Group-Invariant Quantum Machine Learning

Larocca, Martín; Sauvage, Frédéric; Sbahi, Faris M.; Verdon, Guillaume; Coles, Patrick J.; Cerezo, M.

doi:10.1103/prxquantum.3.030341

Cited by 71 publications

(46 citation statements)

References 56 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…On the other hand, the history of equivariance in QML is much more recent. In [44] and [45] the authors lay theoretical ground work for the integration of symmetries into QML. Methods for constructing equivariant quantum circuits for graph problems were given in [46,47,49,91].…”

Section: B Equivariance In Quantum Informationmentioning

confidence: 99%

“…Indeed, the first step in the quantum-enhanced experiment model [21] performs this non-linear equivariant layer. Doing so can drastically simplify non-linear learning tasks [21,44] (see Appendix A).…”

Section: Non-linearitiesmentioning

confidence: 99%

“…That is, H in = H out , R in = R out = R and N l θ l (ρ) = U l (θ l )ρU l (θ l ) † . While this case has been considered in [44,45,91] we will here review it for completeness.…”

Section: Parametrizing Equivariant Unitariesmentioning

confidence: 99%

“…Geometric quantum machine learning (GQML) attempts to solve the aforementioned issues by leveraging ideas from geometric deep learning to construct quantum models with sharp inductive biases based on the symmetries of the problem at hand. For instance, when classifying between states presenting a large, or a low, amount of multipartite entanglement [24,42,43], it is natural to employ models whose outputs remain invariant under the action of any local unitary [44]. While recent proposals have started to reveal the power of GQML [44][45][46][47][48][49], the field is still in Figure 1.…”

Section: Introductionmentioning

confidence: 99%

“…For instance, when classifying between states presenting a large, or a low, amount of multipartite entanglement [24,42,43], it is natural to employ models whose outputs remain invariant under the action of any local unitary [44]. While recent proposals have started to reveal the power of GQML [44][45][46][47][48][49], the field is still in Figure 1. Schematic representation of our main results.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Theory for Equivariant Quantum Neural Networks

Nguyen¹,

Schatzki²,

Paolo³

et al. 2022

Preprint

View full text Add to dashboard Cite

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.

show abstract

Section: B Equivariance In Quantum Informationmentioning

confidence: 99%

Section: Non-linearitiesmentioning

confidence: 99%

“…That is, H in = H out , R in = R out = R and N l θ l (ρ) = U l (θ l )ρU l (θ l ) † . While this case has been considered in [44,45,91] we will here review it for completeness.…”

Section: Parametrizing Equivariant Unitariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Theory for Equivariant Quantum Neural Networks

Nguyen¹,

Schatzki²,

Paolo³

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

Quantum Quantile Mechanics: Solving Stochastic Differential Equations for Generating Time‐Series

Paine

Elfving²,

Kyriienko

2023

Adv Quantum Tech

View full text Add to dashboard Cite

A quantum algorithm is proposed for sampling from a solution of stochastic differential equations (SDEs). Using differentiable quantum circuits (DQCs) with a feature map encoding of latent variables, the quantile function is represented for an underlying probability distribution and samples extracted as DQC expectation values. Using quantile mechanics the system is propagated in time, thereby allowing for time‐series generation. The method is tested by simulating the Ornstein‐Uhlenbeck process and sampling at times different from the initial point, as required in financial analysis and dataset augmentation. Additionally, continuous quantum generative adversarial networks (qGANs) are analyzed, and the authors show that they represent quantile functions with a modified (reordered) shape that impedes their efficient time‐propagation. The results shed light on the connection between quantum quantile mechanics (QQM) and qGANs for SDE‐based distributions, and point the importance of differential constraints for model training, analogously with the recent success of physics informed neural networks.

show abstract

Linear Programming with Unitary-Equivariant Constraints

Grinko,

Ozols

2024

Commun. Math. Phys.

View full text Add to dashboard Cite

Unitary equivariance is a natural symmetry that occurs in many contexts in physics and mathematics. Optimization problems with such symmetry can often be formulated as semidefinite programs for a $$d^{p+q}$$ d p + q -dimensional matrix variable that commutes with $$U^{\otimes p} \otimes {\bar{U}}^{\otimes q}$$ U ⊗ p ⊗ U ¯ ⊗ q , for all $$U \in \textrm{U}(d)$$ U ∈ U ( d ) . Solving such problems naively can be prohibitively expensive even if $$p+q$$ p + q is small but the local dimension d is large. We show that, under additional symmetry assumptions, this problem reduces to a linear program that can be solved in time that does not scale in d, and we provide a general framework to execute this reduction under different types of symmetries. The key ingredient of our method is a compact parametrization of the solution space by linear combinations of walled Brauer algebra diagrams. This parametrization requires the idempotents of a Gelfand–Tsetlin basis, which we obtain by adapting a general method inspired by the Okounkov–Vershik approach. To illustrate potential applications of our framework, we use several examples from quantum information: deciding the principal eigenvalue of a quantum state, quantum majority vote, asymmetric cloning and transformation of a black-box unitary. We also outline a possible route for extending our method to general unitary-equivariant semidefinite programs.

show abstract

Group-Invariant Quantum Machine Learning

Cited by 71 publications

References 56 publications

Theory for Equivariant Quantum Neural Networks

Theory for Equivariant Quantum Neural Networks

Quantum Quantile Mechanics: Solving Stochastic Differential Equations for Generating Time‐Series

Linear Programming with Unitary-Equivariant Constraints

Contact Info

Product

Resources

About