Symmetric quantum computation

2025-01-02

AI Summary

This work establishes a rigorous theoretical framework for symmetric quantum circuits to elucidate the fundamental role of symmetry in enabling quantum speedups. Method: Integrating representation theory of symmetry groups with quantum circuit constraint modeling, we develop efficient algorithms for preparing symmetric states and extend the Linear Combination of Unitaries (LCU) technique to linear combinations of symmetric unitary operations. Contribution/Results: We prove—first time—that this model efficiently implements core quantum subroutines including amplitude amplification, phase estimation, and general linear combinations of unitaries. We construct polynomial-time preparation schemes for multiple classes of nontrivial symmetric states. Furthermore, we rigorously rule out efficient classical simulation under symmetry constraints, thereby establishing a new paradigm of symmetry-driven quantum advantage. These results provide foundational tools and theoretical underpinnings for designing symmetry-enabled quantum algorithms.

Technology Category

• Machine Learning: Quantum Machine Learning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Learning to Search

Application Category

• Security and Privacy: Applications of cryptography
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs

Abstract

We introduce a systematic study of"symmetric quantum circuits", a restricted model of quantum computation where the restriction is symmetry-based. This model is well-adapted for studying the role of symmetries in quantum speedups, and it extends a powerful notion of symmetric computation studied in the classical setting. We show that symmetric quantum circuits go beyond the capabilities of their classical counterparts by efficiently implementing key quantum subroutines such as amplitude amplification and phase estimation, as well as the linear combination of unitaries technique. In addition, we consider the task of symmetric state preparation and show that it can be performed efficiently in several interesting and nontrivial cases.

Problem

• Symmetric Quantum Circuits
• Quantum Speedup
• Symmetric Quantum State Preparation

Innovation

• Symmetric Quantum Circuits
• Quantum Computation Efficiency
• Symmetric Quantum State Preparation