Bounds in the Projective Unitary Group with Respect to Global Phase Invariant Metric

2025-10-10

AI Summary

This work addresses geometric coding on the projective unitary group $PU(n)$ under a global-phase-invariant metric. Methodologically, it integrates differential geometry, measure theory, and coding theory, leveraging the Gilbert–Varshamov and Hamming bounds alongside random sampling simulations to characterize the manifold’s volume measure, asymptotic behavior of geodesic balls, and codebook structure. Contributions include: (i) tight upper and lower bounds on the kissing radius and an improved Hamming bound for $PU(n)$; (ii) distortion-rate function bounds for uniformly distributed sources; and (iii) closed-form minimum-distance calculations for projection Pauli, Clifford, and diagonal Clifford hierarchy gate groups, with simulation-validated predictions for covering radius and quantization distortion. These results establish novel theoretical tools and performance benchmarks for quantum error-correcting code design and geometric quantum information encoding.

Technology Category

• Machine Learning: Quantum Machine Learning
• Reasoning under Uncertainty: Uncertainty Representations
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Security and Privacy: Large-scale security measurements
• 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: Web performance, measurement, and characterization

Abstract

We consider a global phase-invariant metric in the projective unitary group PUn, relevant for universal quantum computing. We obtain the volume and measure of small metric ball in PUn and derive the Gilbert-Varshamov and Hamming bounds in PUn. In addition, we provide upper and lower bounds for the kissing radius of the codebooks in PUn as a function of the minimum distance. Using the lower bound of the kissing radius, we find a tight Hamming bound. Also, we establish bounds on the distortion-rate function for quantizing a source uniformly distributed over PUn. As example codebooks in PUn, we consider the projective Pauli and Clifford groups, as well as the projective group of diagonal gates in the Clifford hierarchy, and find their minimum distances. For any code in PUn with given cardinality we provide a lower bound of covering radius. Also, we provide expected value of the covering radius of randomly distributed points on PUn, when cardinality of code is sufficiently large. We discuss codebooks at various stages of the projective Clifford + T and projective Clifford + S constructions in PU2, and obtain their minimum distance, distortion, and covering radius. Finally, we verify the analytical results by simulation.

Problem

• Establish bounds on quantum error-correcting codes in projective unitary groups
• Analyze geometric properties of quantum codes using phase-invariant metrics
• Evaluate code performance through minimum distance and covering radius

Innovation

• Global phase-invariant metric in projective unitary group
• Bounds on kissing radius and distortion-rate function
• Analysis of codebooks in projective Clifford hierarchy