Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Speeding up Learning Quantum States through Group Equivariant
  Convolutional Quantum Ans\"atze   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.

10
The categories of this paper are: q-bio
27
The categories of this paper are: comp-gas
30
The categories of this paper are: q-alg
22
The categories of this paper are: chao-dyn
36
The categories of this paper are: cmp-lg
0
The categories of this paper are: hep-ph
8
The categories of this paper are: hep-ex
9
The categories of this paper are: nlin
15
The categories of this paper are: nucl-ex
34
The categories of this paper are: chem-ph
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The categories of this paper are: cond-mat
31
The categories of this paper are: ao-sci
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The categories of this paper are: adap-org
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The categories of this paper are: q-fin
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The categories of this paper are: astro-ph
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The categories of this paper are: patt-sol