Few measurement shots challenge generalization in learning to classify entanglement

TL;DR

This paper identifies an instance of this possibly general issue by focusing on the classification of maximally entangled vs. separable states, and introduces an estimator based on classical shadows that performs better in the big data, few copy regime.

Abstract

The ability to extract general laws from a few known examples depends on the complexity of the problem and on the amount of training data. In the quantum setting, the learner's generalization performance is further challenged by the destructive nature of quantum measurements that, together with the no-cloning theorem, limits the amount of information that can be extracted from each training sample. In this paper we focus on hybrid quantum learning techniques where classical machine-learning methods are paired with quantum algorithms and show that, in some settings, the uncertainty coming from a few measurement shots can be the dominant source of errors. We identify an instance of this possibly general issue by focusing on the classification of maximally entangled vs. separable states, showing that this toy problem becomes challenging for learners unaware of entanglement theory. Finally, we introduce an estimator based on classical shadows that performs better in the big data, few copy regime. Our results show that the naive application of classical machine-learning methods to the quantum setting is problematic, and that a better theoretical foundation of quantum learning is required.

Figures (3)

• Figure 1: (a) Region with success rate higher than 99% in classifying a new state, not present in the training set, vs. number of training pairs $N$ and number of shots $S$, for different dimensions $d=2,4,8,16$. (b) Comparison between success rate in classifying a new state and $N$, for $d=8,16,32$. Solid lines show numerical simulations with exactly computed expectation values ($S\to\infty$), dashed lines use $S=2^{14}=16384$ shots, while dotted lines (mostly overlapping with the dashed ones) also use $S=2^{14}$ shots, but then directly construct the "analytical" classifier ($B_{\mathrm{obs}}$, the unbiased estimator of $B^{}_{(2)}$ from Eq. (\ref{['eq:opt B']})) from the measurement results, rather than finding it using a support vector machine.
• Figure 2: Success rate in learning to classify a new state, not present in the training set, vs number of shots $S$ and number of training pairs $N$ for different dimensions $d=4,8,16,32$.
• Figure 3: (a) Performance of the learning approach, as in Fig. \ref{['fig:success']}(a), but with the swap test replaced by the shadow overlap estimator with $N_U$ unitaries and $N_M$. For $d=16$ (not shown) the maximum success rate is 96%, which is below the threshold of 99%. (b) Same data of Fig. \ref{['fig:success']}(a), but reshaped to have $\log_2(NS)$ in the vertical axis.

Theorems & Definitions (4)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4