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arXiv:2403.13927 · Nature Physics

Noise-induced shallow circuits and the absence of barren plateaus

Antonio Anna Mele, Armando Angrisani, Soumik Ghosh, Sumeet Khatri, Jens Eisert, Daniel Stilck França, Yihui Quek

Published Apr 2, 2026• 12 citations
arXivHTMLPDF

Abstract

Without a successful implementation of fault-tolerant quantum error correction, calculations on quantum computers are subject to noise that limits their capabilities. Here, motivated by realistic near-term hardware considerations, we study the impact of uncorrected local noise on logical quantum circuits. We first show that, in the task of estimating observable expectation values, any noise effectively truncates most quantum circuits to logarithmic depth. We then prove that quantum circuits under any non-unital noise do not exhibit barren plateaus for cost functions composed of local observables. However, by using the effective shallowness, we also design an efficient classical algorithm to estimate observable expectation values within any constant additive accuracy, with high probability over the choice of the circuit, in any circuit architecture. Taken together, our results establish that, unless we carefully engineer quantum circuits to take advantage of the noise, noisy quantum circuits are unlikely to offer an advantage over shallow ones for algorithms that output observable expectation value estimates, such as many variational quantum machine learning proposals.

DOI
10.1038/s41567-026-03245-z

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Without a successful implementation of fault-tolerant quantum error correction, calculations on quantum computers are subject to noise that limits their capabilities. Here, motivated by realistic near-term hardware considerations, we study the impact of uncorrected local noise on logical quantum circuits. We first show that, in the task of estimating observable expectation values, any noise effectively truncates most quantum circuits to logarithmic depth. We then prove that quantum circuits under any non-unital noise do not exhibit barren plateaus for cost functions composed of local observables. However, by using the effective shallowness, we also design an efficient classical algorithm to estimate observable expectation values within any constant additive accuracy, with high probability over the choice of the circuit, in any circuit architecture. Taken together, our results establish that, unless we carefully engineer quantum circuits to take advantage of the noise, noisy quantum circuits are unlikely to offer an advantage over shallow ones for algorithms that output observable expectation value estimates, such as many variational quantum machine learning proposals.
Quantum error mitigation has been proposed as a means to combat unwanted and unavoidable errors in near-term quantum computing without the heavy resource overheads required by fault-tolerant schemes. Recently, error mitigation has been successfully applied to reduce noise in near-term applications. In this work, however, we identify strong limitations to the degree to which quantum noise can be effectively 'undone' for larger system sizes. Our framework rigorously captures large classes of error-mitigation schemes in use today. By relating error mitigation to a statistical inference problem, we show that even at shallow circuit depths comparable to those of current experiments, a superpolynomial number of samples is needed in the worst case to estimate the expectation values of noiseless observables, the principal task of error mitigation. Notably, our construction implies that scrambling due to noise can kick in at exponentially smaller depths than previously thought. Noise also impacts other near-term applications by constraining kernel estimation in quantum machine learning, causing an earlier emergence of noise-induced barren plateaus in variational quantum algorithms and ruling out exponential quantum speed-ups in estimating expectation values in the presence of noise or preparing the ground state of a Hamiltonian.
We present a polynomial-time classical algorithm for estimating expectation values of arbitrary observables on typical quantum circuits under any incoherent local noise, including non-unital or dephasing. Although previous research demonstrated that some carefully designed quantum circuits affected by non-unital noise cannot be efficiently simulated, we show that this does not apply to average-case circuits, as these can be efficiently simulated using Pauli-path methods. Specifically, we prove that, with high probability over the circuit gates’ choice, Pauli propagation algorithms with tailored truncation strategies achieve an inversely polynomially small simulation error. This result holds for arbitrary circuit topologies and for any local noise, under the assumption that the distribution of each circuit layer is invariant under single-qubit random gates. Under the same minimal assumptions, we also prove that most noisy circuits can be truncated to an effective logarithmic depth for the task of estimating expectation values of observables, thus generalizing prior results to a significantly broader class of circuit ensembles. We further numerically validate our algorithm with simulations on a 6 × 6 lattice of qubits under the effects of amplitude damping and dephasing noise, as well as real-time dynamics on an 11 × 11 lattice of qubits affected by amplitude damping.
