5 research outputs found
Reversible Effects as Inverse Arrows
Reversible computing models settings in which all processes can be reversed.
Applications include low-power computing, quantum computing, and robotics. It
is unclear how to represent side-effects in this setting, because conventional
methods need not respect reversibility. We model reversible effects by adapting
Hughes' arrows to dagger arrows and inverse arrows. This captures several
fundamental reversible effects, including serialization and mutable store
computations. Whereas arrows are monoids in the category of profunctors, dagger
arrows are involutive monoids in the category of profunctors, and inverse
arrows satisfy certain additional properties. These semantics inform the design
of functional reversible programs supporting side-effects.Comment: 15 pages; corrected Example 3.
With a Few Square Roots, Quantum Computing is as Easy as {\Pi}
Rig groupoids provide a semantic model of \PiLang, a universal classical
reversible programming language over finite types. We prove that extending rig
groupoids with just two maps and three equations about them results in a model
of quantum computing that is computationally universal and equationally sound
and complete for a variety of gate sets. The first map corresponds to an
root of the identity morphism on the unit . The second map
corresponds to a square root of the symmetry on . As square roots are
generally not unique and can sometimes even be trivial, the maps are
constrained to satisfy a nondegeneracy axiom, which we relate to the Euler
decomposition of the Hadamard gate. The semantic construction is turned into an
extension of \PiLang, called \SPiLang, that is a computationally universal
quantum programming language equipped with an equational theory that is sound
and complete with respect to the Clifford gate set, the standard gate set of
Clifford+T restricted to qubits, and the computationally universal
Gaussian Clifford+T gate set
With a Few Square Roots, Quantum Computing Is as Easy as Pi
Rig groupoids provide a semantic model of Π, a universal classical reversible programming language over finite types. We prove that extending rig groupoids with just two maps and three equations about them results in a model of quantum computing that is computationally universal and equationally sound and complete for a variety of gate sets. The first map corresponds to an 8th root of the identity morphism on the unit 1. The second map corresponds to a square root of the symmetry on 1+1. As square roots are generally not unique and can sometimes even be trivial, the maps are constrained to satisfy a nondegeneracy axiom, which we relate to the Euler decomposition of the Hadamard gate. The semantic construction is turned into an extension of Π, called √Π, that is a computationally universal quantum programming language equipped with an equational theory that is sound and complete with respect to the Clifford gate set, the standard gate set of Clifford+T restricted to ≤2 qubits, and the computationally universal Gaussian Clifford+T gate set