High-dimensional quantum Schur transforms

TL;DR

<3-5 sentence high-level summary>This work addresses gaps in the quantum Schur transform by (i) correcting Krovi’s high-dimensional approach and (ii) developing a GT-basis–aware framework that achieves a robust Schur transform with asymptotically favorable scaling. The authors establish a corrected three-stage Krovi circuit (preprocessing, QFT over $\mathrm{S}_n$, and a blockwise isometry $V$) that yields Gelfand--Tsetlin bases for both $\mathrm{S}_n$ and $\mathrm{U}_d$ with total gate complexity $\tilde{O}(n^4)$ and space $\tilde{O}(n^2)$, and they show the compression-based high-dimensional BCH circuit reduces $d$-dependence to $\tilde{O}(\min(n^5, d^4 n))$. The analysis heavily relies on $F$-symbols, split-versus-SYT basis changes, and the Schur--Weyl duality framework. These results strengthen algorithmic foundations for Schur--Weyl duality in quantum information and improve practical regimes where the local dimension exceeds the number of subsystems.

Abstract

The quantum Schur transform has become a foundational quantum algorithm, yet even after two decades since the seminal 2005 paper by Bacon, Chuang, and Harrow (BCH), some aspects of the transform remain insufficiently understood. Moreover, an alternative approach proposed by Krovi in 2018 was recently found to contain a crucial error. In this paper, we present a corrected version of Krovi's algorithm along with a detailed treatment of the high-dimensional version of the BCH Schur transform. This high-dimensional focus makes the two versions of the transform practical for regimes where the number of qudits $n$ is smaller than the local dimension $d$, with Krovi's algorithm scaling as $\widetilde{O}(n^4)$ and BCH as $\widetilde{O}(\min(n^5,nd^4))$. Our work addresses a key gap in the literature, strengthening the algorithmic foundations of a wide range of results that rely on Schur--Weyl duality in quantum information theory and quantum computation.

Figures (17)

• Figure 1: A change in the fusion order is described by $F$-symbols. On the left we see a general situation when fusion happens with multiplicities, while on the right--fusion is multiplicity free, which is relevant for our application.
• Figure 2: Krovi's quantum Schur transform consists of three steps: preprocessing $\mathrm{P}$, Quantum Fourier Transform $\mathrm{QFT}$ over $\mathop{\mathrm{S}}\nolimits_n$ and compression (inverse) isometry $V^\dagger$. The state after $\mathrm{QFT}$ lies in ${\rm im} V$ (Lemma 9), so $V^\dagger$ is reversible.
• Figure 3: Tensor network contraction between Young--Yamanouchi basis vector labelled by $T \in \mathrm{SYT}(\lambda)$ and a split basis vector $\widetilde{M} \in \mathrm{SSYT}(\lambda,\mu)$. The example is drawn for $n=5$ qudit system, where $T^5 = \widetilde{M}_2 = \lambda$ with $\mu = (2,3)$. The sequence of $F$-moves transforms the diagram on the left to the diagram on the right, which corresponds to trivial contraction.
• Figure 4: Inductive decomposition of preparation isometry $\mathrm{P} \equiv \mathop{\mathrm{Prep}}\nolimits$. Notice that some auxiliary registers are omitted for clarity of the picture.
• Figure 5: A quantum circuit for an isometry $\mathop{\mathrm{Prep}}\nolimits_n$ utilizes two auxiliary registers ($\lvert{\cdot }\rangle_{p}\in \mathcal{H}_d$ and $\lvert{\cdot }\rangle_{u}\in \mathcal{H}_2$) that store information about the position $e_n$ and uniqueness $c_n$ of $x_n$. It can be further decomposed into six subroutines: $A, B, C, D, E,$ and $H$. \ref{['fig:subroutibesABCD']} presents the quantum circuits for the aforementioned subroutines and provides an explanation of their roles in implementing $\mathop{\mathrm{Prep}}\nolimits_n$.

Theorems & Definitions (19)

• Theorem 1: Corrected version of Krovi’s Schur transform algorithm
• proof
• Lemma 2: kawano2016quantum
• Example 3
• Remark 4
• Lemma 5: le1987new
• Lemma 6: Grand orthogonality relations
• Corollary 7
• Lemma 8
• proof