Sample complexity of learning parametric quantum circuits

Abstract

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 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 $\tilde{O}(n^{c+1})$. 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 practice.

Generalization in quantum machine learning from few training data

“…Next, we give a comparison to previously known results. Some prior works have studied the generalization capabilities of quantum models, among them the classical learning-theoretic approaches of [81][82][83][84][85][86][87][88][89] ; the more geometric perspective of 17,18 ; and the informationtheoretic technique of 20,37 . Independently of this work, Ref.…”

Out-of-distribution generalization for learning quantum dynamics

“…Thus, when training using training data D Q (N ), the out-of-distribution risk R P (α opt ) of the optimized parameters α opt after training is controlled in terms of the optimized training cost C D Q (N ) (α opt ) and the indistribution generalization error gen Q,D Q (N ) (α opt ). We can now bound the in-distribution generalization error using already known QML in-distribution generalization bounds [11][12][13][14][15][16][17][18][19][20][21][22][23] (or, indeed, any such bounds that are derived in the future). As a concrete example of guarantees that can be obtained this way, we combine Corollary 1 with an in-distribution generalization bound established in [20] to prove:…”

Expressivity of Variational Quantum Machine Learning on the Boolean Cube

“…As an example of the potential applicability of VQML, VOLUME 4, 2016 it has been observed that these models can be used to solve a variety of financial problems, such as fraud detection and creditworthiness determination [24]- [26]. There has been an active line of studies to characterize the expressivity [27]- [29], the generalizability [30]- [38], and the trainability [39]- [41] of VQML models. However, the specific case of quantum models with discrete-valued inputs has not been investigated as extensively [42], [43].…”

“…In Appendix A-B, we present a concrete example of the unitary operators in Equation ( 54) that produces a linear quantum model on a single qubit whose output is expressible as a nontrivial linear combination of all Fourier basis elements for B 3 . This alternative operator, in the case where V k are not trainable, could be used in place of U 3,1 in Equation (38) in the multiqubit case. However, the degree of freedom that the trainable part of the model, O θ , has in choosing the coefficients of the χ P is limited when compared to the ensemble approach.…”