Two main challenges preventing efficient training of variational quantum algorithms and quantum machine learning models are local minima and barren plateaus. Typically, barren plateaus are associated with deep circuits, while shallow circuits have been shown to suffer from suboptimal local minima. We point out a simple mechanism that creates exponentially many low-quality local minima specifically in the barren plateau regime. These local minima are trivial solutions that optimize only a few terms in the loss function, leaving the rest on their barren plateaus. More precisely, we show the existence of approximate local minima, optimizing a single loss term, and conjecture the existence of exact local minima, optimizing only a logarithmic fraction of all loss function terms. One implication of our findings is that many, if not most, regions with large gradients are vicinities of poor local minima. This highlights additional challenges in finding efficient initialization strategies.
Two main challenges preventing efficient training of variational quantum algorithms and quantum machine learning models are local minima and barren plateaus. Typically, barren plateaus are associated with deep circuits, while shallow circuits have been shown to suffer from suboptimal local minima. We point out a simple mechanism that creates exponentially many low-quality local minima specifically in the barren plateau regime. These local minima are trivial solutions that optimize only a few terms in the loss function, leaving the rest on their barren plateaus. More precisely, we show the existence of approximate local minima, optimizing a single loss term, and conjecture the existence of exact local minima, optimizing only a logarithmic fraction of all loss function terms. One implication of our findings is that many, if not most, regions with large gradients are vicinities of poor local minima. This highlights additional challenges in finding efficient initialization strategies.
One of the most important properties of classical neural networks is how surprisingly trainable they are, though their training algorithms typically rely on optimizing complicated, nonconvex loss functions. Previous results have shown that unlike the case in classical neural networks, variational quantum models are often not trainable. The most studied phenomenon is the onset of barren plateaus in the training landscape of these quantum models, typically when the models are very deep. This focus on barren plateaus has made the phenomenon almost synonymous with the trainability of quantum models. Here, we show that barren plateaus are only a part of the story. We prove that a wide class of variational quantum models-which are shallow, and exhibit no barren plateaus-have only a superpolynomially small fraction of local minima within any constant energy from the global minimum, rendering these models untrainable if no good initial guess of the optimal parameters is known. We also study the trainability of variational quantum algorithms from a statistical query framework, and show that noisy optimization of a wide variety of quantum models is impossible with a sub-exponential number of queries. Finally, we numerically confirm our results on a variety of problem instances. Though we exclude a wide variety of quantum algorithms here, we give reason for optimism for certain classes of variational algorithms and discuss potential ways forward in showing the practical utility of such algorithms.
Abstract We define laziness to describe a large suppression of variational parameter updates for neural networks, classical or quantum. In the quantum case, the suppression is exponential in the number of qubits for randomized variational quantum circuits. We discuss the difference between laziness and barren plateau in quantum machine learning created by quantum physicists in McClean et al (2018 Nat. Commun. 9 1–6) for the flatness of the loss function landscape during gradient descent. We address a novel theoretical understanding of those two phenomena in light of the theory of neural tangent kernels. For noiseless quantum circuits, without the measurement noise, the loss function landscape is complicated in the overparametrized regime with a large number of trainable variational angles. Instead, around a random starting point in optimization, there are large numbers of local minima that are good enough and could minimize the mean square loss function, where we still have quantum laziness, but we do not have barren plateaus. However, the complicated landscape is not visible within a limited number of iterations, and low precision in quantum control and quantum sensing. Moreover, we look at the effect of noises during optimization by assuming intuitive noise models, and show that variational quantum algorithms are noise-resilient in the overparametrization regime. Our work precisely reformulates the quantum barren plateau statement towards a precision statement and justifies the statement in certain noise models, injects new hope toward near-term variational quantum algorithms, and provides theoretical connections toward classical machine learning. Our paper provides conceptual perspectives about quantum barren plateaus, together with discussions about the gradient descent dynamics in Liu et al (2023 Phys. Rev. Lett. 130 150601).
