Effective Quantum Certification via Linear Homotopy Types

Abstract: The intricacies of realistic — namely: of classically controlled and (topologically) error-protected — quantum algorithms arguably make computer-assisted verification a practical necessity; and yet a satisfactory theory of dependent quantum data types had been missing, certainly one that would be aware of topological error-protection.

To solve this problem we present Linear homotopy type theory (LHoTT) as a programming and certification language for quantum computers with classical control and topologically protected quantum gates, focusing on (1.) its categorical semantics, which is a homotopy-theoretic extension of that of Proto-Quipper and a parameterized extension of Abramsky et al.'s quantum protocols, (2.) its expression of quantum measurement as a computational effect induced from dependent linear type formation and reminiscent of Lee at al.'s dynamic lifting monad but recovering the interacting systems of Coecke et al.'s "classical structures" monads.

Namely, we have recently shown that classical dependent type theory in its novel but mature full-blown form of Homotopy Type Theory (HoTT) is naturally a certification language for realistic topological logic gates. But given that categorical semantics of HoTT is famously provided by parameterized homotopy theory, we had argued earlier [Sc14] for a quantum enhancement LHoTT of classical HoTT, now with semantics in parameterized stable homotopy theory. This linear homotopy type theory LHoTT has meanwhile been formally described; here we explain it as the previously missing certified quantum language with monadic dynamic lifting, as announced in.

Concretely, we observe that besides its support, inherited from HoTT, for topological logic gates, LHoTT intrinsically provides a system of monadic computational effects which realize what in algebraic topology is known as the ambidextrous form of Grothendieck’s "Motivic Yoga"; and we show how this naturally serves to code quantum circuits subject to classical control implemented via computational effects. Logically this emerges as a linearly-typed quantum version of epistemic modal logic inside LHoTT, which besides providing a philosophically satisfactory formulation of quantum measurement, makes the language validate the quantum programming language axioms proposed by Staton; notably the deferred measurement principle is verified by LHoTT.

Finally we indicate the syntax of a domain-specific programming language QS (an abbreviation both for "Quantum Systems" and for "QS^0 -modules" aka spectra) which sugars LHoTT to a practical quantum programming language with all these features; and we showcase QS-pseudocode for simple forms of key algorithm classes, such as quantum teleportation, quantum error-correction and repeat-until-success quantum gates.

(This is joint work with D. J. Myers, M. Riley and H. Sati. Slides will be available at: ncatlab.org/schreiber/show/Quantum+Certification+via+Linear+Homotopy+Types)

Computer sciencealgebraic topologycategory theorylogic

Audience: learners

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