Quantum phase classification via partial tomography-based quantum hypothesis testing

TL;DR

This work proposes a classification algorithm based on the quantum Neyman-Pearson test, which is theoretically optimal for distinguishing between two quantum states, and achieves lower classification error probabilities and significantly reduces the training cost compared to the QCNN and the recently developed classical machine learning algorithm enhanced with quantum data.

Abstract

Quantum phase classification is a fundamental problem in quantum many-body physics, traditionally approached using order parameters or quantum machine learning techniques such as quantum convolutional neural networks (QCNNs). However, these methods often require extensive prior knowledge of the system or large numbers of quantum state copies for reliable classification. In this work, we propose a classification algorithm based on the quantum Neyman-Pearson test, which is theoretically optimal for distinguishing between two quantum states. While directly constructing the optimal test for many-body systems via full state tomography is intractable due to the exponential growth of the Hilbert space, we introduce a partitioning strategy that applies hypothesis tests to subsystems rather than the entire state, effectively reducing the required number of quantum state copies while maintaining classification accuracy. We validate our approach through numerical simulations, demonstrating its advantages over conventional methods, including the order parameter-based classifier, the QCNN, and the recently developed classical machine learning algorithm enhanced with quantum data. Our results show that the proposed method achieves lower classification error probabilities with fewer required quantum state copies compared to all of these baselines, while also reducing the training cost relative to the QCNN and the classical machine learning algorithm enhanced with quantum data, and further decreasing the classical computational time in comparison with the latter. We additionally demonstrate scalability of our method in numerical experiments up to systems with 81 qubits. These findings highlight the potential of quantum hypothesis testing as a powerful tool for quantum phase classification, particularly in experimental settings where quantum measurements are combined with classical post-processing.

Figures (12)

• Figure 1: Schematic depiction of our method in the (a) training step and (b) test step. Steps within the blue boxes represent quantum processes, while those within the green boxes represent classical processes. In the training step (a), we perform partial tomography on all the quantum many-body states $\{\rho^{(i)}\}$ in a training dataset, obtaining their $k$-RDMs. In the test step (b), for each test state $\rho_{\text{test}}^{(i)}$, the approximate quantum Neyman-Pearson test is conducted to obtain a prediction $y^{(i)}$.
• Figure 2: Quantum phase diagram of the ground state of the Hamiltonian in Eq. (\ref{['eq:cluster-Ising']}), along with the training and test data for (a) Trivial vs. FM and (b) Trivial vs. SPT cases. The 20 black dots represent the training data, while the test data consist of 100 randomly selected points within the red box near the phase boundary.
• Figure 3: Type-I and Type-II error probabilities, $\alpha_n$ and $\beta_n$, for the test data in the order parameter, Exact QCNN, QCNN, and our method on $L=27$ qubits in the Trivial vs. FM case and Trivial vs. SPT case. Panel (a) shows the error probabilities $\alpha_{1}$ and $\beta_{1}$ for a single-copy test dataset ($n = 1$), whereas Panel (b) shows the number of test data copies $n$ required to achieve $\beta_n$ under the condition $\alpha_n \leq 5\,\%$.
• Figure 4: Learning curves for our method and the QCNN on $L = 15$ qubits in the Trivial vs. FM case and Trivial vs. SPT case. Panels (a) and (b) show the results for our method and the QCNN, respectively. The training shots represent the total number of training data, and the validation loss (MSE) is defined in Eq. (\ref{['eq:validation loss']}).
• Figure 5: Learning curves for our method and the low-weight QCNN on $L = 15$ qubits. Panel (a) corresponds to the Trivial vs. FM case, and Panel (b) to the Trivial vs. SPT case; both are shown in units of $10^3$. The results for our method are the same as those in Fig. \ref{['fig:Comparison with QCNNs']}.

Theorems & Definitions (6)

• Theorem 1: Variance reduction by the majority vote on a $d$-dimensional model
• proof
• Lemma 1: Neyman-Pearson Lemma Neyman1933
• Lemma 2: Karlin1956
• Theorem 2
• proof