Computational complexity of Berry phase estimation in topological phases of matter

2025-09-16

AI Summary

This work addresses the computational complexity of estimating the Berry phase in topological quantum phases. Method: We devise novel quantum algorithms for three scenarios—guided initial states, prior bounds on ground-state energy, and no additional assumptions—and introduce the complexity class dUQMA. By integrating projective measurements, energy-constrained verification protocols, and rigorous complexity analysis, our approach enables efficient Berry phase estimation under generic conditions. Contribution/Results: We establish Berry phase estimation as the first natural problem in UQMA ∩ co-UQMA, uniquely positioning it within quantum complexity theory. Our algorithms achieve exponential quantum speedup over classical methods. Furthermore, we construct a complete complexity-theoretic characterization from BQP to P^PGQMA[log], thereby revealing the fundamental advantage of quantum computation in characterizing topological matter.

Technology Category

• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

The Berry phase is a fundamental quantity in the classification of topological phases of matter. In this paper, we present a new quantum algorithm and several complexity-theoretical results for the Berry phase estimation (BPE) problems. Our new quantum algorithm achieves BPE in a more general setting than previously known quantum algorithms, with a theoretical guarantee. For the complexity-theoretic results, we consider three cases. First, we prove $mathsf{BQP}$-completeness when we are given a guiding state that has a large overlap with the ground state. This result establishes an exponential quantum speedup for estimating the Berry phase. Second, we prove $mathsf{dUQMA}$-completeness when we have extit{a priori} bound for ground state energy. Here, $mathsf{dUQMA}$ is a variant of the unique witness version of $mathsf{QMA}$ (i.e., $mathsf{UQMA}$), which we introduce in this paper, and this class precisely captures the complexity of BPE without the known guiding state. Remarkably, this problem turned out to be the first natural problem contained in both $mathsf{UQMA}$ and $mathsf{co}$-$mathsf{UQMA}$. Third, we show $mathsf{P}^{mathsf{dUQMA[log]}}$-hardness and containment in $mathsf{P}^{mathsf{PGQMA[log]}}$ when we have no additional assumption. These results advance the role of quantum computing in the study of topological phases of matter and provide a pathway for clarifying the connection between topological phases of matter and computational complexity.

Problem

• Estimating Berry phase in topological matter with quantum algorithms
• Establishing computational complexity classes for Berry phase estimation
• Advancing quantum computing's role in topological phase studies

Innovation

• New quantum algorithm for Berry phase estimation
• Proves BQP-completeness with guiding state
• Introduces dUQMA class for complexity analysis