Speeding up Learning Quantum States through Group Equivariant Convolutional Quantum Ans\"atze. (arXiv:2112.07611v2 [quant-ph] UPDATED)

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 …

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