Abstract
Recently we find several candidates of quantum algorithms that may be implementable in near-term devices for estimating the amplitude of a given quantum state, which is a core subroutine in various computing tasks such as the Monte Carlo methods. One of those algorithms is based on the maximum likelihood estimate with parallelized quantum circuits. In this paper, we extend this method so that it incorporates the realistic noise effect, and then give an experimental demonstration on a superconducting IBM Quantum device. The maximum likelihood estimator is constructed based on the model assuming the depolarization noise. We then formulate the problem as a two-parameters estimation problem with respect to the target amplitude parameter and the noise parameter. In particular we show that there exist anomalous target values, where the Fisher information matrix becomes degenerate and consequently the estimation error cannot be improved even by increasing the number of amplitude amplifications. The experimental demonstration shows that the proposed maximum likelihood estimator achieves quantum speedup in the number of queries, though the estimation error saturates due to the noise. This saturated value of estimation error is consistent to the theory, which implies the validity of the depolarization noise model and thereby enables us to predict the basic requirement on the hardware components (particularly the gate error) in quantum computers to realize the quantum speedup in the amplitude estimation task.
Highlights
Ideal and fault-tolerant quantum computers will provide us with game-changing platforms in various area such as security [1], chemical [2,3], and financial engineering [4,5,6,7,8,9,10]
This paper focuses on the quantum amplitude estimation algorithm [19,20], which can be typically applied to speedup the classical Monte Carlo methods [4,5], etc; more precisely, in an ideal setup the quantum amplitude estimation algorithm can quadratically reduce the number of samples and thereby the computation time for Monte Carlo methods
This paper extends the maximum likelihood (ML) method [21] so that it can be used for actual noisy quantum computers, and gives an experimental demonstration on a superconducting quantum device
Summary
Ideal and fault-tolerant quantum computers will provide us with game-changing platforms in various area such as security [1], chemical [2,3], and financial engineering [4,5,6,7,8,9,10]. Thanks to the property of depolarizing noise, we have an explicit form of the Fisher information matrix for discussing the accuracy of estimation and thereby derive a formula for specifying the noise level so that near-quadratic speedup is achieved to reach a given estimation accuracy. Section 2: We take the depolarizing noise model and formulate the two-parameters estimation problem With this noise model we can have the analytic expression of the Fisher information matrix, which gives an asymptotically achievable lower bound of the estimation error. This result is further used to derive a condition of the noise level required to have nearly quadratic speedup to reach a given estimation error. The computational complexity of our proposed algorithm is discussed
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