Speeding Up Learning Quantum States Through Group Equivariant Convolutional Quantum Ansätze
Han Zheng, Zimu Li, Junyu Liu, Sergii Strelchuk, Risi Kondor · PRX Quantum · 2023
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.