Without large quantum computers to empirically evaluate performance, theoretical frameworks such as the quantum statistical query (QSQ) are a primary tool to study quantum algorithms for learning classical functions and search for quantum advantage in machine learning tasks. However, we only understand quantum advantage in this model at two extremes: either exponential advantages for uniform input distributions or no advantage for arbitrary distributions. Our work helps close the gap between these two regimes by designing an efficient quantum algorithm for learning periodic neurons in the QSQ model over a variety of non-uniform distributions and the first explicit treatment of real-valued functions. We prove that this problem is hard not only for classical gradient-based algorithms, which are the workhorses of machine learning, but also for a more general class of SQ algorithms, establishing an exponential quantum advantage.
Without large quantum computers to empirically evaluate performance, theoretical frameworks such as the quantum statistical query (QSQ) are a primary tool to study quantum algorithms for learning classical functions and search for quantum advantage in machine learning tasks. However, we only understand quantum advantage in this model at two extremes: either exponential advantages for uniform input distributions or no advantage for arbitrary distributions. Our work helps close the gap between these two regimes by designing an efficient quantum algorithm for learning periodic neurons in the QSQ model over a variety of non-uniform distributions and the first explicit treatment of real-valued functions. We prove that this problem is hard not only for classical gradient-based algorithms, which are the workhorses of machine learning, but also for a more general class of SQ algorithms, establishing an exponential quantum advantage.
An overarching milestone of quantum machine learning (QML) is to demonstrate the advantage of QML over all possible classical learning methods in accelerating a common type of learning task, as represented by supervised learning from classical data. However, the provable advantages of QML in supervised learning have been known so far only for the learning tasks designed for using the advantage of specific quantum algorithms, i.e., Shor’s algorithms. Here, we explicitly construct an unprecedentedly broader family of supervised learning tasks with classical data to offer the provable advantage of QML based on general quantum computational advantages, progressing beyond Shor’s algorithms. Our learning task is feasibly achievable by executing a general class of functions that can be computed efficiently in polynomial time for a large fraction of inputs by arbitrary quantum algorithms, but not by any classical algorithm. We prove the hardness of achieving this learning task for any possible polynomial-time classical learning method. We also clarify protocols for preparing the classical data to demonstrate this learning task in experiments. These results provide a way to exploit a variety of quantum advantages in computing functions for the realization of provably advantageous QML.
It has been shown that the apparent advantage of some quantum machine learning algorithms may be efficiently replicated using classical algorithms with suitable data access -- a process known as dequantization. Existing works on dequantization compare quantum algorithms which take copies of an n-qubit quantum state ∣x⟩=∑ixi∣i⟩ as input to classical algorithms which have sample and query (SQ) access to the vector x. In this note, we prove that classical algorithms with SQ access can accomplish some learning tasks exponentially faster than quantum algorithms with quantum state inputs. Because classical algorithms are a subset of quantum algorithms, this demonstrates that SQ access can sometimes be significantly more powerful than quantum state inputs. Our findings suggest that the absence of exponential quantum advantage in some learning tasks may be due to SQ access being too powerful relative to quantum state inputs. If we compare quantum algorithms with quantum state inputs to classical algorithms with access to measurement data on quantum states, the landscape of quantum advantage can be dramatically different. We remark that when the quantum states are constructed from exponential-size classical data, comparing SQ access and quantum state inputs is appropriate since both require exponential time to prepare.
Symmetry underlies many of the most effective classical and quantum learning algorithms, yet whether quantum learners can gain a fundamental advantage under symmetry-imposed structures remains an open question. Based on evidence that classical statistical query ($\SQ$) frameworks have revealed exponential query complexity in learning symmetric function classes, we ask: can quantum learning algorithms exploit the problem symmetry better? In this work, we investigate the potential benefits of symmetry within the quantum statistical query ($\QSQ$) model, which is a natural quantum analog of classical $\SQ$. Our results uncover three distinct phenomena: (i) we obtain an exponential separation between $\QSQ$ and $\SQ$ on a permutation-invariant function class; (ii) we establish query complexity lower bounds for $\QSQ$ learning that match, up to constant factors, the corresponding classical $\SQ$ lower bounds for most commonly studied symmetries; however, the potential advantages may occur under highly skewed orbit distributions; and (iii) we further identify a tolerance-based separation exists, where quantum learners succeed at noise levels that render classical $\SQ$ algorithms ineffective. Together, these findings provide insight into when symmetry can enable quantum advantage in learning.
One of the key challenges in quantum machine learning is finding relevant machine learning tasks with a provable quantum advantage. A natural candidate for this is learning unknown Hamiltonian dynamics. Here, we tackle the supervised learning version of this problem, where we are given random examples of the inputs to the dynamics as classical data, paired with the expectation values of some observable after the time evolution, as corresponding output labels. The task is to replicate the corresponding input-output function. We prove that this task can yield provable exponential classical-quantum learning advantages under common complexity assumptions in natural settings. To design our quantum learning algorithms, we introduce a new method, which we term \textit{\subroutine}~algorithm for parametrized circuit functions, and which may be of independent interest. Furthermore, we discuss the limitations of generalizing our method to arbitrary quantum dynamics while maintaining provable guarantees. We explain that significant generalizations are impossible under certain complexity-theoretic assumptions, but nonetheless, we provide a heuristic kernel method, where we trade-off provable correctness for broader applicability.
Abstract Quantum neural network (QNN), or equivalently, the variational quantum circuits with a gradient-based classical optimizer, has been broadly applied to many experimental proposals for noisy intermediate scale quantum (NISQ) devices. However, the learning capability of QNN remains largely unknown due to the non-convex optimization landscape, the measurement error, and the unavoidable gate noise introduced by NISQ machines. In this study, we theoretically explore the learnability of QNN from the perspective of the trainability and generalization. Particularly, we derive the convergence performance of QNN under the NISQ setting, and identify classes of computationally hard concepts that can be efficiently learned by QNN. Our results demonstrate that large gate noise, few quantum measurements, and deep circuit depth will lead to poor convergence rates of QNN towards the empirical risk minimization. Moreover, we prove that any concept class, which is efficiently learnable by a restricted quantum statistical query (QSQ) learning model, can also be efficiently learned by QNN. Since the restricted QSQ learning model can tackle certain problems such as parity learning with a runtime speedup, our result suggests that QNN established on NISQ devices will retain the quantum advantage. Our work provides the theoretical guidance for developing advanced QNNs and opens up avenues for exploring quantum advantages using NISQ devices.
We consider the learnability of the quantum neural network (QNN) built on the variational hybrid quantum-classical scheme, which remains largely unknown due to the non-convex optimization landscape, the measurement error, and the unavoidable gate errors introduced by noisy intermediate-scale quantum (NISQ) machines. Our contributions in this paper are multi-fold. First, we derive the utility bounds of QNN towards empirical risk minimization, and show that large gate noise, few quantum measurements, and deep circuit depth will lead to the poor utility bounds. This result also applies to the variational quantum circuits with gradient-based classical optimization, and can be of independent interest. We then prove that QNN can be treated as a differentially private (DP) model. Thirdly, we show that if a concept class can be efficiently learned by QNN, then it can also be effectively learned by QNN even with gate noise. This result implies the same learnability of QNN whether it is implemented on noiseless or noisy quantum machines. We last exhibit that the quantum statistical query (QSQ) model can be effectively simulated by noisy QNN. Since the QSQ model can tackle certain tasks with runtime speedup, our result suggests that the modified QNN implemented on NISQ devices will retain the quantum advantage. Numerical simulations support the theoretical results.