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arXiv:2210.14894

Learning to predict arbitrary quantum processes

Hsin-Yuan Huang, Sitan Chen, John P. Preskill

Published Oct 26, 2022• 1 citation
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Abstract

We present an efficient machine learning (ML) algorithm for predicting any unknown quantum process $\mathcal{E}$ over $n$ qubits. For a wide range of distributions $\mathcal{D}$ 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~$\mathcal{E}$, with a small average error over input states drawn from $\mathcal{D}$. 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.

DOI
10.48550/arxiv.2210.14894

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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.
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We investigate the problem of predicting the output behavior of unknown quantum channels. Given query access to an nnn-qubit channel EEE and an observable OOO, we aim to learn the mapping \begin{equation*} ρ\mapsto \mathrm{Tr}(O E[ρ]) \end{equation*} to within a small error for most ρρρ sampled from a distribution DDD. Previously, Huang, Chen, and Preskill proved a surprising result that even if EEE is arbitrary, this task can be solved in time roughly nO(log⁡(1/ε))n^{O(\log(1/ε))}nO(log(1/ε)), where εεε is the target prediction error. However, their guarantee applied only to input distributions DDD invariant under all single-qubit Clifford gates, and their algorithm fails for important cases such as general product distributions over product states ρρρ. In this work, we propose a new approach that achieves accurate prediction over essentially any product distribution DDD, provided it is not "classical" in which case there is a trivial exponential lower bound. Our method employs a "biased Pauli analysis," analogous to classical biased Fourier analysis. Implementing this approach requires overcoming several challenges unique to the quantum setting, including the lack of a basis with appropriate orthogonality properties. The techniques we develop to address these issues may have broader applications in quantum information.
Abstract. Despite fundamental interests in learning quantum circuits, the existence of a computationally efficient algorithm for learning shallow quantum circuits remains an open question. Because shallow quantum circuits can generate distributions that are classically hard to sample from, existing learning algorithms do not apply. In this work, we present a polynomial-time classical algorithm for learning the description of any unknown [Formula: see text]-qubit shallow quantum circuit [Formula: see text] (with arbitrary unknown architecture) within a small diamond distance using single-qubit measurement data on the output states of [Formula: see text]. We also provide a polynomial-time classical algorithm for learning the description of any unknown [Formula: see text]-qubit state [Formula: see text] prepared by a shallow quantum circuit [Formula: see text] (on a 2D lattice) within a small trace distance using single-qubit measurements on copies of [Formula: see text]. Our approach uses a quantum circuit representation based on local inversions and a technique to combine these inversions. This circuit representation yields an optimization landscape that can be efficiently navigated and enables efficient learning of quantum circuits that are classically hard to simulate.