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 $\operatorname{SU}(d)$- and $S_n$-irrep bases and to
establish the $S_n$-equivariant Convolutional Quantum Alternating Ans\"atze
($S_n$-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 $\operatorname{SU}(d)$
symmetry. We present numerical simulations to showcase the effectiveness of the
ans\"atze 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.

我们提出了 $S_n$- 等变量子卷积电路的理论框架，利用傅里叶空间神经结构等方法速度超指数级地计算 $S_n$- 傅里叶系数矩阵元，证明了该框架的普适性，并进行了在量子机器学习和优化方面的应用与数值模拟，这是第一次将著名的 Okounkov-Vershik 表示理论应用于机器学习和量子物理。