Quantum computers hold unprecedented potentials for machine learning applications. Here, we prove that physical quantum circuits are PAC (probably approximately correct) learnable on a quantum computer via empirical risk minimization: to learn a quantum circuit with at most $n^c$ gates and each gate acting on a constant number of qubits, the sample complexity is bounded by $\tilde{O}(n^{c+1})$. In particular, we explicitly construct a family of variational quantum circuits with $O(n^{c+1})$ elementary gates arranged in a fixed pattern, which can represent all physical quantum circuits consisting of at most $n^c$ elementary gates. Our results provide a valuable guide for quantum machine learning in both theory and experiment.
Quantum computers hold unprecedented potentials for machine learning applications. Here, we prove that physical quantum circuits are PAC (probably approximately correct) learnable on a quantum computer via empirical risk minimization: to learn a quantum circuit with at most nc gates and each gate acting on a constant number of qubits, the sample complexity is bounded by O~(nc+1). In particular, we explicitly construct a family of variational quantum circuits with O(nc+1) elementary gates arranged in a fixed pattern, which can represent all physical quantum circuits consisting of at most nc elementary gates. Our results provide a valuable guide for quantum machine learning in both theory and experiment.
Abstract Quantum computers hold unprecedented potentials for machine learning applications. Here, we prove that physical quantum circuits are probably approximately correct learnable on a quantum computer via empirical risk minimization: to learn a parametric quantum circuit with at most n c gates and each gate acting on a constant number of qubits, the sample complexity is bounded by O ~ ( n c + 1 ) . In particular, we explicitly construct a family of variational quantum circuits with O(n c+1) elementary gates arranged in a fixed pattern, which can represent all physical quantum circuits consisting of at most n c elementary gates. Our results provide a valuable guide for quantum machine learning in both theory and practice.
A central principle in quantum machine learning is that an ansatz should be expressive enough to represent the quantum data of interest. Yet, the expressibility is statistically meaningful only insofar as it can be learned from finitely many copies of an unknown quantum state. In this work, we develop an information-theoretic Occam theory for quantum data generated by finite-size quantum circuits. For the class Sn,G of n-qubit pure states preparable with at most G two-qubit gates, a metric-entropy argument gives the realizable sample law Θ(G/ε2) in the circuit-limited regime. For an arbitrary source ρ^, we introduce the best G-gate approximation error dG(ρ^) and the approximate circuit complexity Cη(ρ^). We prove an agnostic quantum Occam theorem: with M copies, one can learn up to the best G-gate approximation error plus a statistical penalty O(G/M). We then remove the need to know G in advance through an adaptive model-selection theorem whose oracle inequality selects the circuit complexity justified by the data. Matching lower bounds yield a sample-supported expressibility law: at trace-distance accuracy ε, M samples can support only Gsupported≃Mε2 gates, up to logarithmic factors and tomography saturation at 2n. Thus, the circuit complexity becomes an adaptive statistical resource rather than a static promise. Our framework turns bounded circuit complexity into a model-selection principle for quantum machine learning.
In this paper, we study the quantum learnability of constant-depth classical circuits under the uniform distribution and in the distribution-independent framework of probably approximately correct (PAC) learning. In order to attain our results, we establish connections between quantum learning and quantum-secure cryptosystems. We then achieve the following results. 1. Hardness of PAC learning AC0 and TC0 under the uniform distribution. Our first result concerns the concept class TC0 (resp., AC0), the class of constant-depth, polynomial-sized circuits with unbounded fan-in majority gates (resp., AND,OR,NOT gates). We show the following: if there exists no quantum (quasi-)polynomial-time algorithm to solve the ring-learning with errors (RLWE) problem, then there exists no (quasi-)polynomial-time quantum learning algorithm for TC0; and if there exists no 2O(d1/η)-time quantum algorithm to solve RLWE with dimension d=O(polylogn) (for every constant η>2), then there exists no O(nlogνn)-time quantum learning algorithm for poly(n)-sized AC0 circuits (for a constant ν>0), matching the classical upper bound of Linial, Mansour and Nisan [J. ACM, 40 (1993), pp. 607--620], where the learning algorithms are under the uniform distribution (even with access to quantum membership queries). The main technique in these results uses an explicit family of pseudorandom functions that are believed to be quantum-secure to construct concept classes that are hard to learn quantumly under the uniform distribution. 2. Hardness of learning TC20 in the PAC setting. Our second result shows that if there exists no quantum polynomial-time algorithm for the LWE problem, then there exists no polynomial-time quantum-PAC learning algorithm for the class TC20, i.e., depth-2 TC0 circuits. The main technique in this result is to establish a connection between the quantum security of public-key encryption schemes and the learnability of a concept class that consists of decryption functions of the cryptosystem. Our results show that quantum resources do not give an exponential improvement to learning constant-depth polynomial-sized neural networks. This also gives a strong (conditional) negative answer to one of the “Ten Semi-Grand Challenges for Quantum Computing Theory" raised by Aaronson https://www.scottaaronson.com/writings/qchallenge.html, 2005.
Given a quantum circuit, a quantum computer can sample the output distribution exponentially faster in the number of bits than classical computers. A similar exponential separation has yet to be established in generative models through quantum sample learning: given samples from an n-qubit computation, can we learn the underlying quantum distribution using models with training parameters that scale polynomial in n under a fixed training time? We study four kinds of generative models: Deep Boltzmann machine (DBM), Generative Adversarial Networks (GANs), Long Short-Term Memory (LSTM) and Autoregressive GAN, on learning quantum data set generated by deep random circuits. We demonstrate the leading performance of LSTM in learning quantum samples, and thus the autoregressive structure present in the underlying quantum distribution from random quantum circuits. Both numerical experiments and a theoretical proof in the case of the DBM show exponentially growing complexity of learning-agent parameters required for achieving a fixed accuracy as n increases. Finally, we establish a connection between learnability and the complexity of generative models by benchmarking learnability against different sets of samples drawn from probability distributions of variable degrees of complexities in their quantum and classical representations.
We generalize the PAC (probably approximately correct) learning model to the quantum world by generalizing the concepts from classical functions to quantum processes, defining the problem of \emph{PAC learning quantum process}, and study its sample complexity. In the problem of PAC learning quantum process, we want to learn an ϵ-approximate of an unknown quantum process c∗ from a known finite concept class C with probability 1−δ using samples {(x1,c∗(x1)),(x2,c∗(x2)),…}, where {x1,x2,…} are computational basis states sampled from an unknown distribution D and {c∗(x1),c∗(x2),…} are the (possibly mixed) quantum states outputted by c∗. The special case of PAC-learning quantum process under constant input reduces to a natural problem which we named as approximate state discrimination, where we are given copies of an unknown quantum state c∗ from an known finite set C, and we want to learn with probability 1−δ an ϵ-approximate of c∗ with as few copies of c∗ as possible. We show that the problem of PAC learning quantum process can be solved with O(ϵ2log∣C∣+log(1/δ)) samples when the outputs are pure states and O(ϵ2log3∣C∣(log∣C∣+log(1/δ))) samples if the outputs can be mixed. Some implications of our results are that we can PAC-learn a polynomial sized quantum circuit in polynomial samples and approximate state discrimination can be solved in polynomial samples even when concept class size ∣C∣ is exponential in the number of qubits, an exponentially improvement over a full state tomography.
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