Schreiber The Quantum Monadology

An article that we are finalizing:

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• Hisham Sati, $\;$ Urs Schreiber:

  
  

  The Quantum Monadology

  
  

  download:

  • pdf

  • arXiv:2310.15735

  
  

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 $H\mathbb{C}$-modules and further to set-indexed complex vector spaces, as discussed in the companion article [EoS]. 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) 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 language with classical control, dynamic lifting and topological effects (discussed in the companion article [TQP]).

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