Abstract In variational quantum algorithms the parameters of a parameterized quantum circuit are optimized in order to minimize a cost function that encodes the solution of the problem. The barren plateau phenomenon manifests as an exponentially vanishing dependence of the cost function with respect to the variational parameters, and thus hampers the optimization process. We discuss how, and in which sense, the phenomenon of noise-induced barren plateaus emerges in parameterized quantum circuits with a layered noise model. Previous results have shown the existence of noise-induced barren plateaus in the presence of local Pauli noise (Wang et al 2021 Nat. Commun. 12 6961). We extend these results analytically to arbitrary completely-positive trace preserving maps in two cases: (1) when a parameter-shift rule holds, (2) when the parameterized quantum circuit at each layer forms a unitary 2-design. The second example shows how highly expressive unitaries give rise not only to standard barren plateaus (McClean et al 2018 Nat. Commun. 9 4812), but also to noise-induced ones. In the second part of the paper, we study numerically the emergence of noise-induced barren plateaus in QAOA circuits focusing on the case of MaxCut problems on d -regular graphs and amplitude damping noise.
Abstract In variational quantum algorithms the parameters of a parameterized quantum circuit are optimized in order to minimize a cost function that encodes the solution of the problem. The barren plateau phenomenon manifests as an exponentially vanishing dependence of the cost function with respect to the variational parameters, and thus hampers the optimization process. We discuss how, and in which sense, the phenomenon of noise-induced barren plateaus emerges in parameterized quantum circuits with a layered noise model. Previous results have shown the existence of noise-induced barren plateaus in the presence of local Pauli noise (Wang et al 2021 Nat. Commun. 12 6961). We extend these results analytically to arbitrary completely-positive trace preserving maps in two cases: (1) when a parameter-shift rule holds, (2) when the parameterized quantum circuit at each layer forms a unitary 2-design. The second example shows how highly expressive unitaries give rise not only to standard barren plateaus (McClean et al 2018 Nat. Commun. 9 4812), but also to noise-induced ones. In the second part of the paper, we study numerically the emergence of noise-induced barren plateaus in QAOA circuits focusing on the case of MaxCut problems on d -regular graphs and amplitude damping noise.
Variational quantum algorithms (VQAs) hold much promise but face the challenge of exponentially small gradients. Unmitigated, this barren plateau (BP) phenomenon leads to an exponential training overhead for VQAs. Perhaps the most pernicious are noise-induced barren plateaus (NIBPs), a type of unavoidable BP arising from open system effects, which have so far been shown to exist for unital noise maps. Here, we generalize the study of NIBPs to more general completely positive, trace-preserving maps, investigating the existence of NIBPs in the unital case and a class of non-unital maps we call Hilbert-Schmidt (HS)-contractive. The latter includes amplitude damping. We identify the associated phenomenon of noise-induced limit sets (NILS) of the VQA cost function and prove its existence for both unital and HS-contractive non-unital noise maps. Along the way, we extend the parameter shift rule of VQAs to the noisy setting. We provide rigorous bounds in terms of the relevant variables that give rise to NIBPs and NILSs, along with numerical simulations of the depolarizing and amplitude-damping maps that illustrate our analytical results.
Variational Quantum Algorithms (VQAs) may be a path to quantum advantage on Noisy Intermediate-Scale Quantum (NISQ) computers. A natural question is whether noise on NISQ devices places fundamental limitations on VQA performance. We rigorously prove a serious limitation for noisy VQAs, in that the noise causes the training landscape to have a barren plateau (i.e., vanishing gradient). Specifically, for the local Pauli noise considered, we prove that the gradient vanishes exponentially in the number of qubits n if the depth of the ansatz grows linearly with n. These noise-induced barren plateaus (NIBPs) are conceptually different from noise-free barren plateaus, which are linked to random parameter initialization. Our result is formulated for a generic ansatz that includes as special cases the Quantum Alternating Operator Ansatz and the Unitary Coupled Cluster Ansatz, among others. For the former, our numerical heuristics demonstrate the NIBP phenomenon for a realistic hardware noise model.
Abstract Variational quantum algorithms rely on gradient based optimization to iteratively minimize a cost function evaluated by measuring output(s) of a quantum processor. A barren plateau is the phenomenon of exponentially vanishing gradients in sufficiently expressive parametrized quantum circuits. It has been established that the onset of a barren plateau regime depends on the cost function, although the particular behavior has been demonstrated only for certain classes of cost functions. Here we derive a lower bound on the variance of the gradient, which depends mainly on the width of the circuit causal cone of each term in the Pauli decomposition of the cost function. Our result further clarifies the conditions under which barren plateaus can occur.
Variational quantum algorithms (VQAs) are a promising paradigm for quantum advantage, yet their trainability is severely hampered by barren plateaus (BPs). Several recent works have proposed dynamic parameterized quantum circuits (DPQCs), which interleave unitary layers with parameterized CPTP maps, such as engineered dissipation, feedforward gadgets, and periodic resets, as a possible strategy for mitigating BPs. We unify this class of circuits into a formalization for DPQCs.We identify constraints on the nature and the structure of DPQCs if they are to prevent a significant number of parameters from becoming untrainable. Using purification and Pauli-path analysis, we further identify a mechanism by which the cost function can remain anti-concentrated even when many parameters remain untrainable. Our analysis reveals ways to design DPQCs that do not have an exponentially concentrated cost function, and our results suggest that BP mitigation via DPQCs is at least as hard as designing BP-free unitaries.
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.