Robin Kaarsgaard

Quantum computing improves substantially on known classical algorithms for various important problems, but the nature of the relationship between quantum and classical computing is not yet fully understood. This relationship can be clarified by free models, that add to classical computing just enough physical principles to represent quantum computing and no more. Here we develop an axiomatisation of quantum computing that replaces the standard continuous postulates with a small number of discrete equations, as well as a free model that replaces the standard linear-algebraic model with a category-theoretical one. The axioms and model are based on reversible classical computing, isolate quantum advantage in the ability to take certain well-behaved square roots, and link to various quantum computing hardware platforms. This approach allows combinatorial optimisation, including brute force computer search, to optimise quantum computations. The free model may be interpreted as a programming language for quantum computers, that has the same expressivity and computational universality as the standard model, but additionally allows automated verification and reasoning.

Quantum computing improves substantially on known classical algorithms for various important problems, but the nature of the relationship between quantum and classical computing is not yet fully understood. This relationship can be clarified by free models, that add to classical computing just enough physical principles to represent quantum computing and no more. Here, we develop an axiomatization of quantum computing that replaces the standard continuous postulates with a small number of discrete equations, as well as a free model that replaces the standard linear-algebraic model with a category-theoretical one. The axioms and model are based on reversible classical computing, isolate quantum advantage in the ability to take certain well-behaved square roots, and link to various quantum computing hardware platforms. This approach allows combinatorial optimization, including brute force computer search, to optimize quantum computations. The free model may be interpreted as a programming language for quantum computers, that has the same expressivity and computational universality as the standard model, but additionally allows automated verification and reasoning.

Quantum computing offers advantages over classical computation, yet the precise features that set the two apart remain unclear. In the standard quantum circuit model, adding a 1-qubit basis-changing gate—commonly chosen to be the Hadamard gate—to a universal set of classical reversible gates yields computationally universal quantum computation. However, the computational behaviours enabled by this addition are not fully characterised. We give such a characterisation by introducing a small quantum programming language extending the universal classical reversible programming language Π with a single primitive corresponding to the Hadamard gate. The language comes equipped with a sound and complete categorical semantics that is specified by a purely equational theory. Completeness is shown by means of a novel finite presentation, and a corresponding synthesis algorithm, for the groups of orthogonal matrices with entries in the ring ℤ[1/√2].

We construct a computationally universal quantum programming language Quantum Π from two copies of Π , the internal language of rig groupoids. The first step constructs a pure (measurement-free) term language by interpreting each copy of Π in a generalisation of the category Unitary in which every morphism is “rotated” by a particular angle, and the two copies are amalgamated using a free categorical construction expressed as a computational effect. The amalgamated language only exhibits quantum behaviour for specific values of the rotation angles, a property which is enforced by imposing a small number of equations on the resulting category. The second step in the construction introduces measurements by layering an additional computational effect.

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.

We provide a universal construction of the category of finite-dimensional C*-algebras and completely positive trace-nonincreasing maps from the rig category of finite-dimensional Hilbert spaces and unitaries. This construction, which can be applied to any dagger rig category, is described in three steps, each associated with their own universal property, and draws on results from dilation theory in finite dimension. In this way, we explicitly construct the category that captures hybrid quantum/classical computation with possible nontermination from the category of its reversible foundations. We discuss how this construction can be used in the design and semantics of quantum programming languages.