Equivalence of quantum barren plateaus to cost concentration and narrow gorges

Arrasmith, Andrew; Holmes, Zoë; Cerezo, M.; Coles, Patrick J.

doi:10.1088/2058-9565/ac7d06

Cited by 58 publications

(32 citation statements)

References 66 publications

(100 reference statements)

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“…Despite its promise, there are still several challenges one needs to address before unlocking the full potential of QML. In particular, a growing amount of evidence suggests that models with little-to-no inductive biases have poor trainability and generalization, greatly limiting their scalability [26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41].…”

Section: Introductionmentioning

confidence: 99%

Theory for Equivariant Quantum Neural Networks

Nguyen¹,

Schatzki²,

Paolo³

et al. 2022

Preprint

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Most currently used quantum neural network architectures have little-to-no inductive biases, leading to trainability and generalization issues. Inspired by a similar problem, recent breakthroughs in classical machine learning address this crux by creating models encoding the symmetries of the learning task. This is materialized through the usage of equivariant neural networks whose action commutes with that of the symmetry. In this work, we import these ideas to the quantum realm by presenting a general theoretical framework to understand, classify, design and implement equivariant quantum neural networks. As a special implementation, we show how standard quantum convolutional neural networks (QCNN) can be generalized to group-equivariant QCNNs where both the convolutional and pooling layers are equivariant under the relevant symmetry group. Our framework can be readily applied to virtually all areas of quantum machine learning, and provides hope to alleviate central challenges such as barren plateaus, poor local minima, and sample complexity.

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Section: Introductionmentioning

confidence: 99%

Theory for Equivariant Quantum Neural Networks

Nguyen¹,

Schatzki²,

Paolo³

et al. 2022

Preprint

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show abstract

“…[11] introduced the phenomenon of "barren plateaus" or regions of non-informative gradients in cost function landscapes resulting from employing PQCs that form approximate 2-designs. Several works extended this work, connecting particular circuit structures to cost function landscape features such as barren plateaus and narrow gorges [12] that hinder algorithm performance. For a better understanding and evaluation of PQCs, Ref.…”

Section: *Our Workmentioning

confidence: 98%

“…where η is the step size or learning rate and L(θ) is the objective function to be minimized. For the four circuit types considered, we have the following Fubini-Study metrics assuming (12) and naively calculating them. Matrix elements that are included in the block-diagonal approximation to the metric tensor are in blue.…”

Section: Quantum Natural Gradient Descentmentioning

confidence: 99%

“…LDCA block while the third and fourth qubits are also entangled using a two-qubit block. Overall, we have 2 two-qubit subsystems, each of which we know the metric tensor, call them g (12) and g (34) . However, suppose we add a static/nonparametric entangler (e.g.…”

Section: Decomposition Problemmentioning

confidence: 99%

“…CNOT) to the second and third qubits. How is the metric tensor of the four-qubit system g (1234) constructed, compared to structures of g (12) and g (34) ? The idea being, rather than anaylzing the n qubit geometry from the ground up in order to understand PQCs for n qubits, can we use the simpler geometry of 2 qubits to bootstrap our way to understanding higher qubit number PQCs?…”

Section: Decomposition Problemmentioning

confidence: 99%

See 2 more Smart Citations

Connecting geometry and performance of two-qubit parameterized quantum circuits

Katabarwa

Sim

Koh

et al. 2022

Quantum

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Parameterized quantum circuits (PQCs) are a central component of many variational quantum algorithms, yet there is a lack of understanding of how their parameterization impacts algorithm performance. We initiate this discussion by using principal bundles to geometrically characterize two-qubit PQCs. On the base manifold, we use the Mannoury-Fubini-Study metric to find a simple equation relating the Ricci scalar (geometry) and concurrence (entanglement). By calculating the Ricci scalar during a variational quantum eigensolver (VQE) optimization process, this offers us a new perspective to how and why Quantum Natural Gradient outperforms the standard gradient descent. We argue that the key to the Quantum Natural Gradient's superior performance is its ability to find regions of high negative curvature early in the optimization process. These regions of high negative curvature appear to be important in accelerating the optimization process.

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A General Approach to Dropout in Quantum Neural Networks

Scala,

Ceschini,

Panella

et al. 2023

Adv Quantum Tech

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In classical machine learning (ML), “overfitting” is the phenomenon occurring when a given model learns the training data excessively well, and it thus performs poorly on unseen data. A commonly employed technique in ML is the so called “dropout,” which prevents computational units from becoming too specialized, hence reducing the risk of overfitting. With the advent of quantum neural networks (QNNs) as learning models, overfitting might soon become an issue, owing to the increasing depth of quantum circuits as well as multiple embedding of classical features, which are employed to give the computational nonlinearity. Here, a generalized approach is presented to apply the dropout technique in QNN models, defining and analyzing different quantum dropout strategies to avoid overfitting and achieve a high level of generalization. This study allows to envision the power of quantum dropout in enabling generalization, providing useful guidelines on determining the maximal dropout probability for a given model, based on overparametrization theory. It also highlights how quantum dropout does not impact the features of the QNN models, such as expressibility and entanglement. All these conclusions are supported by extensive numerical simulations and may pave the way to efficiently employing deep quantum machine learning (QML) models based on state‐of‐the‐art QNNs.

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Equivalence of quantum barren plateaus to cost concentration and narrow gorges

Cited by 58 publications

References 66 publications

Theory for Equivariant Quantum Neural Networks

Theory for Equivariant Quantum Neural Networks

Connecting geometry and performance of two-qubit parameterized quantum circuits

A General Approach to Dropout in Quantum Neural Networks

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