Higher-Order Programs with Indefinite Causal Orders: a Linear Approach to Coherent Control of Quantum Processes

2026-07-10

AI Summary

Existing quantum programming languages struggle to correctly model indefinite causal order—such as in the quantum switch—when non-unitary operations involving measurements are present, and they lack physically realizable formal support for coherent control. This work proposes a higher-order quantum functional language that, for the first time, extends indefinite causal order to general quantum channels with measurements within a linear type-theoretic framework. It introduces a synchronized measurement mechanism and type rules that go beyond standard linearity, guaranteeing that all well-typed programs are physically implementable. The language supports full first-order quantum channels and a broad class of second-order QC–QC circuits—including the quantum switch—and is equipped with small-step operational semantics, denotational semantics, and static verification in the causal category Caus[CPM]. The type system is proven sound and naturally extends to a nonlinear setting with recursion.

Abstract

Processes with indefinite causal orders (ICOs), such as the quantum switch, are higher-order quantum processes that superpose the order in which quantum operations are performed. Such coherent control yields computational advantages but is not faithfully captured by existing quantum programming languages: either they are restricted to the unitary case, and thus cannot combine ICOs with measurement, or they treat coherent control nonlinearly. In both cases, they do not realize the full computational power of ICOs. We introduce a higher-order quantum functional language that supports general quantum computation, not merely the permutation of channels, and whose linear type system allows quantum control to be well-defined beyond the unitary case, on arbitrary quantum channels. We equip this language with a small-step operational semantics that synchronizes measurement outcomes across superposed branches, using device references and a memory function. We also give a denotational semantics by means of completely positive maps. With linearity as the only constraint, some well-typed terms would denote unphysical maps. We therefore impose a typing discipline that goes beyond linearity, and interpret programs in the causal category Caus[CPM], under which every well-typed program is physically meaningful, a property that can be checked statically and efficiently. We prove soundness, and study the language's expressive power: it can express every quantum channel at first order, and at second order a large subclass of the so-called quantum circuits with quantum control (QC-QCs), containing the quantum switch. Last but not least, we show that this language is well-designed enough to be extended to the nonlinear setting with recursion.

Problem

• indefinite causal order
• higher-order quantum processes
• quantum programming languages
• coherent control
• quantum channels

Innovation

• indefinite causal order
• linear type system
• quantum functional language
• coherent control
• causal category