9 research outputs found
Classical Control, Quantum Circuits and Linear Logic in Enriched Category Theory
We describe categorical models of a circuit-based (quantum) functional
programming language. We show that enriched categories play a crucial role.
Following earlier work on QWire by Paykin et al., we consider both a simple
first-order linear language for circuits, and a more powerful host language,
such that the circuit language is embedded inside the host language. Our
categorical semantics for the host language is standard, and involves cartesian
closed categories and monads. We interpret the circuit language not in an
ordinary category, but in a category that is enriched in the host category. We
show that this structure is also related to linear/non-linear models. As an
extended example, we recall an earlier result that the category of W*-algebras
is dcpo-enriched, and we use this model to extend the circuit language with
some recursive types
Quantum Programming with Inductive Datatypes: Causality and Affine Type Theory
Inductive datatypes in programming languages allow users to define useful
data structures such as natural numbers, lists, trees, and others. In this
paper we show how inductive datatypes may be added to the quantum programming
language QPL. We construct a sound categorical model for the language and by
doing so we provide the first detailed semantic treatment of user-defined
inductive datatypes in quantum programming. We also show our denotational
interpretation is invariant with respect to big-step reduction, thereby
establishing another novel result for quantum programming. Compared to
classical programming, this property is considerably more difficult to prove
and we demonstrate its usefulness by showing how it immediately implies
computational adequacy at all types. To further cement our results, our
semantics is entirely based on a physically natural model of von Neumann
algebras, which are mathematical structures used by physicists to study quantum
mechanics
Quantum Programming with Inductive Datatypes
Inductive datatypes in programming languages allow users to define useful data structures such as natural numbers, lists, trees, and others. In this paper we show how inductive datatypes may be added to the quantum programming language QPL. We construct a sound categorical model for the language and by doing so we provide the first detailed semantic treatment of user-defined inductive datatypes in quantum programming. We also show our denotational interpretation is invariant with respect to big-step reduction, thereby establishing another novel result for quantum programming. Compared to classical programming, this property is considerably more difficult to prove and we demonstrate its usefulness by showing how it immediately implies computational adequacy at all types. To further cement our results, our semantics is entirely based on a physically natural model of von Neumann algebras, which are mathematical structures used by physicists to study quantum mechanics
Infinite-Dimensionality in Quantum Foundations: W*-algebras as Presheaves over Matrix Algebras
In this paper, W*-algebras are presented as canonical colimits of diagrams of
matrix algebras and completely positive maps. In other words, matrix algebras
are dense in W*-algebras
Convexity and order in probabilistic call-by-name FPC
Kegelspitzen are mathematical structures coined by Keimel and Plotkin, in order to encompass the structure of a convex set and the structure of a dcpo. In this paper, we ask ourselves what are Kegelspitzen the model of. We adopt a categorical viewpoint and show that Kegelspitzen model stochastic matrices onto a category of domains. Consequently, Kegelspitzen form a denotational model of pPCF, an abstract functional programming language for probabilistic computing. We conclude the present work with a discussion of the interpretation of (probabilistic) recursive types, which are types for entities which might contain other entities of the same type, such as lists and trees
Hybrid divide-and-conquer approach for tree search algorithms
As we are entering the era of real-world small quantum computers, finding
applications for these limited devices is a key challenge. In this vein, it was
recently shown that a hybrid classical-quantum method can help provide
polynomial speed-ups to classical divide-and-conquer algorithms, even when only
given access to a quantum computer much smaller than the problem itself. In
this work we study the hybrid divide-and-conquer method in the context of tree
search algorithms, and extend it by including quantum backtracking, which
allows better results than previous Grover-based methods. Further, we provide
general criteria for polynomial speed-ups in the tree search context, and
provide a number of examples where polynomial speed ups, using arbitrarily
smaller quantum computers, can still be obtained. This study possible speed-ups
for the well known algorithm of DPLL and prove threshold-free speed-ups for the
tree search subroutines of the so-called PPSZ algorithm - which is the core of
the fastest exact Boolean satisfiability solver - for certain classes of
formulas. We also provide a simple example where speed-ups can be obtained in
an algorithm-independent fashion, under certain well-studied
complexity-theoretical assumptions. Finally, we briefly discuss the fundamental
limitations of hybrid methods in providing speed-ups for larger problems.Comment: 48 pages, 13 figure