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arXiv:2402.04992 · Physical review. E

Challenges and opportunities in the supervised learning of quantum circuit expectation values

Simone Cantori, Sebastiano Pilati

Published Jul 2, 2025• 0 citations
arXivHTMLPDF

Abstract

Recently, deep neural networks have been proven capable of predicting output expectation values of certain random quantum circuits via a supervised learning approach. Here we investigate the potential of this possible approach to the emulation of quantum circuits, testing both its limitations and the successful applications where it might outperform conventional direct simulation methods. The testbeds we consider are circuits often employed in variational quantum algorithms, featuring layers of cnot gates alternated with single-qubit random rotations. On the one hand, we find that the computational cost of supervised learning scales exponentially with the interlayer variance of the random angles. This allows for entering a promising regime for quantum advantage, where quantum computers could easily outperform classical neural networks. On the other hand, circuits featuring only interqubit angle variations are easily emulated. In fact, thanks to a suitable scalable design, the trained networks accurately predict the expectation values of larger and deeper circuits than those used for training, even reaching circuit sizes, which, as we numerically show, are computationally intractable for the most common simulation libraries of state-vector and tensor-network algorithms. A repository of test data in the intractable regime is provided. We also analyze the most common metrics for the entanglement content, the expressibility, and the classical computational cost of quantum circuits, finding that they do not distinguish the easy from the hard circuit configurations.

DOI
10.1103/9bq5-sqhd

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Recently, deep neural networks have been proven capable of predicting output expectation values of certain random quantum circuits via a supervised learning approach. Here we investigate the potential of this possible approach to the emulation of quantum circuits, testing both its limitations and the successful applications where it might outperform conventional direct simulation methods. The testbeds we consider are circuits often employed in variational quantum algorithms, featuring layers of cnot gates alternated with single-qubit random rotations. On the one hand, we find that the computational cost of supervised learning scales exponentially with the interlayer variance of the random angles. This allows for entering a promising regime for quantum advantage, where quantum computers could easily outperform classical neural networks. On the other hand, circuits featuring only interqubit angle variations are easily emulated. In fact, thanks to a suitable scalable design, the trained networks accurately predict the expectation values of larger and deeper circuits than those used for training, even reaching circuit sizes, which, as we numerically show, are computationally intractable for the most common simulation libraries of state-vector and tensor-network algorithms. A repository of test data in the intractable regime is provided. We also analyze the most common metrics for the entanglement content, the expressibility, and the classical computational cost of quantum circuits, finding that they do not distinguish the easy from the hard circuit configurations.
Abstract Predicting the output of quantum circuits is a hard computational task that plays a pivotal role in the development of universal quantum computers. Here we investigate the supervised learning of output expectation values of random quantum circuits. Deep convolutional neural networks (CNNs) are trained to predict single-qubit and two-qubit expectation values using databases of classically simulated circuits. These circuits are built using either a universal gate set or a continuous set of rotations plus an entangling gate, and they are represented via properly designed encodings of these gates. The prediction accuracy for previously unseen circuits is analyzed, also making comparisons with small-scale quantum computers available from the free IBM Quantum program. The CNNs often outperform these quantum devices, depending on the circuit depth, on the network depth, and on the training set size. Notably, our CNNs are designed to be scalable. This allows us exploiting transfer learning and performing extrapolations to circuits larger than those included in the training set. These CNNs also demonstrate remarkable resilience against noise, namely, they remain accurate even when trained on (simulated) expectation values averaged over very few measurements.
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Deep learning is one of the most successful and far-reaching strategies used in machine learning today. However, the scale and utility of neural networks is still greatly limited by the current hardware used to train them. These concerns have become increasingly pressing as conventional computers quickly approach physical limitations that will slow performance improvements in years to come. For these reasons, scientists have begun to explore alternative computing platforms, like quantum computers, for training neural networks. In recent years, variational quantum circuits have emerged as one of the most successful approaches to quantum deep learning on noisy intermediate scale quantum devices. We propose a hybrid quantum-classical neural network architecture where each neuron is a variational quantum circuit. We empirically analyze the performance of this hybrid neural network on a series of binary classification data sets using a simulated universal quantum computer and a state of the art universal quantum computer. On simulated hardware, we observe that the hybrid neural network achieves roughly 10% higher classification accuracy and 20% better minimization of cost than an individual variational quantum circuit. On quantum hardware, we observe that each model only performs well when the qubit and gate count is sufficiently small.
Supervised machine learning is emerging as a powerful computational tool to predict the properties of complex quantum systems at a limited computational cost. In this article, we quantify how accurately deep neural networks can learn the properties of disordered quantum systems as a function of the system size. We implement a scalable convolutional network that can address arbitrary system sizes. This network is compared with a recently introduced extensive convolutional architecture [Mills et al., Chem. Sci. 10, 4129 (2019)2041-652010.1039/C8SC04578J] and with conventional dense networks with all-to-all connectivity. The networks are trained to predict the exact ground-state energies of various disordered systems, namely, a continuous-space single-particle Hamiltonian for cold-atoms in speckle disorder, and different setups of a quantum Ising chain with random couplings, including one with only short-range interactions and one augmented with a long-range term. In all testbeds we consider, the scalable network retains high accuracy as the system size increases. Furthermore, we demonstrate that the network scalability enables a transfer-learning protocol, whereby a pretraining performed on small systems drastically accelerates the learning of large-system properties, allowing reaching high accuracy with small training sets. In fact, with the scalable network one can even extrapolate to sizes larger than those included in the training set, accurately reproducing the results of state-of-the-art quantum Monte Carlo simulations.
Abstract We investigate the potential of combining the computational power of noisy quantum computers and of classical scalable convolutional neural networks (CNNs). The goal is to accurately predict exact expectation values of parameterized quantum circuits representing the Trotter-decomposed dynamics of quantum Ising models. By incorporating (simulated) noisy expectation values alongside circuit structure information, our CNNs effectively capture the underlying relationships between circuit architecture and output behaviour, enabling, via transfer learning, also predictions for circuits with more qubits than those included in the training set. Notably, thanks to the quantum information, our CNNs succeed even when supervised learning based only on classical descriptors fails. Furthermore, they outperform a popular error mitigation scheme, namely, zero-noise extrapolation, demonstrating that the synergy between quantum and classical computational tools leads to higher accuracy compared with quantum-only or classical-only approaches. By tuning the noise strength, we explore the crossover from a computationally powerful classical CNN assisted by quantum noisy data, towards rather precise quantum computations, further error-mitigated via classical deep learning.