The vast and complicated many-qubit state space forbids us to comprehensively capture the dynamics of modern quantum computers via classical simulations or quantum tomography. Recent progress in quantum learning theory prompts a crucial question: can linear properties of a many-qubit circuit with d tunable RZ gates and G − d Clifford gates be efficiently learned from measurement data generated by varying classical inputs? In this work, we prove that the sample complexity scaling linearly in d is required to achieve a small prediction error, while the corresponding computational complexity may scale exponentially in d. To address this challenge, we propose a kernel-based method leveraging classical shadows and truncated trigonometric expansions, enabling a controllable trade-off between prediction accuracy and computational overhead. Our results advance two crucial realms in quantum computation: the exploration of quantum algorithms with practical utilities and learning-based quantum system certification. We conduct numerical simulations to validate our proposals across diverse scenarios, encompassing quantum information processing protocols, Hamiltonian simulation, and variational quantum algorithms up to 60 qubits. It is currently unclear whether machine learning model could efficiently learn the dynamics of many-qubit bounded-gate quantum circuits. This work reveals the learnability of predicting linear properties of large- scale quantum circuits and proposes a learning algorithm that enables resource-efficient solutions for many practical quantum applications.
The vast and complicated many-qubit state space forbids us to comprehensively capture the dynamics of modern quantum computers via classical simulations or quantum tomography. Recent progress in quantum learning theory prompts a crucial question: can linear properties of a many-qubit circuit with d tunable RZ gates and G − d Clifford gates be efficiently learned from measurement data generated by varying classical inputs? In this work, we prove that the sample complexity scaling linearly in d is required to achieve a small prediction error, while the corresponding computational complexity may scale exponentially in d. To address this challenge, we propose a kernel-based method leveraging classical shadows and truncated trigonometric expansions, enabling a controllable trade-off between prediction accuracy and computational overhead. Our results advance two crucial realms in quantum computation: the exploration of quantum algorithms with practical utilities and learning-based quantum system certification. We conduct numerical simulations to validate our proposals across diverse scenarios, encompassing quantum information processing protocols, Hamiltonian simulation, and variational quantum algorithms up to 60 qubits. It is currently unclear whether machine learning model could efficiently learn the dynamics of many-qubit bounded-gate quantum circuits. This work reveals the learnability of predicting linear properties of large- scale quantum circuits and proposes a learning algorithm that enables resource-efficient solutions for many practical quantum applications.
While quantum state tomography is notoriously hard, most states hold little interest to practically minded tomographers. Given that states and unitaries appearing in nature are of bounded gate complexity, it is natural to ask if efficient learning becomes possible. In this work, we prove that to learn a state generated by a quantum circuit with G two-qubit gates to a small trace distance, a sample complexity scaling linearly in G is necessary and sufficient. We also prove that the optimal query complexity to learn a unitary generated by G gates to a small average-case error scales linearly in G . While sample-efficient learning can be achieved, we show that under reasonable cryptographic conjectures, the computational complexity for learning states and unitaries of gate complexity G must scale exponentially in G . We illustrate how these results establish fundamental limitations on the expressivity of quantum machine-learning models and provide new perspectives on no-free-lunch theorems in unitary learning. Together, our results answer how the complexity of learning quantum states and unitaries relate to the complexity of creating these states and unitaries. Published by the American Physical Society 2024
Machine learning algorithms based on parametrized quantum circuits are prime candidates for near-term applications on noisy quantum computers. In this direction, various types of quantum machine learning models have been introduced and studied extensively. Yet, our understanding of how these models compare, both mutually and to classical models, remains limited. In this work, we identify a constructive framework that captures all standard models based on parametrized quantum circuits: that of linear quantum models. In particular, we show using tools from quantum information theory how data re-uploading circuits, an apparent outlier of this framework, can be efficiently mapped into the simpler picture of linear models in quantum Hilbert spaces. Furthermore, we analyze the experimentally-relevant resource requirements of these models in terms of qubit number and amount of data needed to learn. Based on recent results from classical machine learning, we prove that linear quantum models must utilize exponentially more qubits than data re-uploading models in order to solve certain learning tasks, while kernel methods additionally require exponentially more data points. Our results provide a more comprehensive view of quantum machine learning models as well as insights on the compatibility of different models with NISQ constraints.
