phi}\rangle $ G k | Φ 〉 , with the number k of operations of the Grover operator G (the Grover number) and the shot number determined adaptively. This paper investigates the bias in IQAE. Through the numerical experiments to simulate IQAE, we reveal that the estimate by IQAE is biased and the bias is enhanced for some specific values of a. We see that the termination criterion in IQAE that the estimated accuracy of â falls below the threshold is a source of the bias. Besides, we observe that kfink_{\mathrm{fin}}kfin​ k fin , the Grover number in the final round, and ffinf_{\mathrm{fin}}ffin​ f fin , a quantity affecting the probability distribution of measurement outcomes in the final round, are the key factors to determine the bias, and the bias enhancement for specific values of a is due to the skewed distribution of (kfin,ffin)(k_{\mathrm{fin}},f_{\mathrm{fin}})(kfin​,ffin​) ( k fin , f fin ) . We also present a bias mitigation method: just re-executing the final round with the Grover number and the shot number fixed.
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.
In quantum computation, amplitude estimation is a fundamental subroutine that is utilized in various quantum algorithms. A general important task of such estimation problems is to characterize the estimation lower bound, which is referred to as quantum Cram\'er-Rao bound (QCRB), and to construct an optimal estimator that achieves QCRB. This paper studies the amplitude estimation in the presence of depolarizing noise with unknown intensity. The main difficulty in this problem is that the optimal measurement depends on both the unknown quantum state and the amplitude we aim to estimate. To deal with these issues, we utilize the variational quantum circuits to approximate the (unknown) optimal measurement basis combined with the two-step adaptive estimation strategy which was proposed in the quantum estimation theory. We numerically show that the proposed method can nearly attain the QCRB.
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.