Quantum error mitigation has been proposed as a means to combat unwanted and unavoidable errors in near-term quantum computing without the heavy resource overheads required by fault tolerant schemes. Recently, error mitigation has been successfully applied to reduce noise in near-term applications. In this work, however, we identify strong limitations to the degree to which quantum noise can be effectively `undone' for larger system sizes. Our framework rigorously captures large classes of error mitigation schemes in use today. By relating error mitigation to a statistical inference problem, we show that even at shallow circuit depths comparable to the current experiments, a superpolynomial number of samples is needed in the worst case to estimate the expectation values of noiseless observables, the principal task of error mitigation. Notably, our construction implies that scrambling due to noise can kick in at exponentially smaller depths than previously thought. They also impact other near-term applications, constraining kernel estimation in quantum machine learning, causing an earlier emergence of noise-induced barren plateaus in variational quantum algorithms and ruling out exponential quantum speed-ups in estimating expectation values in the presence of noise or preparing the ground state of a Hamiltonian.
Variational quantum algorithms are promising candidates for near-term quantum computing but can be hindered by barren plateaus, where gradients vanish exponentially and optimization becomes intractable. Noise-Induced Barren Plateaus (NIBP) are particularly concerning because they are predicted to arise generically from noise accumulation, independent of system size, circuit structure, and observable locality. We experimentally investigate NIBP on IBM quantum hardware. Using Information Content Landscape Analysis (ICLA), we efficiently estimate gradient norms for variational circuits ranging from 8 to 102 qubits, up to hundreds of parameters and circuit runtimes of hundreds of microseconds. Contrary to NIBP expectations, we observe that gradient magnitudes saturate beyond a characteristic runtime rather than decaying exponentially. Classical simulations of the 8-qubit case under noiseless, depolarizing, amplitude-damping, and dephasing noise models support this behavior. Consistent with recent theory, our results show that T1T_1T1​-dominated non-unital noise inhibits the emergence of NIBP. Our analysis suggest that average calibration metrics may be insufficient to predict variational algorithm performance.
Variational quantum algorithms (VQAs) optimize the parameters θ\boldsymbol{\theta}θ of a quantum neural network V(θ)V(\boldsymbol{\theta})V(θ) to minimize a cost function CCC. 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)V(\boldsymbol{\theta})V(θ) is a hardware-efficient ansatz composed of blocks forming local 2-designs. Our first result states that defining CCC in terms of global observables leads to an exponentially vanishing gradient (i.e., a barren plateau) even when V(θ)V(\boldsymbol{\theta})V(θ) is shallow. This implies that several VQAs in the literature must revise their proposed cost functions. On the other hand, our second result states that defining CCC with local observables leads to at worst a polynomially vanishing gradient, so long as the depth of V(θ)V(\boldsymbol{\theta})V(θ) is O(log⁡n)\mathcal{O}(\log n)O(logn). Taken together, our results establish a connection between locality and trainability. Finally, we illustrate these ideas with large-scale simulations, up to 100 qubits, of a particular VQA known as quantum autoencoders.
Abstract Although quantum computers are believed to be more powerful than classical ones, a convincing experimental demonstration of this fact remains elusive. Proposed schemes either rely on unproven complexity-theoretic hardness assumptions, and/or require universal, fault-tolerant scalable quantum computers to implement. This has motivated the study of restricted models of computation. Constant-depth quantum circuits are known to be more powerful than unbounded fan-in classical ( AC0{{\mathsf{AC}}}^{0}AC0 AC 0 -)circuits. Here we ask if this advantage persists in the presence of noise and under locality constraints. We present a computational problem for which every instance can be solved with near-certainty, despite noise, by a constant-depth quantum circuit with local operations in 3D. In contrast, every AC0{{\mathsf{AC}}}^{0}AC0 AC 0 -circuit of size smaller than a certain (sub)exponential fails with near-certainty on a uniformly random instance. This constitutes a proposal with built-in fault-tolerance to experimentally observe the strongest known complexity-theoretic separation between classical and quantum computation.