Barren plateaus-regions of exponentially small gradients-are a major obstacle for training variational quantum algorithms on non-trivial problem sizes. Existing results analyze specific causes in isolation, such as random deep circuits or global cost functions, but lack a unified picture that integrates circuit design, data encoding, cost structure, and noise. In this work, we develop a comprehensive theory of barren plateaus for parametrized quantum circuits. We prove that three mechanisms jointly control gradient scaling: (1) ansatz-induced concentration, where highly expressive or sufficiently deep circuits approximate unitary 2-designs causing gradient variance to scale as O(2 −n); (2) data-encoding-induced concentration, where certain feature maps wash out class-conditional structure; and (3) probleminduced concentration, where cost functions effectively average over exponentially many degrees of freedom. For each mechanism, we derive explicit scaling laws for the gradient variance as a function of qubit count n, depth L, and encoding complexity. We show that appropriately constrained expressibility, locality, and symmetry breaking can prevent gradients from vanishing exponentially and obtain conditions under which gradients remain at worst polynomially small. This yields architectural and initialization design principles for plateau-resistant circuits, including shallow, problem-informed ansätze, layer-wise training, and structured data encodings. Numerical experiments on up to 100 qubits for VQE, QAOA, and quantum classifiers confirm our theoretical predictions and demonstrate 10-100× improvements in effective gradient magnitudes over standard architectures. Our framework subsumes existing barren plateau results and provides a practical toolkit for diagnosing and mitigating trainability issues in variational quantum algorithms.
Barren plateaus are a notorious problem in the optimization of variational quantum algorithms and pose a critical obstacle in the quest for more efficient quantum machine learning algorithms. Many potential reasons for barren plateaus have been identified but few solutions have been proposed to avoid them in practice. Existing solutions are mainly focused on the initialization of unitary gate parameters without taking into account the changes induced by input data. In this paper, we propose an alternative strategy which initializes the parameters of a unitary gate by drawing from a beta distribution. The hyperparameters of the beta distribution are estimated from the data. To further prevent barren plateau during training we add a novel perturbation at every gradient descent step. Taking these ideas together, we empirically show that our proposed framework significantly reduces the possibility of a complex quantum neural network getting stuck in a barren plateau.
Barren Plateaus are a formidable challenge for hybrid quantum-classical algorithms that lead to flat plateaus in the loss function landscape making it difficult to take advantage of the expressive power of parameterized quantum circuits with gradient-based methods. Like in classical neural network models, parameterized quantum circuits suffer the same vanishing gradient issue due to large parameter spaces with non-convex landscapes. In this review, we present an overview of the different genesis for barren plateaus, mathematical formalisms of common themes around barren plateaus, and dives into gradients. The central objective is to provide a conceptual perspective between classical and quantum interpretations of vanishing gradients as well as dive into techniques involving cost functions, entanglement, and initialization strategies to mitigate barren plateaus. Addressing barren plateaus paves the way towards feasibility of many classically intractable applications for quantum simulation, optimization, chemistry, and quantum machine learning.
The barren plateau phenomenon, characterized by loss gradients that vanish exponentially with system size, poses a challenge to scaling variational quantum algorithms. Here we explore the potential of warm starts, whereby one initializes closer to a solution in the hope of enjoying larger loss variances. Focusing on an iterative variational method for learning shorter-depth circuits for quantum we conduct a case study to elucidate the potential and limitations of warm starts. We start by proving that the iterative variational algorithm will exhibit substantial (at worst vanishing polynomially in system size) gradients in a small region around the initializations at each time step. Convexity guarantees for these regions are then established, suggesting trainability for polynomial-size time steps. However, our study highlights scenarios where a good minimum shifts outside the region with trainability guarantees. Our analysis leaves open the question whether such minima jumps necessitate optimization across barren plateau landscapes or whether there exist gradient flows, i.e., fertile valleys away from the plateau with substantial gradients, that allow for training. While our main focus is on this case study of variational quantum simulation, we end by discussing how our results work in other iterative settings.