Scalable bayesian shadow tomography for quantum property estimation with set transformers

TL;DR

Quantum state characterization faces scalability limits when reconstructing full density matrices. This work presents a scalable Bayesian shadow tomography framework that directly estimates scalar properties by fusing classical shadows with a permutation-invariant set transformer and residual learning to approximate the Bayesian posterior mean without full state reconstruction. The method supports non-adaptive measurements, encodes outcomes into fixed-dimensional features, and achieves lower mean-squared error than classical shadows on GHZ fidelity and second-order Renyi entropy, including in the few-copy regime. It also provides permutation-invariant architectures, calibration strategies, and concrete Pauli/Clifford encoding schemes, offering a path to practical large-scale quantum property estimation.

Abstract

A scalable Bayesian machine learning framework is introduced for estimating scalar properties of an unknown quantum state from measurement data, which bypasses full density matrix reconstruction. This work is the first to integrate the classical shadows protocol with a permutation-invariant set transformer architecture, enabling the approach to predict and correct bias in existing estimators to approximate the true Bayesian posterior mean. Measurement outcomes are encoded as fixed-dimensional feature vectors, and the network outputs a residual correction to a baseline estimator. Scalability to large quantum systems is ensured by the polynomial dependence of input size on system size and number of measurements. On Greenberger-Horne-Zeilinger state fidelity and second-order Rényi entropy estimation tasks -- using random Pauli and random Clifford measurements -- this Bayesian estimator always achieves lower mean squared error than classical shadows alone, with more than a 99\% reduction in the few copy regime.

Figures (9)

• Figure 1: General framework for residual Bayesian estimation of functions of quantum states. A single experimental realization, comprising $N$ measurement outcomes, is encoded into $N$ feature vectors characterizing the outcomes. A PI network is trained to infer a quantity $x$ that depends on the measurement data and corrects the bias introduced by a baseline non-Bayesian estimator, which outputs a point estimate $F$.
• Figure 2: MSE comparison between classical shadow estimation and the proposed Bayesian estimation. Displayed are DFE results for the GHZ state using (a) random Pauli measurements with $N=10$, (b) random Pauli measurements with $N=100$, (c) random Clifford measurements with $N=10$, and (d) random Clifford measurements with $N=100$.
• Figure 3: MSE comparison between classical shadow estimation and the proposed Bayesian estimation under bit-flip noise with parameter $\lambda=0.1$. Displayed are DFE results for the GHZ state using (a) random Pauli measurements with $N=10$, (b) random Pauli measurements with $N=100$, (c) random Clifford measurements with $N=10$, and (d) random Clifford measurements with $N=100$.
• Figure 4: MSE comparison between classical shadow estimation and the proposed Bayesian estimation. Displayed are entanglement entropy estimation results for the GHZ state using (a) random Pauli measurements with $N=10$, (b) random Pauli measurements with $N=100$, (c) random Clifford measurements with $N=10$, and (d) random Clifford measurements with $N=100$.
• Figure 5: MSE comparison between classical shadow estimation and the proposed Bayesian estimation under bit-flip noise with parameter $\lambda=0.1$. Displayed are entanglement entropy estimation results for the GHZ state using (a) random Pauli measurements with $N=10$, (b) random Pauli measurements with $N=100$, (c) random Clifford measurements with $N=10$, and (d) random Clifford measurements with $N=100$.

Theorems & Definitions (12)

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• Lemma 4.1: Informal
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• Remark 5.1
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