The mixed Schur transform: efficient quantum circuit and applications

• URL: http://arxiv.org/abs/2310.01613v1
• Date: Mon, 2 Oct 2023 20:03:56 GMT
• Title: The mixed Schur transform: efficient quantum circuit and applications
• Authors: Quynh T. Nguyen
• Abstract summary: The Schur transform is an important primitive in quantum information and theoretical physics. We give a generalization of its quantum circuit implementation due to Bacon, Chuang, and Harrow (SODA 2007) We show how the mixed Schur transform enables efficient implementation of unitary-equivariant channels in various settings.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The Schur transform, which block-diagonalizes the tensor representation $U^{\otimes n}$ of the unitary group $\mathbf{U}_d$ on $n$ qudits, is an important primitive in quantum information and theoretical physics. We give a generalization of its quantum circuit implementation due to Bacon, Chuang, and Harrow (SODA 2007) to the case of mixed tensor $U^{\otimes n} \otimes \bar{U}^{\otimes m}$, where $\bar{U}$ is the dual representation. This representation is the symmetry of unitary-equivariant channels, which find various applications in quantum majority vote, multiport-based teleportation, asymmetric state cloning, black-box unitary transformations, etc. The "mixed" Schur transform contains several natural extensions of the representation theory used in the Schur transform, in which the main ingredient is a duality between the mixed tensor representations and the walled Brauer algebra. Another element is an efficient implementation of a "dual" Clebsch-Gordan transform for $\bar{U}$. The overall circuit has complexity $\widetilde{O} ((n+m)d^4)$. Finally, we show how the mixed Schur transform enables efficient implementation of unitary-equivariant channels in various settings and discuss other potential applications, including an extension of permutational quantum computing that includes partial transposes.

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