  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.
