Variational quantum algorithms

Cerezo, M.; Arrasmith, Andrew; Babbush, Ryan; Benjamin, Simon C.; Endo, Suguru; Fujii, Keisuke; McClean, Jarrod R.; Mitarai, Kosuke; Yuan, Xiao; Cincio, Łukasz; Coles, Patrick J.

doi:10.1038/s42254-021-00348-9

Cited by 1,619 publications

(1,103 citation statements)

References 163 publications

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“…Here, the training set is given by S = {ρ x , y x } where y x are the true labels, and y(θ, ρ x ) are the labels predicted by the Quantum Neural Network. In addition, the cost of several Variational Quantum Algorithms is covered by (1). In this case, the cost takes a simpler form, where the training set contains a single state (S = 1) and the cost is C(θ) = Tr[OV (θ)ρV † (θ)], with O a Hermitian operator [3][4][5][6][7][8][9][10][11][12][13][14].…”

Section: Cost Functionmentioning

confidence: 99%

“…Parameterized quantum circuits offer a flexible paradigm for programming Noisy Intermediate Scale Quantum (NISQ) computers. These circuits are utilized in both Variational Quantum Algorithms (VQAs) [1][2][3][4][5][6][7][8][9][10][11][12][13][14] and Quantum Neural Networks (QNNs) [15][16][17][18]. Both VQA and QNN approaches involve efficiently evaluating a cost function C(θ) or its gradient ∇C(θ) on a quantum computer.…”

Section: Introductionmentioning

confidence: 99%

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Effect of barren plateaus on gradient-free optimization

Arrasmith

Cerezo

Czarnik

et al. 2021

Quantum

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151

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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: Cost Functionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Effect of barren plateaus on gradient-free optimization

Arrasmith

Cerezo

Czarnik

et al. 2021

Quantum

Self Cite

151

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“…In the majority of our numerical simulations, we consider a QCBM with 3 qubits. This corresponds to a discrete target distribution p which takes 2 3 values. We generally also assume that the target distribution corresponds to a particular instantiation of the QCBM, for a fixed number of layers, D p .…”

Section: Numerical Resultsmentioning

confidence: 99%

“…where we have defined H x := |x x|. We can thus write the reverse KL in the form of a generic cost function (see, e.g., [3]) as…”

Section: Local Cost Functionsmentioning

confidence: 99%

“…One of the most challenging technological questions of our time is whether existing quantum computers can achieve quantum advantage in tasks of practical interest. Variational quantum algorithms (VQAs), which are well suited to the constraints imposed by existing devices, have emerged as the leading strategy for achieving such a quantum advantage [1][2][3][4].…”

Section: Introductionmentioning

confidence: 99%

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F-Divergences and Cost Function Locality in Generative Modelling with Quantum Circuits

Leadbeater

Louis

Coyle

et al. 2021

Entropy

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Generative modelling is an important unsupervised task in machine learning. In this work, we study a hybrid quantum-classical approach to this task, based on the use of a quantum circuit born machine. In particular, we consider training a quantum circuit born machine using f-divergences. We first discuss the adversarial framework for generative modelling, which enables the estimation of any f-divergence in the near term. Based on this capability, we introduce two heuristics which demonstrably improve the training of the born machine. The first is based on f-divergence switching during training. The second introduces locality to the divergence, a strategy which has proved important in similar applications in terms of mitigating barren plateaus. Finally, we discuss the long-term implications of quantum devices for computing f-divergences, including algorithms which provide quadratic speedups to their estimation. In particular, we generalise existing algorithms for estimating the Kullback–Leibler divergence and the total variation distance to obtain a fault-tolerant quantum algorithm for estimating another f-divergence, namely, the Pearson divergence.

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Perspectives of quantum computing for chemical engineering

et al. 2022

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Quantum computing has been attracting public attention recently. This interest is driven by the advancements in hardware, software, and algorithms required for its successful usage and the promise that it entails the potential acceleration of computational tasks compared to classical computing. This perspective article presents a short review on quantum computing, how this computational approach solves problems, and three fields that quantum computing can potentially impact the most while relevant to chemical engineering: computational chemistry, optimization, and machine learning. Here, we present a series of chemical engineering applications, the developments, potential improvements with respect to classical computing, and challenges that quantum computing faces for each of these fields. This article intends to provide a clear picture of the challenges and potential advantages that quantum technology may yield for chemical engineering, together with an invitation for our colleagues to join us in the adoption and development of quantum computing.

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Variational quantum algorithms

Cited by 1,619 publications

References 163 publications

Effect of barren plateaus on gradient-free optimization

Effect of barren plateaus on gradient-free optimization

F-Divergences and Cost Function Locality in Generative Modelling with Quantum Circuits

Perspectives of quantum computing for chemical engineering

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