We present an efficient machine-learning (ML) algorithm for predicting any unknown quantum process E over n qubits.For a wide range of distributions D on arbitrary n-qubit states, we show that this ML algorithm can learn to predict any local property of the output from the unknown process E, with a small average error over input states drawn from D. The ML algorithm is computationally efficient even when the unknown process is a quantum circuit with exponentially many gates.Our algorithm combines efficient procedures for learning properties of an unknown state and for learning a low-degree approximation to an unknown observable.The analysis hinges on proving new norm inequalities, including a quantum analogue of the classical Bohnenblust-Hille inequality, which we derive by giving an improved algorithm for optimizing local Hamiltonians.Numerical experiments on predicting quantum dynamics with evolution time up to 10 6 and system size up to 50 qubits corroborate our proof.Overall, our results highlight the potential for ML models to predict the output of complex quantum dynamics much faster than the time needed to run the process itself.
Quantum resource theory has sharpened our understanding of the intrinsic complexity of quantum systems, particularly their classical simulability. However, it remains unclear which quantum resource governs the classical learnability of quantum circuits, especially beyond the regime of efficient classical simulation. Here we close this knowledge gap by studying the expectation-value functions of families of tunable quantum circuits, with many applications in digital quantum simulation, quantum metrology, and quantum-system characterization. Specifically, we introduce a new resource measure, the dynamical stabilizer entropy (\DSE), which quantifies how broadly an expectation-value function is distributed across its frequency modes. By relating \DSE to operator stabilizer entropy, we establish a computational phase diagram that compares classical simulators with quantum-data-assisted classical surrogates. We first determine the \DSE-dependent learnability boundary of this diagram by deriving bounds on the sample complexity and runtime of classical surrogates, and by developing a \DSE-guided surrogate. We then complete the diagram by proving, under standard complexity-theoretic assumptions, the existence of circuit families that can be efficiently learned by this surrogate but cannot be efficiently emulated from their circuit descriptions alone. Numerical experiments on random and structured circuits with up to 80 qubits support the predicted \DSE-dependent computational landscape. These results establish a quantitative resource-theoretic framework for delineating the boundary between classical simulation and learning, motivate resource measures linking quantum resources to learnability, and guide the design of learning-based algorithms for scalable quantum systems beyond the reach of direct classical simulation.
Quantum processors may enhance machine learning by mapping high-dimensional data onto quantum systems for processing. Conventional feature maps, for encoding data onto a quantum circuit are currently impractical, as the number of entangling gates scales quadratically with the dimension of the dataset and the number of qubits. In this work, we introduce a quantum feature map designed to handle high-dimensional data with a significantly reduced number of qubits and entangling operations. Our approach preserves essential data characteristics while promoting computational efficiency, as evidenced by extensive experiments on benchmark datasets that demonstrate a marked improvement in both accuracy and resource utilization when using our feature map as a kernel for characterization, as compared to state-of-the-art quantum feature maps. Our noisy simulation results, combined with lower resource requirements, highlight our map's ability to function within the constraints of noisy intermediate-scale quantum devices. Through numerical simulations and small-scale implementation on a superconducting circuit quantum computing platform, we demonstrate that our scheme performs on par or better than a set of classical algorithms for classification. While quantum kernels are typically stymied by exponential concentration, our approach is affected with a slower rate with respect to both the number of qubits and features, which allows practical applications to remain within reach. Our findings herald a promising avenue for the practical implementation of quantum machine learning algorithms on near future quantum computing platforms.
We present an efficient machine learning (ML) algorithm for predicting any unknown quantum process E over n qubits. For a wide range of distributions D on arbitrary n-qubit states, we show that this ML algorithm can learn to predict any local property of the output from the unknown process~E, with a small average error over input states drawn from D. The ML algorithm is computationally efficient even when the unknown process is a quantum circuit with exponentially many gates. Our algorithm combines efficient procedures for learning properties of an unknown state and for learning a low-degree approximation to an unknown observable. The analysis hinges on proving new norm inequalities, including a quantum analogue of the classical Bohnenblust-Hille inequality, which we derive by giving an improved algorithm for optimizing local Hamiltonians. Numerical experiments on predicting quantum dynamics with evolution time up to 106 and system size up to 50 qubits corroborate our proof. Overall, our results highlight the potential for ML models to predict the output of complex quantum dynamics much faster than the time needed to run the process itself.