5 research outputs found
Linear Dependent Type Theory for Quantum Programming Languages
Modern quantum programming languages integrate quantum resources and
classical control. They must, on the one hand, be linearly typed to reflect the
no-cloning property of quantum resources. On the other hand, high-level and
practical languages should also support quantum circuits as first-class
citizens, as well as families of circuits that are indexed by some classical
parameters. Quantum programming languages thus need linear dependent type
theory. This paper defines a general semantic structure for such a type theory
via certain fibrations of monoidal categories. The categorical model of the
quantum circuit description language Proto-Quipper-M by Rios and Selinger
(2017) constitutes an example of such a fibration, which means that the
language can readily be integrated with dependent types. We then devise both a
general linear dependent type system and a dependently typed extension of
Proto-Quipper-M, and provide them with operational semantics as well as a
prototype implementation
Quantum and Reality
Formalizations of quantum information theory in category theory and type
theory, for the design of verifiable quantum programming languages, need to
express its two fundamental characteristics: (1) parameterized linearity and
(2) metricity. The first is naturally addressed by dependent-linearly typed
languages such as Proto-Quipper or, following our recent observations: Linear
Homotopy Type Theory (LHoTT). The second point has received much attention
(only) in the form of semantics in "dagger-categories", where operator adjoints
are axiomatized but their specification to Hermitian adjoints still needs to be
imposed by hand.
We describe a natural emergence of Hermiticity which is rooted in principles
of equivariant homotopy theory, lends itself to homotopically-typed languages
and naturally connects to topological quantum states classified by twisted
equivariant KR-theory. Namely, we observe that when the complex numbers are
considered as a monoid internal to Z/2-equivariant real linear types, via
complex conjugation, then (finite-dimensional) Hilbert spaces do become
self-dual objects among internally-complex Real modules.
The point is that this construction of Hermitian forms requires of the
ambient linear type theory nothing further than a negative unit term of tensor
unit type. We observe that just such a term is constructible in LHoTT, where it
interprets as an element of the infinity-group of units of the sphere spectrum,
tying the foundations of quantum theory to homotopy theory. We close by
indicating how this allows for encoding (and verifying) the unitarity of
quantum gates and of quantum channels in quantum languages embedded into LHoTT.Comment: 10 pages, some figure
Entanglement of Sections: The pushout of entangled and parameterized quantum information
Recently Freedman & Hastings asked for a mathematical theory that would unify
quantum entanglement/tensor-structure with parameterized/bundle-structure via
their amalgamation (a hypothetical pushout) along bare quantum (information)
theory. As a proposed answer to this question, we first make precise a form of
the relevant pushout diagram in monoidal category theory. Then we prove that
the pushout produces what is known as the *external* tensor product on vector
bundles/K-classes, or rather on flat such bundles (flat K-theory), i.e., those
equipped with monodromy encoding topological Berry phases. The bulk of our
result is a further homotopy-theoretic enhancement of the situation to the
"derived category" (infinity-category) of flat infinity-vector bundles
("infinity-local systems") equipped with the "derived functor" of the external
tensor product. Concretely, we present an integral model category of simplicial
functors into simplicial K-chain complexes which conveniently presents the
infinity-category of parameterized HK-module spectra over varying base spaces
and is equipped with homotopically well-behaved external tensor product
structure. In concluding we indicate how this model category serves as
categorical semantics for the linear-multiplicative fragment of Linear Homotopy
Type Theory (LHoTT), which is thus exhibited as a universal quantum programming
language. This is the context in which we recently showed that topological
anyonic braid quantum gates are native objects in LHoTT.Comment: 71 pages, various figure
The Quantum Monadology
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.Comment: 120 pages, various figure
Linear Dependent Type Theory for Quantum Programming Languages
Modern quantum programming languages integrate quantum resources and
classical control. They must, on the one hand, be linearly typed to reflect the
no-cloning property of quantum resources. On the other hand, high-level and
practical languages should also support quantum circuits as first-class
citizens, as well as families of circuits that are indexed by some classical
parameters. Quantum programming languages thus need linear dependent type
theory. This paper defines a general semantic structure for such a type theory
via certain fibrations of monoidal categories. The categorical model of the
quantum circuit description language Proto-Quipper-M by Rios and Selinger
(2017) constitutes an example of such a fibration, which means that the
language can readily be integrated with dependent types. We then devise both a
general linear dependent type system and a dependently typed extension of
Proto-Quipper-M, and provide them with operational semantics as well as a
prototype implementation