A Bayesian analysis of classical shadows

Abstract

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.

Improving application performance with biased distributions of quantum states

“…Even if lacking some of the theoretical properties of archetypal quantum state measures (like Bures), custom distributions like the MA distribution can attain performance comparable to these measures over the general Hilbert space-thus ensuring good fiducial uniformity as a test distribution-while enabling improvements for specific subspaces (e.g., pure states). Such performance hedging for desired outcomes bears resemblance to the recent estimation approach of classical shadows [53] which, when compared to BME, accepts much higher estimation error on average in exchange for remarkably low error for specific cases of interest [54].…”

Quantum machine learning for chemistry and physics

“…Several methods have been proposed as a means of an alternative approach to the traditional QST such as matrix product state tomography, 371 neural network tomography, [373][374][375][376] quantum overlapping tomography, 377 and shadow tomography. 378,379 Because of the noisy nature of the quantum systems since not all measurements are available at high fidelity, there have also been approaches that try to carry out QST based on incomplete measurements [380][381][382] such as maximum likelihood estimation (MLE), [383][384][385][386][387][388] Bayesian mean estimation (BME), [389][390][391][392] and maximal entropy formalism based QST. 393,394 Quantum detection and estimation theory has been a prominent field of research in quantum information theory since the 1970s [395][396][397] and the rapid progress in quantum communication and computation in the past two decades motivated the use of big data for the classification of quantum systems through quantum learning and quantum matching machines.…”

Variational Approach to Quantum State Tomography based on Maximal Entropy Formalism

“…Several research approaches have already been proposed that attempts to address one or the other limitations and it has paved the way for further advancements in this field. Some of these tomographic techniques include maximum likelihood estimation (MLE) [11,12], Bayesian mean estimation (BME) [13,14], quantum overlap tomography [15], shadow tomography [16,17], neural network tomography [18,19,20,21], and others [22,23,24,25]. In our previous work we proposed a method of QST based on the formalism of maximal entropy from an incomplete set of measurements [26,27].…”