High-dimensional quantum Schur transforms

2025-09-26

AI Summary

This work addresses theoretical gaps and algorithmic deficiencies in high-dimensional quantum Schur transforms: existing studies lack a rigorous understanding of Schur transforms for qudit systems with dimension $d > 2$, and Krovi’s (2018) construction contains a critical error. We first correct the fundamental flaw in Krovi’s algorithm and systematically establish a comprehensive theoretical framework for quantum Schur transforms in arbitrary dimensions. Specifically, we extend and refine the Bacon–Chuang–Harrow scheme to $d$-dimensional qudits and $n$-partite systems, significantly improving practicality when $n < d$. Methodologically, we integrate representation theory, Schur–Weyl duality, and optimized quantum circuit design, yielding two efficient algorithms with gate complexities $O(n^4)$ and $O(min(n^5, n d^4))$, respectively. Our results fill a foundational gap in Schur–Weyl duality-based algorithms for high-dimensional quantum computation and substantially enhance scalability in many-body quantum information processing and quantum group invariant computation.

Technology Category

• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

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.

Problem

• Correcting Krovi's flawed quantum Schur transform algorithm
• Providing detailed analysis of high-dimensional BCH Schur transform
• Addressing algorithmic gaps for quantum information applications

Innovation

• Corrected Krovi's algorithm for Schur transform
• Detailed high-dimensional BCH Schur transform treatment
• Practical implementation for qudits when n < d