The mixed Schur transform: efficient quantum circuit and applications

TL;DR

The paper introduces an efficient quantum circuit for the mixed Schur transform, generalizing previous work to include dual representations, enabling applications in quantum channels and extending permutational quantum computing.

Contribution

It develops a generalized mixed Schur transform circuit based on duality with the walled Brauer algebra, expanding the scope of quantum symmetry transformations.

Findings

Circuit complexity is $ ilde{O}((n+m)d^4)$.

Enables efficient implementation of unitary-equivariant channels.

Extends permutational quantum computing to include partial transposes.

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.