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.

arXiv:2101.02138PRX QuantumJan 24, 2022571 citations

Experimental demonstration of the absence of noise-induced barren plateaus using information content landscape analysis

Sebastian Schmitt, Linus Ekstrøm, Alberto Bottarelli +1 more

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 $T_1$ -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 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 $T_1$-dominated non-unital noise inhibits the emergence of NIBP. Our analysis suggest that average calibration metrics may be insufficient to predict variational algorithm performance.

quant-pharXiv:2602.22851Feb 27, 20260 citations

Beyond unital noise in variational quantum algorithms: noise-induced barren plateaus and limit sets

Phattharaporn Singkanipa, Daniel A. Lidar

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) 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.

arXiv:2402.08721QuantumJan 30, 202517 citations

A milestone in quantum computing

Stephen D. Bartlett

NatureAug 3, 20164 citations

T_1

-dominated non-unital noise inhibits the emergence of NIBP. Our analysis suggest that average calibration metrics may be insufficient to predict variational algorithm performance.