Quantum algorithms for group convolution, cross-correlation, and equivariant transformations

Castelazo, Grecia; Nguyen, Quynh T.; De Palma, Giacomo; Englund, Dirk; Lloyd, Seth; Kiani, Bobak T.

doi:10.48550/arxiv.2109.11330

Cited by 3 publications

(3 citation statements)

References 26 publications

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“…Although imposing invariance at the map level effectively results in global invariance of the model, this is quite restrictive. A more relaxed approach towards the construction of group invariant models involves the concept of equivariance [7][8][9]88] which is now defined.…”

Section: Equivariant Quantum Neural Networkmentioning

confidence: 99%

Group-Invariant Quantum Machine Learning

Larocca,

Sauvage,

Sbahi

et al. 2022

Preprint

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Quantum Machine Learning (QML) models are aimed at learning from data encoded in quantum states. Recently, it has been shown that models with little to no inductive biases (i.e., with no assumptions about the problem embedded in the model) are likely to have trainability and generalization issues, especially for large problem sizes. As such, it is fundamental to develop schemes that encode as much information as available about the problem at hand. In this work we present a simple, yet powerful, framework where the underlying invariances in the data are used to build QML models that, by construction, respect those symmetries. These so-called group-invariant models produce outputs that remain invariant under the action of any element of the symmetry group G associated to the dataset. We present theoretical results underpinning the design of G-invariant models, and exemplify their application through several paradigmatic QML classification tasks including cases when G is a continuous Lie group and also when it is a discrete symmetry group. Notably, our framework allows us to recover, in an elegant way, several well known algorithms for the literature, as well as to discover new ones. Taken together, we expect that our results will help pave the way towards a more geometric and group-theoretic approach to QML model design.

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Section: Equivariant Quantum Neural Networkmentioning

confidence: 99%

Group-Invariant Quantum Machine Learning

Larocca,

Sauvage,

Sbahi

et al. 2022

Preprint

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show abstract

“…Finally, we note that the case of finite groups and regular representations (i.e., when the intermediate representations are chosen to be R reg : G → C[G] corresponding to the group action on its own group algebra) has been studied in the classical literature under the name of homogeneous ENNs [14]. In this case, any equivariant map is a group convolution [11], which can be realized as a unitary operator embedding the classical convolution kernel by the quantum algorithms in [116]. Combining this with quantum algorithms for polynomial transformations of quantum states [117,118] allows one to quantize classical homogeneous ENNs.…”

Section: Intermediate Representations As Hyperparametersmentioning

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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“…In fact, achieving a runtime logarithmic in N for dense and full-rank matrices is generally challenging unless there are specific symmetries or structures inherent in the matrix. For example, quantum algorithms have been developed to achieve polylogarithmic complexity in N for Toepliz systems [6], Hankel matrices [7], and linear group convolutions [8] which all feature some inherent structure.…”

Section: Introductionmentioning

confidence: 99%

Block-encoding dense and full-rank kernels using hierarchical matrices: applications in quantum numerical linear algebra

Nguyen¹,

Kiani²,

Lloyd³

2022

Preprint

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Kernel matrices, which arise from discretizing a kernel function k(x, x ), have a variety of applications in mathematics and engineering. Classically, the celebrated fast multipole method was designed to perform matrix multiplication on kernel matrices of dimension N in time almost linear in N by using techniques later generalized into the linear algebraic framework of hierarchical matrices. In light of this success, we propose a quantum algorithm for efficiently performing matrix operations on hierarchical matrices by implementing a quantum block-encoding of the hierarchical matrix structure. When applied to many kernel matrices, our quantum algorithm can solve quantum linear systems of dimension N in time O(κ polylog( N ε )), where κ and ε are the condition number and error bound of the matrix operation. This runtime is exponentially faster than any existing quantum algorithms for implementing dense kernel matrices. Finally, we discuss possible applications of our methodology in solving integral equations or accelerating computations in N-body problems.

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Quantum algorithms for group convolution, cross-correlation, and equivariant transformations

Cited by 3 publications

References 26 publications

Group-Invariant Quantum Machine Learning

Group-Invariant Quantum Machine Learning

Theory for Equivariant Quantum Neural Networks

Block-encoding dense and full-rank kernels using hierarchical matrices: applications in quantum numerical linear algebra

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