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arXiv:2110.04258 · Physical Review A

Noisy quantum amplitude estimation without noise estimation

Tomoki Tanaka, Shumpei Uno, Tamiya Onodera, Naoki Yamamoto, Yohichi Suzuki

Published Jan 6, 2022• 11 citations
arXivHTMLPDF

Abstract

Many quantum algorithms contain an important subroutine---the quantum amplitude estimation. As the name implies, this is essentially the parameter estimation problem and thus can be handled via the established statistical estimation theory. However, this problem has an intrinsic difficulty that the system, i.e., the real quantum computing device, inevitably introduces unknown noise; the probability distribution model then has to incorporate many nuisance noise parameters, resulting that the construction of an optimal estimator becomes inefficient and difficult. For this problem we apply the theory of nuisance parameters (more specifically, the parameter orthogonalization method) to precisely compute the maximum likelihood estimator for only the target amplitude parameter by removing the other nuisance noise parameters. That is, we can estimate the amplitude parameter without estimating the noise parameters. We validate the parameter orthogonalization method in a numerical simulation and study the performance of the estimator in the experiment using a real superconducting quantum device.

DOI
10.1103/physreva.105.012411

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Many quantum algorithms contain an important subroutine---the quantum amplitude estimation. As the name implies, this is essentially the parameter estimation problem and thus can be handled via the established statistical estimation theory. However, this problem has an intrinsic difficulty that the system, i.e., the real quantum computing device, inevitably introduces unknown noise; the probability distribution model then has to incorporate many nuisance noise parameters, resulting that the construction of an optimal estimator becomes inefficient and difficult. For this problem we apply the theory of nuisance parameters (more specifically, the parameter orthogonalization method) to precisely compute the maximum likelihood estimator for only the target amplitude parameter by removing the other nuisance noise parameters. That is, we can estimate the amplitude parameter without estimating the noise parameters. We validate the parameter orthogonalization method in a numerical simulation and study the performance of the estimator in the experiment using a real superconducting quantum device.
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.
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.
Amplitude Estimation (AE) is a critical subroutine in many quantum algorithms, allowing for a quadratic speedup in various applications like those involving estimating statistics of various functions as in financial Monte Carlo simulations. Much work has gone into devising methods to efficiently estimate the amplitude of a quantum state without expensive operations like the Quantum Fourier Transform (QFT), which is especially prohibitive given the constraints of current NISQ devices. Newer methods have reduced the number of operations required on a quantum computer and are the most promising near-term implementations of the AE subroutine. While it remains to be seen the exact circuit requirements for a quantum advantage in applications relying on AE, it is necessary to continue to benchmark the algorithm's performance on current quantum computers and the circuit costs associated with such subroutines. Given these considerations, we expand on results from previous experiments in using Maximum Likelihood Estimation (MLE) to approximate the amplitude of a quantum state and provide empirical upper bounds on the current feasible circuit depths for AE on a superconducting quantum computer. Our results show that MLE using optimally-compiled circuits can currently outperform naive sampling for up to 3 Grover Iterations with a circuit depth of 131, which is higher than reported in other experimental results. This functional benchmark is one of many that will be continually monitored against current quantum hardware to measure the necessary progress towards quantum advantage.
Quantum amplitude estimation is a key subroutine in a number of powerful quantum algorithms, including quantum-enhanced Monte Carlo simulation and quantum machine learning. Maximum-likelihood quantum amplitude estimation (MLQAE) is one of a number of recent approaches that employ much simpler quantum circuits than the original algorithm based on quantum phase estimation. In this article, we deepen the analysis of MLQAE to put the algorithm in a more prescriptive form, including scenarios where quantum circuit depth is limited. In the process, we observe and explain particular ranges of `exceptional' values of the target amplitude for which the algorithm fails to achieve the desired precision. We then propose and numerically validate a heuristic modification to the algorithm to overcome this problem, bringing the algorithm even closer to being useful as a practical subroutine on near- and mid-term quantum hardware.
Abstract In parameter estimation, nuisance parameters refer to parameters that are not of interest but nevertheless affect the precision of estimating other parameters of interest. For instance, the strength of noises in a probe can be regarded as a nuisance parameter. Despite its long history in classical statistics, the nuisance parameter problem in quantum estimation remains largely unexplored. The goal of this article is to provide a systematic review of quantum estimation in the presence of nuisance parameters, and to supply those who work in quantum tomography and quantum metrology with tools to tackle relevant problems. After an introduction to the nuisance parameter and quantum estimation theory, we explicitly formulate the problem of quantum state estimation with nuisance parameters. We extend quantum Cramér–Rao bounds to the nuisance parameter case and provide a parameter orthogonalization tool to separate the nuisance parameters from the parameters of interest. In particular, we put more focus on the case of one-parameter estimation in the presence of nuisance parameters, as it is most frequently encountered in practice.
The recent quantum age has allowed quantum algorithms to perform in the real world. However, they have significant performance errors in noisy environments. To investigate this issue, we conduct an analysis and simulation to evaluate maximum likelihood quantum amplitude estimation (MLQAE) circuit depths under various conditions: (1) ideal, (2) theoretical noise model, and (3) real noise model. These show variant performance errors, especially in a high-qubit circuit with high error rates, even when using a low-noise model.