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arXiv:2007.02537 · Physical Review Letters

Quantum State Learning via Single-Shot Measurements

Sang Min Lee, Hee Su Park, Jinhyoung Lee, Jaewan Kim, Jeongho Bang

Published Apr 30, 2021• 7 citations
arXivHTMLPDF

Abstract

A novel machine-learning algorithm based on single-shot measurements, named single-shot measurement learning, is demonstrated achieving the theoretical optimal accuracy. The method is at least as efficient as existing tomographic schemes and computationally much less demanding. The merits are attributed to the inclusion of weighted randomness in the learning rule governing the exploration of diverse learning routes. These advantages are explored experimentally by a linear-optical setup that is designed to draw the fullest potential of the proposed method. The experimental results show an unprecedented high level of accuracy for qubit-state learning and reproduction exhibiting (nearly) optimal infidelity scaling, O(N^{-0.983}), for the number N of unknown state copies, down to <10^{-5} without any compensation for experimental nonidealities. Extension to high dimensions is discussed with simulation results.

DOI
10.1103/physrevlett.126.170504

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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.
Quantum state tomography (QST) has been the traditional method for characterization of an unknown state. Recently, many direct measurement methods have been implemented to reconstruct the state in a resource efficient way. In this Letter, we present an interferometric method, in which any qubit state, whether mixed or pure, can be inferred from the visibility, phase shift, and average intensity of an interference pattern using a single-shot measurement-hence, we call it quantum state interferography. This provides us with a "black box" approach to quantum state estimation, wherein, between the incidence of the photon and extraction of state information, we are not changing any conditions within the setup, thus giving us a true single shot estimation of the quantum state. In contrast, standard QST requires at least two measurements for pure state qubit and at least three measurements for mixed state qubit reconstruction. We then go on to show that QSI is more resource efficient than QST for quantification of entanglement in pure bipartite qubits. We experimentally implement our method with high fidelity using the polarization degree of freedom of light. An extension of the scheme to pure states involving d-1 interferograms for d-dimensional systems is also presented. Thus, the scaling gain is even more dramatic in the qudit scenario for our method, where, in contrast, standard QST, without any assumptions, scales roughly as d^{2}.
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The constantly increasing dimensionality of artificial quantum systems demands for highly efficient methods for their characterization and benchmarking. Conventional quantum tomography fails for larger systems due to the exponential growth of the required number of measurements. The conceptual solution for this dimensionality curse relies on a simple idea---a complete description of a quantum state is excessive and can be discarded in favor of experimentally accessible information about the system. The probably approximately correct learning theory has been recently successfully applied to a problem of building accurate predictors for the measurement outcomes using a data set which scales only linearly with the number of qubits. Here, we present a constructive and numerically efficient protocol which learns a tensor network model of an unknown quantum system. We discuss the limitations and the scalability of the proposed method.
Learning the unknown Hamiltonian governing the dynamics of a quantum many-body system is a challenging task. In this manuscript, we propose a possible strategy based on repeated measurements on a single time-dependent state. We prove that the accuracy of the learning process is maximized for states that are delocalized in the Hamiltonian eigenbasis. This implies that delocalization is a quantum resource for Hamiltonian learning, that can be exploited to select optimal initial states for learning algorithms. We investigate the error scaling of our reconstruction with respect to the number of measurements, and we provide examples of our learning algorithm on simulated quantum systems.
Estimation of the expectation value of observables is a key subroutine in quantum computing and is also the bottleneck of the performance of many near-term quantum algorithms. Many methods have been proposed to reduce the number of measurements needed for this task by designing measurement schemes that decide the measurements to perform; however, these schemes are usually constructed from hand-crafted heuristics, which limits the measurement efficiency they can achieve. In this paper, we propose a framework for learning measurement schemes directly from the observable, using machine learning techniques including stochastic gradient descent and a two time-scale update rule. As a concrete realization of this framework, we introduce Composite-Locally Biased Classical Shadow (C-LBCS), which learns a mixture of locally-biased classical shadows and their mixing weights end-to-end. We numerically demonstrate C-LBCS on molecular systems up to CO2\mathrm{CO}_2CO2​ (30 qubits) and show that C-LBCS outperforms the previous state-of-the-art methods despite its simplicity. We believe our approach opens up a reliable and scalable path toward efficient observable estimation on large quantum systems.