Parametrized quantum circuits serve as ansatze for solving variational problems and provide a flexible paradigm for the programming of near-term quantum computers.Ideally, such ansatze should be highly expressive, so that a close approximation of the desired solution can be accessed.On the other hand, the ansatz must also have sufficiently large gradients to allow for training.Here, we derive a fundamental relationship between these two essential properties: expressibility and trainability.This is done by extending the well-established barren plateau phenomenon, which holds for ansatze that form exact 2-designs, to arbitrary ansatze.Specifically, we calculate the variance in the cost gradient in terms of the expressibility of the ansatz, as measured by its distance from being a 2-design.Our resulting bounds indicate that highly expressive ansatze exhibit flatter cost landscapes and therefore will be harder to train.Furthermore, we provide numerics illustrating the effect of expressibility on gradient scalings and we discuss the implications for designing strategies to avoid barren plateaus.
Parametrized quantum circuits serve as ansatze for solving variational problems and provide a flexible paradigm for the programming of near-term quantum computers.Ideally, such ansatze should be highly expressive, so that a close approximation of the desired solution can be accessed.On the other hand, the ansatz must also have sufficiently large gradients to allow for training.Here, we derive a fundamental relationship between these two essential properties: expressibility and trainability.This is done by extending the well-established barren plateau phenomenon, which holds for ansatze that form exact 2-designs, to arbitrary ansatze.Specifically, we calculate the variance in the cost gradient in terms of the expressibility of the ansatz, as measured by its distance from being a 2-design.Our resulting bounds indicate that highly expressive ansatze exhibit flatter cost landscapes and therefore will be harder to train.Furthermore, we provide numerics illustrating the effect of expressibility on gradient scalings and we discuss the implications for designing strategies to avoid barren plateaus.
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.
Variational quantum algorithms, which combine highly expressive parameterized quantum circuits (PQCs) and optimization techniques in machine learning, are one of the most promising applications of a near-term quantum computer. Despite their huge potential, the utility of variational quantum algorithms beyond tens of qubits is still questioned. One of the central problems is the trainability of PQCs. The cost function landscape of a randomly initialized PQC is often too flat, asking for an exponential amount of quantum resources to find a solution. This problem, dubbed barren plateaus , has gained lots of attention recently, but a general solution is still not available. In this paper, we solve this problem for the Hamiltonian variational ansatz (HVA), which is widely studied for solving quantum many-body problems. After showing that a circuit described by a time-evolution operator generated by a local Hamiltonian does not have exponentially small gradients, we derive parameter conditions for which the HVA is well approximated by such an operator. Based on this result, we propose an initialization scheme for the variational quantum algorithms and a parameter-constrained ansatz free from barren plateaus.
Parameterized quantum circuits used as variational ansatzes are emerging as promising tools to tackle complex problems ranging from quantum chemistry to combinatorial optimization.These variational quantum circuits can suffer from the well-known curse of barren plateaus, which is characterized by an exponential vanishing of the cost-function gradient with the system size, making training unfeasible for practical applications.Since a generic quantum circuit cannot be simulated efficiently, the determination of its trainability is an important problem.Here we find an efficient method to compute the gradient of the cost function and its variance for a wide class of variational quantum circuits.Our scheme relies on our proof of an exact mapping from randomly initialized circuits to a set of Clifford circuits that can be efficiently simulated on a classical computer by virtue of the celebrated Gottesman-Knill theorem.This method is scalable and can be used to certify trainability for variational quantum circuits and explore design strategies that can overcome the barren-plateau problem.As illustrative examples, we show results with up to 100 qubits.
Variational quantum algorithms (VQAs) optimize the parameters θ of a parametrized quantum circuit V(θ) to minimize a cost function C. While VQAs may enable practical applications of noisy quantum computers, they are nevertheless heuristic methods with unproven scaling. Here, we rigorously prove two results, assuming V(θ) is an alternating layered ansatz composed of blocks forming local 2-designs. Our first result states that defining C in terms of global observables leads to exponentially vanishing gradients (i.e., barren plateaus) even when V(θ) is shallow. Hence, several VQAs in the literature must revise their proposed costs. On the other hand, our second result states that defining C with local observables leads to at worst a polynomially vanishing gradient, so long as the depth of V(θ) is [Formula: see text]. Our results establish a connection between locality and trainability. We illustrate these ideas with large-scale simulations, up to 100 qubits, of a quantum autoencoder implementation.
The barren plateau phenomenon; where cost function gradients vanish exponentially with system size; remains a fundamental obstacle to training variational quantum circuits (VQCs) at scale. We demonstrate, both theoretically and numerically, that embedding partial differential equation (PDE) constraints into the VQC loss function provides a natural and effective mitigation mechanism against barren plateaus. We derive analytical gradient variance lower bounds showing that physics-constrained loss functions composed of local PDE residuals evaluated at spatial collocation points inherit the favorable polynomial scaling of local cost functions, while additionally benefiting from constraint-induced landscape narrowing that concentrates gradient information. Systematic numerical experiments on the one-dimensional heat equation, Burgers' equation, and the Saint-Venant shallow water equations quantify the gradient variance across 4-8 qubits and 1-5 layer depths, comparing global cost, local cost, PDE-constrained, and PDE-constrained with structured ansatz configurations. We find that PDE-constrained circuits exhibit favorable gradient variance scaling with system size, with the physics constraints creating a stabilizing effect that resists exponential gradient vanishing. Entanglement entropy analysis reveals that structured ansatze operate in a sub-maximal entanglement regime consistent with trainability. Convergence experiments confirm that physics-constrained VQCs achieve lower loss values in fewer epochs. These results establish PDE constraints as a principled, physically motivated strategy for designing trainable variational quantum circuits, with direct implications for quantum physics-informed neural networks and variational quantum simulation.
The training of a parameterized model largely depends on the landscape of the underlying loss function. In particular, vanishing gradients are a central bottleneck in the scalability of variational quantum algorithms (VQAs), and are known to arise in various ways. However, a caveat of most existing gradient bound results is the requirement of t-design circuit assumptions that are typically not satisfied in practice. In this work, we loosen these assumptions altogether and derive tight upper and lower bounds on loss and gradient concentration for a large class of parameterized quantum circuits and arbitrary observables, which are significantly stronger than prior work. Moreover, we show that these bounds, as well as the variance of the loss itself, can be estimated efficiently and classically-providing practical tools to study the loss landscapes of VQA models, including verifying whether or not a circuit/observable induces barren plateaus. In particular, our results can readily be leveraged to rule out barren plateaus for a realistic class of ansätze and mixed observables, namely, observables containing a non-vanishing local term. This insight has direct implications for hybrid Quantum Generative Adversarial Networks (qGANs). We prove that designing the discriminator appropriately leads to 1-local weights that stay constant in the number of qubits, regardless of discriminator depth. This implies that qGANs with appropriately chosen generators do not suffer from barren plateaus even at scale-making them a promising candidate for applications in generative quantum machine learning. We demonstrate this result by training a qGAN to learn a 2D mixture of Gaussian distributions with up to 16 qubits, and provide numerical evidence that global contributions to the gradient, while initially exponentially small, may kick in substantially over the course of training.