Speeding up Learning Quantum States with the help of Group Equivariant Convolutional Quantum Ansätze

Zapraszamy w piątek 04.03.2022r. o godz. 12:15

Seminarium ONLINE pt.: Speeding up Learning Quantum States with the help of Group Equivariant Convolutional Quantum Ansätze (Speaker: dr Sergii Strelchuk,

 Department of Applied Mathematics and Theoretical Physics, University of Cambridge)

Place: https://zoom.us/j/93636884042?pwd=TksrY0xaMEczY2k0RDRqajFMV1lxdz09

 Abstract:. In this talk, I will discuss one of the key properties which are responsible for the unreasonable success of classical convolutional neural networks – equivariance. It states that if the input to the neural network is shifted, then its activations translate accordingly. Developing the corresponding notion for discrete representational spaces used to describe finite-dimensional quantum systems is challenging. We generalize this notion by introducing a new framework for Sn-equivariant quantum convolutional circuits, building on and significantly generalizing Permutational Quantum Computing (PQC) formalism.

We demonstrate how to effectively apply the celebrated Okounkov-Vershik's representation theory in machine learning and quantum physics :  (1) we show how to gain a super-exponential speedup in computing the matrix elements of Sn-Fourier coefficients compared to the best known classical Fast Fourier Transform (FFT) over the symmetric group. (2) we prove that Sn Convolutional Quantum Ansätze are dense, thus expressible within each Sn-irrep block, which may serve as a universal model for potential future quantum machine learning and optimization applications. (3) we get a new proof (which is of distinctly representation-theoretic flavour) of the universality of the Quantum Approximate Optimization Algorithm. (4) our framework can be naturally applied to a wide array of problems with global SU(d) (for any integer d) symmetry. (5) We show that our ansätze are highly effective numerically by providing numerical solutions to the problem of the sign structure of the ground state of the J1-J2 antiferromagnetic Heisenberg model on the rectangular and Kagome lattices