The method of classical shadows heralds unprecedented opportunities for quantum estimation with limited measurements [H.-Y. Huang, R. Kueng, and J. Preskill, Nat. Phys. 16, 1050 (2020)]. Yet its relationship to established quantum tomographic approaches, particularly those based on likelihood models, remains unclear. In this article, we investigate classical shadows through the lens of Bayesian mean estimation (BME). In direct tests on numerical data, BME is found to attain significantly lower error on average, but classical shadows prove remarkably more accurate in specific situations -- such as high-fidelity ground truth states -- which are improbable in a fully uniform Hilbert space. We then introduce an observable-oriented pseudo-likelihood that successfully emulates the dimension-independence and state-specific optimality of classical shadows, but within a Bayesian framework that ensures only physical states. Our research reveals how classical shadows effect important departures from conventional thinking in quantum state estimation, as well as the utility of Bayesian methods for uncovering and formalizing statistical assumptions.
Locally-biased classical shadows allow rapid estimation of energies of quantum Hamiltonians. Recently, derandomised classical shadows have emerged claiming to be even more accurate. This accuracy comes at a cost of introducing classical computing resources into the energy estimation procedure. This present note shows, by adding a fraction of this classical computing resource to the locally-biased classical shadows setting, that the modified algorithm, termed Adaptive Pauli Shadows is state-of-the-art for energy estimation.
Classical optimisation of reduced density matrix estimations with classical shadows using N-representability conditions under shot noise considerations
Distinguishing Ordered Phases using Machine Learning and Classical Shadows
Universidade de São Paulo（圣保罗大学）；Federal University of Rio Grande do Norte（北里奥格兰德联邦大学）；Universidade Federal de Alagoas（阿拉戈阿斯联邦大学）；Technology Innovation Institute（技术创新研究所）
Obtaining precise estimates of quantum observables is a crucial step of variational quantum algorithms. We consider the problem of estimating expectation values of molecular Hamiltonians, obtained on states prepared on a quantum computer. We propose a novel estimator for this task, which is locally optimised with knowledge of the Hamiltonian and a classical approximation to the underlying quantum state. Our estimator is based on the concept of classical shadows of a quantum state, and has the important property of not adding to the circuit depth for the state preparation. We test its performan…
Variational quantum algorithms (VQAs) are the quantum analog of classical neural networks (NNs). A VQA consists of a parameterized quantum circuit (PQC) which is composed of multiple layers of ansatzes (simpler PQCs, which are an analogy of NN layers) that differ only in selections of parameters. Previous work has identified the alternating layered ansatz as potentially a new standard ansatz in near-term quantum computing. Indeed, shallow alternating layered VQAs are easy to implement and have been shown to be both trainable and expressive. In this work, we introduce a training algorithm with …
Full quantum tomography of high-dimensional quantum systems is experimentally infeasible due to the exponential scaling of the number of required measurements on the number of qubits in the system. However, several ideas were proposed recently for predicting the limited number of features for these states, or estimating the expectation values of operators, without the need for full state reconstruction. These ideas go under the general name of shadow tomography. Here we provide an experimental demonstration of property estimation based on classical shadows proposed in [H.-Y. Huang, R. Kueng, J…
The quantum component in uncertainty relation can be naturally characterized by the quantum coherence of a quantum state, which is of paramount importance in quantum information science. Here, we experimentally investigate quantum uncertainty relations construed with relative entropy of coherence, $l_1$ norm of coherence and coherence of formation. In stead of quantum state tomographic technology, we employ the classical shadow algorithm for the detection of lower bounds in quantum uncertainty relations. With an all optical setup, we prepare a family of quantum states whose purity can be fully…
Demonstrating and Benchmarking Classical Shadows for Lindblad Tomography
Center for Quantum Devices, Niels Bohr Institute, University of Copenhagen（哥本哈根大学尼尔斯·玻尔研究所量子设备中心）；Center for the Mathematics of Quantum Theory, MATH Department, University of Copenhagen（哥本哈根大学数学系量子理论数学中心）；NNF Quantum Computing Programme, Niels Bohr Institute, University of Copenhagen（哥本哈根大学尼尔斯·玻尔研究所NNF量子计算项目）；Department of Microtechnology and Nanoscience, Chalmers University of Technology（查尔姆斯理工大学微技术和纳米科学系）
Verification Complexity and Extension of Classical Shadows
Paderborn University（帕德博恩大学）；Freie Universität Berlin（柏林自由大学）；Singapore Institute of Technology（新加坡科技研究局理工学院）；Agency for Science, Technology and Research (A*STAR)（科学技术研究局）；Singapore University of Technology and Design（新加坡科技设计大学）
Exploiting near-term quantum computers and achieving practical value is a considerable and exciting challenge. Most prominent candidates as variational algorithms typically aim to find the ground state of a Hamiltonian by minimising a single classical (energy) surface which is sampled from by a quantum computer. Here we introduce a method we call CoVaR, an alternative means to exploit the power of variational circuits: We find eigenstates by finding joint roots of a polynomially growing number of properties of the quantum state as covariance functions between the Hamiltonian and an operator po…
Variational quantum algorithms are promising algorithms for achieving quantum advantage on near-term devices. The quantum hardware is used to implement a variational wave function and measure observables, whereas the classical computer is used to store and update the variational parameters. The optimization landscape of expressive variational ansätze is however dominated by large regions in parameter space, known as barren plateaus, with vanishing gradients which prevents efficient optimization. In this work we propose a general algorithm to avoid barren plateaus in the initialization and thro…
We propose a tomographic protocol for estimating any $ k $-body reduced density matrix ($ k $-RDM) of an $ n $-mode fermionic state, a ubiquitous step in near-term quantum algorithms for simulating many-body physics, chemistry, and materials. Our approach extends the framework of classical shadows, a randomized approach to learning a collection of quantum-state properties, to the fermionic setting. Our sampling protocol uses randomized measurement settings generated by a discrete group of fermionic Gaussian unitaries, implementable with linear-depth circuits. We prove that estimating all $ k $…
Shadow models of a quantum model for cloud cover and the influence of finite sampling noise
Deutsches Zentrum für Luft- und Raumfahrt, Institut für Physik der Atmosphäre（德国航空航天中心，大气物理研究所）；University of Bremen, Institute of Environmental Physics (IUP)（不来梅大学，环境物理研究所）
Classical shadows with arbitrary group representations
Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室）；University of Melbourne（墨尔本大学）；Quantinuum；University of Texas at Austin（德克萨斯大学奥斯汀分校）；University of Florida（佛罗里达大学）；Xanadu；QuSoft；University of Amsterdam（阿姆斯特丹大学）；Institute for Logic, Language and Computation, University of Amsterdam（阿姆斯特丹大学逻辑、语言与计算研究所）；Korteweg-de Vries Institute for Mathematics, University of Amsterdam（阿姆斯特丹大学科特韦格-德弗里斯数学研究所）；Quantum Science Center（量子科学中心）
