Compiling Quantum Lambda-Terms into Circuits via the Geometry of Interaction

2026-02-19

AI Summary

This work addresses the challenge of efficiently compiling higher-order quantum λ-terms—featuring higher-order control flow—into executable quantum circuits. To this end, it introduces Girard’s Geometry of Interaction framework into quantum compilation for the first time, integrating it with a linear quantum λ-calculus and a type system to simultaneously preserve classical computation and generate efficient, complete sequences of quantum operations. The core contribution is a novel compilation methodology that enables efficient circuit synthesis for quantum programs with higher-order control flow, supported by a formal type system that precisely characterizes the subset of λ-terms amenable to efficient compilation.

Technology Category

• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Qualitative Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions

Abstract

We present an algorithm turning any term of a linear quantum $λ$-calculus into a quantum circuit. The essential ingredient behind the proposed algorithm is Girard's geometry of interaction, which, differently from its well-known uses from the literature, is here leveraged to perform as much of the classical computation as possible, at the same time producing a circuit that, when evaluated, performs all the quantum operations in the underlying $λ$-term. We identify higher-order control flow as the primary obstacle towards efficient solutions to the problem at hand. Notably, geometry of interaction proves sufficiently flexible to enable efficient compilation in many cases, while still supporting a total compilation procedure. Finally, we characterize through a type system those $λ$-terms for which compilation can be performed efficiently.

Problem

• quantum lambda-calculus
• quantum circuit compilation
• higher-order control flow
• geometry of interaction
• efficient compilation

Innovation

• quantum lambda-calculus
• geometry of interaction
• quantum circuit compilation
• higher-order control flow
• type system