Speeding up Learning Quantum States through Group Equivariant Convolutional Quantum Ansätze

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 (PQC) 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 (Ph.D. Thesis 2005 p.160) on the equivalence between operatorname{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 (YJM) 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 Quantum Approximate Optimization Algorithm (QAOA) and verifies that 4-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 J₁--J₂ antiferromagnetic Heisenberg model on the rectangular and Kagome lattices. Our work provides the first application of the celebrated Okounkov-Vershik's 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.