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arXiv:2310.16921 · Quantum

On the connection between least squares, regularization, and classical shadows

Zhihui Zhu, Joseph M. Lukens, Brian T. Kirby

Published Aug 29, 2024• 3 citations
arXivHTMLPDF

Abstract

Classical shadows (CS) offer a resource-efficient means to estimate quantum observables, circumventing the need for exhaustive state tomography. Here, we clarify and explore the connection between CS techniques and least squares (LS) and regularized least squares (RLS) methods commonly used in machine learning and data analysis. By formal identification of LS and RLS ``shadows'' completely analogous to those in CS---namely, point estimators calculated from the empirical frequencies of single measurements---we show that both RLS and CS can be viewed as regularizers for the underdetermined regime, replacing the pseudoinverse with invertible alternatives. Through numerical simulations, we evaluate RLS and CS from three distinct angles: the tradeoff in bias and variance, mismatch between the expected and actual measurement distributions, and the interplay between the number of measurements and number of shots per measurement. Compared to CS, RLS attains lower variance at the expense of bias, is robust to distribution mismatch, and is more sensitive to the number of shots for a fixed number of state copies---differences that can be understood from the distinct approaches taken to regularization. Conceptually, our integration of LS, RLS, and CS under a unifying ``shadow'' umbrella aids in advancing the overall picture of CS techniques, while practically our results highlight the tradeoffs intrinsic to these measurement approaches, illuminating the circumstances under which either RLS or CS would be preferred, such as unverified randomness for the former or unbiased estimation for the latter.

DOI
10.22331/q-2024-08-29-1455

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Classical shadows (CS) offer a resource-efficient means to estimate quantum observables, circumventing the need for exhaustive state tomography. Here, we clarify and explore the connection between CS techniques and least squares (LS) and regularized least squares (RLS) methods commonly used in machine learning and data analysis. By formal identification of LS and RLS shadows'' completely analogous to those in CS---namely, point estimators calculated from the empirical frequencies of single measurements---we show that both RLS and CS can be viewed as regularizers for the underdetermined regime, replacing the pseudoinverse with invertible alternatives. Through numerical simulations, we evaluate RLS and CS from three distinct angles: the tradeoff in bias and variance, mismatch between the expected and actual measurement distributions, and the interplay between the number of measurements and number of shots per measurement. Compared to CS, RLS attains lower variance at the expense of bias, is robust to distribution mismatch, and is more sensitive to the number of shots for a fixed number of state copies---differences that can be understood from the distinct approaches taken to regularization. Conceptually, our integration of LS, RLS, and CS under a unifying shadow'' umbrella aids in advancing the overall picture of CS techniques, while practically our results highlight the tradeoffs intrinsic to these measurement approaches, illuminating the circumstances under which either RLS or CS would be preferred, such as unverified randomness for the former or unbiased estimation for the latter.
We introduce a postprocessing technique for classical shadow measurement data that enhances the precision of ground state estimation through high-dimensional subspace expansion; the dimensionality is only limited by the amount of classical postprocessing resources rather than by quantum resources. Crucial steps of our approach are the efficient identification of useful observables from shadow data, followed by our regularized subspace expansion that is designed to be numerically stable even when using noisy data. We analytically investigate noise propagation within our method, and upper bound the statistical fluctuations due to the limited number of snapshots in classical shadows. In numerical simulations, our method can achieve a reduction in the energy estimation errors in many cases, sometimes by more than an order of magnitude. We also demonstrate that our performance improvements are robust against both coherent errors (bad initial state) and gate noise in the state-preparation circuits. Furthermore, performance is guaranteed to be at least as good—and in many cases better—than direct energy estimation without using additional quantum resources, and the approach is thus a very natural alternative for estimating ground state energies directly from classical shadow data.
Classical shadow tomography has become a powerful tool in learning about quantum states prepared on a quantum computer. Recent works have used classical shadows to variationally enforce N -representability conditions on the two-particle reduced density matrix. In this paper, we build upon previous research by choosing an improved estimator within classical shadow tomography and rephrasing the optimization constraints, resulting in an overall enhancement in performance under comparable measurement shot budgets. We further explore the specific regimes where these methods outperform the unbiased estimator of the stand-alone classical shadow protocol and quantify the potential savings in numerical studies.
Classical shadows (CS) has recently emerged as an important framework to efficiently predict properties of an unknown quantum state. A common strategy in CS protocols is to parametrize the basis in which one measures the state by a random group action; many examples of this have been proposed and studied on a case-by-case basis. In this work, we present a unified theory that allows us to simultaneously understand CS protocols based on sampling from general group representations, extending previous approaches that worked in simplified (multiplicity-free) settings. We identify a class of measurement bases which we call "centralizing bases" that allows us to analytically characterize and invert the measurement channel, minimizing classical post-processing costs. We complement this analysis by deriving general bounds on the sample-complexity necessary to obtain estimates of a given precision. Beyond its unification of previous CS protocols, our method allows us to readily generate new protocols based on other groups, or different representations of previously considered ones. For example, we characterize novel shadow protocols based on sampling from the spin and tensor representations of SU(2)\textsf{SU}(2)SU(2), symmetric and orthogonal groups, and the exceptional Lie group G2G_2G2​.
Classical shadows (CS) have emerged as a powerful way to estimate many properties of quantum states based on random measurements and classical post-processing. In their original formulation, they come with optimal (or close to) sampling complexity guarantees for generic states and generic observables. Still, it is natural to expect to even further lower sampling requirements when equipped with a priori knowledge regarding either the underlying state or the observables. Here, we consider the case where such knowledge is provided in terms of symmetries of the unknown state or of the observables. Criterion and guidelines for symmetric shadows are provided. As a concrete example we focus on the case of permutation invariance (PI), and detail constructions of several families of PI-CSs. In particular, building on results obtained in the field of PI quantum tomography, we develop and study shallow PI-CS protocol. Benefits of these symmetric CS are demonstrated compared to established CS protocols showcasing vastly improved performances.
In this paper, we study extended linear regression approaches for quantum state tomography based on regularization techniques. For unknown quantum states represented by density matrices, performing measurements under certain basis yields random outcomes, from which a classical linear regression model can be established. First of all, for complete or over-complete measurement bases, we show that the empirical data can be utilized for the construction of a weighted least squares estimate (LSE) for quantum tomography. Taking into consideration the trace-one condition, a constrained weighted LSE can be explicitly computed, being the optimal unbiased estimation among all linear estimators. Next, for general measurement bases, we show that ℓ2\ell_2ℓ2​-regularization with proper regularization gain provides even lower mean-square error under a cost in bias. The regularization parameter is tuned by two estimators in terms of a risk characterization. Finally, a concise and unified formula is established for the regularization parameter with complete measurement basis under an equivalent regression model, which proves that the proposed tuning estimators are asymptotically optimal as the number of samples grows to infinity under the risk metric. Additionally, numerical examples are provided to validate the established results.