Barren Plateaus Preclude Learning Scramblers

Holmes, Zoë; Arrasmith, Andrew; Yan, Bin; Coles, Patrick J.; Albrecht, Andreas; Sornborger, Andrew

doi:10.1103/physrevlett.126.190501

Cited by 93 publications

(62 citation statements)

References 42 publications

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“…Gradient scaling is one of the few directions of significant progress. The most famous gradient scaling result is the barren plateau phenomenon [19][20][21][22][23][24][25][26][27][28][29][30][31][32][33], whereby the gradient of the cost function shrinks exponentially with the number of qubits. Various issues lead to barren plateaus, such as deep ansatzes that lack structure [19,21,31], global cost functions [20,21], high levels of noise [22,32], scrambling target unitaries [24], and large entanglement [28,29].…”

Section: Introductionmentioning

confidence: 99%

Effect of barren plateaus on gradient-free optimization

Arrasmith

Cerezo

Czarnik

et al. 2021

Quantum

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160

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Barren plateau landscapes correspond to gradients that vanish exponentially in the number of qubits. Such landscapes have been demonstrated for variational quantum algorithms and quantum neural networks with either deep circuits or global cost functions. For obvious reasons, it is expected that gradient-based optimizers will be significantly affected by barren plateaus. However, whether or not gradient-free optimizers are impacted is a topic of debate, with some arguing that gradient-free approaches are unaffected by barren plateaus. Here we show that, indeed, gradient-free optimizers do not solve the barren plateau problem. Our main result proves that cost function differences, which are the basis for making decisions in a gradient-free optimization, are exponentially suppressed in a barren plateau. Hence, without exponential precision, gradient-free optimizers will not make progress in the optimization. We numerically confirm this by training in a barren plateau with several gradient-free optimizers (Nelder-Mead, Powell, and COBYLA algorithms), and show that the numbers of shots required in the optimization grows exponentially with the number of qubits.

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Section: Introductionmentioning

confidence: 99%

Effect of barren plateaus on gradient-free optimization

Arrasmith

Cerezo

Czarnik

et al. 2021

Quantum

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160

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“…Since BPs have been mainly studied in the context of Variational Quantum Algorithms (VQAs) [23,[35][36][37][38][39][40][41][42][43][44], we here briefly recall that in a VQA implementation, the goal is to minimize a linear cost function that is usually of the form…”

Section: B Quantum Landscape Theory and Barren Plateausmentioning

confidence: 99%

“…This field aims to optimize over sets of quantum circuits to discover efficient versions of quantum algorithms for accomplishing various tasks, such as finding ground states [24], solving linear systems of equations [25][26][27], quantum compiling [28,29], factoring [30], and dynamical simulation [31][32][33][34]. In this field, a great deal of effort has been put forward towards analyzing and avoiding the barren plateau phenomenon [35][36][37][38][39][40][41][42][43][44]. When a VQA exhibits a barren plateau, the cost function gradients vanish exponentially with the problem size, leading to an exponentially flat optimization landscape.…”

Section: Introductionmentioning

confidence: 99%

Subtleties in the trainability of quantum machine learning models

Thanasilp¹,

Wang²,

Nghiem³

et al. 2021

Preprint

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A new paradigm for data science has emerged, with quantum data, quantum models, and quantum computational devices. This field, called Quantum Machine Learning (QML), aims to achieve a speedup over traditional machine learning for data analysis. However, its success usually hinges on efficiently training the parameters in quantum neural networks, and the field of QML is still lacking theoretical scaling results for their trainability. Some trainability results have been proven for a closely related field called Variational Quantum Algorithms (VQAs). While both fields involve training a parametrized quantum circuit, there are crucial differences that make the results for one setting not readily applicable to the other. In this work we bridge the two frameworks and show that gradient scaling results for VQAs can also be applied to study the gradient scaling of QML models. Our results indicate that features deemed detrimental for VQA trainability can also lead to issues such as barren plateaus in QML. Consequently, our work has implications for several QML proposals in the literature. In addition, we provide theoretical and numerical evidence that QML models exhibit further trainability issues not present in VQAs, arising from the use of a training dataset. We refer to these as dataset-induced barren plateaus. These results are most relevant when dealing with classical data, as here the choice of embedding scheme (i.e., the map between classical data and quantum states) can greatly affect the gradient scaling.

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“…Identifying such algorithms for heuristic computations requires additional optimization steps, e.g., using the methods of machine learning [29], which would correlate the annealing protocol to a given structured H I . Such methods, however, become inefficient in the limit of maximal complexity with fully random H I because of the emergence of the barren plateau [30]. We leave the question whether they can produce a more efficient protocol than the solvable one in this limit open.…”

Section: Nonadiabatic Grover's Algorithmmentioning

confidence: 99%

Analytical solution for nonadiabatic quantum annealing to arbitrary Ising spin Hamiltonian

Yan,

Sinitsyn

2021

Preprint

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We point to the existence of an analytical solution to a general quantum annealing (QA) problem of finding low energy states of an arbitrary Ising spin Hamiltonian HI by implementing time evolution with a Hamiltonian H(t) = HI + g(t)Ht. We will assume that the nonadiabatic annealing protocol is defined by a specific decaying coupling g(t) and a specific mixing Hamiltonian Ht that make the model analytically solvable arbitrarily far from the adiabatic regime. In specific cases of HI, the solution shows the possibility of a considerable quantum speedup of finding the Ising ground state. We then compare predictions of our solution to results of numerical simulations, and argue that the solvable QA protocol produces the optimal performance in the limit of maximal complexity of the computational problem. Our solution demonstrates for the most complex spin glasses a power-law energy relaxation with the annealing time T and uncorrelated from HI annealing schedule. This proves the possibility for spin glasses of a faster than ∼ 1/ log β T energy relaxation.

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Barren Plateaus Preclude Learning Scramblers

Cited by 93 publications

References 42 publications

Effect of barren plateaus on gradient-free optimization

Effect of barren plateaus on gradient-free optimization

Subtleties in the trainability of quantum machine learning models

Analytical solution for nonadiabatic quantum annealing to arbitrary Ising spin Hamiltonian

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