The Quantum Monadology

TL;DR

The Quantum Monadology addresses the challenge of encoding quantum side effects and measurement within a rigorous programming-language framework. It develops a domain-specific language QS that is embeddable into $LHoTT$ using Grothendieck's motivic yoga of six operations, modeling quantum effects as ambidextrous monads/comonads (notably the quantization modality $Q$ and the indefiniteness modality $\bigcirc_W$) and employing density-matrix formalisms for mixed states. Semantics rely on parameterized module spectra to interpret indexed quantum types and to capture topological quantum effects, dynamic lifting, and classical control. The framework aims to enable formally verifiable universal quantum programming with linear quantum types, classical control, and topological hardware considerations, with the companion TQP article detailing topological gates and implementation avenues.

Abstract

The modern theory of functional programming languages uses monads for encoding computational side-effects and side-contexts, beyond bare-bone program logic. Even though quantum computing is intrinsically side-effectful (as in quantum measurement) and context-dependent (as on mixed ancillary states), little of this monadic paradigm has previously been brought to bear on quantum programming languages. Here we systematically analyze the (co)monads on categories of parameterized module spectra which are induced by Grothendieck's "motivic yoga of operations" -- for the present purpose specialized to HC-modules and further to set-indexed complex vector spaces. Interpreting an indexed vector space as a collection of alternative possible quantum state spaces parameterized by quantum measurement results, as familiar from Proto-Quipper-semantics, we find that these (co)monads provide a comprehensive natural language for functional quantum programming with classical control and with "dynamic lifting" of quantum measurement results back into classical contexts. We close by indicating a domain-specific quantum programming language (QS) expressing these monadic quantum effects in transparent do-notation, embeddable into the recently constructed Linear Homotopy Type Theory (LHoTT) which interprets into parameterized module spectra. Once embedded into LHoTT, this should make for formally verifiable universal quantum programming with linear quantum types, classical control, dynamic lifting, and notably also with topological effects.