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arXiv:2008.09931

Mean-squared-error-based adaptive estimation of pure quantum states and unitary transformations

Rojas, A., Luciano Pereira, Sebastián Niklitschek, A. Delgado

Published Aug 23, 2020• 0 citations
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Abstract

In this article we propose a method to estimate with high accuracy pure quantum states of a single qudit. Our method is based on the minimization of the squared error between the complex probability amplitudes of the unknown state and its estimate. We show by means of numerical experiments that the estimation accuracy of the present method, which is given by the expectation of the squared error on the sample space of estimates, is state independent. Furthermore, the estimation accuracy delivered by our method is close to twice the Gill-Massar lower bound, which represents the best achievable accuracy, for all inspected dimensions. The minimization problem is solved via the concatenation of the Complex simultaneous perturbation approximation, an iterative stochastic optimization method that works within the field of the complex numbers, and Maximum likelihood estimation, a well-known statistical inference method. This can be carried out with the help of a multi-arm interferometric array. In the case of a single qubit, a Mach-Zehnder interferometer suffices. We also show that our estimation procedure can be easily extended to estimate unknown unitary transformations acting on a single qudit. Thereby, the estimation of unitary transformations achieves a higher accuracy than that achieved by processes based on tomographic methods for mixed states.

DOI
10.48550/arxiv.2008.09931

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The accuracy of estimating ddd-dimensional quantum states is limited by the Gill-Massar bound. It can be saturated in the qubit (d=2d=2d=2) scenario using adaptive standard quantum tomography. In higher dimensions, however, this is not the case and the accuracy achievable with adaptive quantum tomography quickly deteriorates with increasing ddd. Moreover, it is not known whether or not the Gill-Massar bound can be reached for an arbitrary ddd. To overcome this limitation, we introduce an adaptive tomographic method that is characterized by a precision that is better than half that of the Gill-Massar bound for any finite dimension. This provides a new achievable accuracy limit for quantum state estimation. We demonstrate the high-accuracy of our method by estimating the state of 10-dimensional quantum systems. With the advent of new technologies capable of high-dimensional quantum information processing, our results become critically relevant as state reconstruction is an essential tool for certifying the proper operation of quantum devices.
We propose a learning method for estimating unknown pure quantum states. The basic idea of our method is to learn a unitary operation $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{U}$ that transforms a given unknown state $|{\ensuremath{\psi}}_{\ensuremath{\tau}}\ensuremath{\rangle}$ to a known fiducial state $|f\ensuremath{\rangle}$. Then, after completion of the learning process, we can estimate and reproduce $|{\ensuremath{\psi}}_{\ensuremath{\tau}}\ensuremath{\rangle}$ based on the learned $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{U}$ and $|f\ensuremath{\rangle}$. To realize this idea, we cast a random-based learning algorithm, called ``single-shot measurement learning,'' in which the learning rule is based on an intuitive and reasonable criterion: the greater the number of success (or failure), the less (or more) changes are imposed. Remarkably, the learning process occurs by means of a single-shot measurement outcome. We demonstrate that our method works effectively, i.e., the learning is completed with a finite number, say NNN, of unknown-state copies. Most surprisingly, our method allows the maximum statistical accuracy to be achieved for large NNN, namely $\ensuremath{\simeq}O({N}^{\ensuremath{-}1})$ scales of average infidelity. It highlights a nontrivial message, that is, a random-based strategy can potentially be as accurate as other standard statistical approaches.
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Identifying an unknown unitary quantum process is a fundamental task in the development of quantum technology. In this paper, we propose an efficient identification algorithm for estimating unitary processes. In the method, input pure states are used and a fast quantum state tomography algorithm is developed to reconstruct the output states. Then the information of the output states is used to estimate the unitary process. The identification algorithm has computational complexity O(d3) for a d-dimensional system. Numerical results show that the proposed identification algorithm is much more efficient than the maximum likelihood estimation method and works well for input mixed states with high purity. An analytical upper bound for the identification error is also provided, and numerical simulations and experimental results on quantum optical systems verify the theoretical results.
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