Speeding Up Learning Quantum States Through Group Equivariant Convolutional Quantum Ansätze

Abstract

We develop a theoretical framework for S_{n}-equivariant convolutional quantum circuits with SU(d) symmetry, building on and significantly generalizing Jordan’s permutational quantum computing formalism based on Schur-Weyl duality connecting both SU(d) and S_{n} actions on qudits. In particular, we utilize the Okounkov-Vershik approach to prove Harrow’s statement on the equivalence between SU⁡(d) and S_{n} irrep bases and to establish the S_{n}-equivariant convolutional quantum alternating ansätze (S_{n}-CQA) using Young-Jucys-Murphy elements. We prove that S_{n}-CQA is able to generate any unitary in any given S_{n} irrep sector, which may serve as a universal model for a wide array of quantum machine-learning problems with the presence of SU⁡(d) symmetry. Our method provides another way to prove the universality of the quantum approximate optimization algorithm and verifies that four-local SU⁡(d)-symmetric unitaries are sufficient to build generic SU⁡(d)-symmetric quantum circuits up to relative phase factors. We present numerical simulations to showcase the effectiveness of the ansätze to find the ground-state energy of the J_{1}-J_{2} antiferromagnetic Heisenberg model on the rectangular and kagome lattices. Our work provides the first application of the celebrated Okounkov-Vershik S_{n} representation theory to quantum physics and machine learning, from which to propose quantum variational ansätze that strongly suggests to be classically intractable tailored towards a specific optimization problem.