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

Group convolutions and cross-correlations, which are equivariant to the actions of group elements, are commonly used in mathematics to analyze or take advantage of symmetries inherent in a given problem setting. Here, we provide efficient quantum algorithms for performing linear group convolutions and cross-correlations on data stored as quantum states. Runtimes for our algorithms are logarithmic in the dimension of the group thus offering an exponential speedup compared to classical algorithms when input data is provided as a quantum state and linear operations are well conditioned. Motivated by the rich literature on quantum algorithms for solving algebraic problems, our theoretical framework opens a path for quantizing many algorithms in machine learning and numerical methods that employ group operations.

[1]  Robin Kothari,et al.  Efficient algorithms in quantum query complexity , 2014 .

[2]  Gabriel Peyré,et al.  Universal Invariant and Equivariant Graph Neural Networks , 2019, NeurIPS.

[3]  A. T. Lonseth,et al.  Sources and Applications of Integral Equations , 1977 .

[4]  Stephan J. Garbin,et al.  Harmonic Networks: Deep Translation and Rotation Equivariance , 2016, 2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR).

[5]  Barnabás Póczos,et al.  Equivariance Through Parameter-Sharing , 2017, ICML.

[6]  Zhen Lin,et al.  Clebsch-Gordan Nets: a Fully Fourier Space Spherical Convolutional Neural Network , 2018, NeurIPS.

[7]  Alexander Russell,et al.  Generic quantum Fourier transforms , 2004, SODA '04.

[8]  Martin Rötteler,et al.  Quantum algorithms for highly non-linear Boolean functions , 2008, SODA '10.

[9]  Kwan Hui Lim,et al.  Hybrid quantum-classical convolutional neural networks , 2019, Science China Physics, Mechanics & Astronomy.

[10]  Martin Rötteler,et al.  Quantum Algorithms for Abelian Difference Sets and Applications to Dihedral Hidden Subgroups , 2016, TQC.

[11]  Su-Juan Qin,et al.  Asymptotic quantum algorithm for the Toeplitz systems , 2016, Physical Review A.

[12]  K. Fujii,et al.  Quantum analog-digital conversion , 2018, Physical Review A.

[13]  Alexander J. Smola,et al.  Deep Sets , 2017, 1703.06114.

[14]  E. Krause,et al.  Taxicab Geometry: An Adventure in Non-Euclidean Geometry , 1987 .

[15]  Sean Hallgren,et al.  Quantum algorithms for some hidden shift problems , 2003, SODA '03.

[16]  J. B. Wang,et al.  Efficient quantum circuits for Toeplitz and Hankel matrices , 2016, 1605.07710.

[17]  Guigang Zhang,et al.  Deep Learning , 2016, Int. J. Semantic Comput..

[18]  Nathan Wiebe,et al.  Fast inversion, preconditioned quantum linear system solvers, and fast evaluation of matrix functions , 2020, Physical Review A.

[19]  Tareq Abed Mohammed,et al.  Understanding of a convolutional neural network , 2017, 2017 International Conference on Engineering and Technology (ICET).

[20]  I. Chuang,et al.  Hamiltonian Simulation by Qubitization , 2016, Quantum.

[21]  Hua Xiang,et al.  Quantum circulant preconditioner for a linear system of equations , 2018, Physical Review A.

[22]  J. B. Wang,et al.  Efficient quantum circuits for dense circulant and circulant like operators , 2016, Royal Society Open Science.

[23]  Arthur Pesah,et al.  Absence of Barren Plateaus in Quantum Convolutional Neural Networks , 2020, Physical Review X.

[24]  Li Li,et al.  Tensor Field Networks: Rotation- and Translation-Equivariant Neural Networks for 3D Point Clouds , 2018, ArXiv.

[25]  Carl E. Rasmussen,et al.  Gaussian processes for machine learning , 2005, Adaptive computation and machine learning.

[26]  Nathanael Perraudin,et al.  DeepSphere: Efficient spherical Convolutional Neural Network with HEALPix sampling for cosmological applications , 2018, Astron. Comput..

[27]  Risi Kondor,et al.  On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups , 2018, ICML.

[28]  Yaron Lipman,et al.  On the Universality of Invariant Networks , 2019, ICML.