Qudit Quantum Programming with Projective Cliffords

2024-07-23

arXiv.org

AI Summary

This work addresses the limitation of existing quantum programming frameworks—confined to qubits and unable to support arbitrary-dimensional qudit projective Clifford operations—by introducing the first qudit-oriented Projective Clifford programming abstraction. Methodologically, it models projective Clifford operations as functions over the qudit Pauli group, develops a categorical semantics based on ℤ_d-linear maps, and designs a corresponding typed λ-calculus. Key contributions include: (i) the first categorical semantics and type system for projective Clifford operations; (ii) algebraic modeling of Pauli encodings with a proof of semantic completeness; and (iii) a sound, composable theoretical foundation for qudit Clifford programming. The framework enables formal verification and structured quantum program construction, providing scalable language-theoretic support for high-dimensional quantum computation.

Technology Category

• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Qualitative Reasoning
• Reasoning under Uncertainty: Probabilistic Programming

Application Category

• Security and Privacy: Data transparency and provenance
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

This paper introduces a novel abstraction for programming quantum operations, specifically projective Cliffords, as functions over the qudit Pauli group. We define a categorical semantics for projective Cliffords based on Pauli encodings in terms of $mathbb{Z}_d$-linear maps. We then introduce a type system and lambda calculus for both $mathbb{Z}_d$-linear maps and projective Cliffords, and prove that these type systems have a sound denotational semantics in terms of the relevant categories. Finally, we explore what it means to program with projective Cliffords through a number of examples and programming constructions.

Problem

• Develops a qudit quantum programming abstraction
• Introduces LambdaPC for projective Clifford operations
• Generalizes Pauli tableaux to any qudit dimension

Innovation

• Qudit programming with projective Cliffords
• LambdaPC type system for Clifford operations
• Generalizes Pauli tableaux to any dimension