Speeding up Learning Quantum States through Group Equivariant Convolutional Quantum Ansätze
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We develop a theoretical framework for S_n-equivariant quantum convolutional circuits, building on and significantly generalizing Jordan's Permutational Quantum Computing (PQC) formalism. We show that quantum circuits are a natural choice for Fourier space neural architectures affording a super-exponential speedup in computing the matrix elements of S_n-Fourier coefficients compared to the best known classical Fast Fourier Transform (FFT) over the symmetric group. In particular, we utilize the Okounkov-Vershik approach to prove Harrow's statement (Ph.D. Thesis 2005 p.160) 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 (YJM) elements. We prove that S_n-CQA are dense, thus expressible within each S_n-irrep block, which may serve as a universal model for potential future quantum machine learning and optimization applications. Our method provides another way to prove the universality of Quantum Approximate Optimization Algorithm (QAOA), from the representation-theoretical point of view. Our framework can be naturally applied to a wide array of problems with global SU(d) symmetry. We present numerical simulations to showcase the effectiveness of the ansätze to find the sign structure of the ground state of the J_1–J_2 antiferromagnetic Heisenberg model on the rectangular and Kagome lattices. Our work identifies quantum advantage for a specific machine learning problem, and provides the first application of the celebrated Okounkov-Vershik's representation theory to machine learning and quantum physics.
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