Axiomatic theory of Higher-Order Quantum Computation

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2019

Higher-order quantum theory is an extension of quantum theory where one introduces transformations whose input and output are transformations, thus generalizing the notion of channels and quantum operations. The generalization then goes recursively, with the construction of a full hierarchy of maps of increasingly higher order. The analysis of special cases already showed that higher-order quantum functions exhibit features that cannot be tracked down to the usual circuits, such as indefinite causal structures, providing provable advantages over circuital maps. The present treatment provides a general framework where this kind of analysis can be carried out in full generality. The hierarchy of higher-order quantum maps is introduced axiomatically with a formulation based on the language of types of transformations. Complete positivity of higher-order maps is derived from the general admissibility conditions instead of being postulated as in previous approaches. The recursive charact...

Time in Physics

2022

Quantum processes with indefinite causal structure emerge when we wonder which are the most general evolutions, allowed by quantum theory, of a set of local systems which are not assumed to be in any particular causal order. These processes can be described within the framework of higher-order quantum theory which, starting from considering maps from quantum transformations to quantum transformations, recursively constructs a hierarchy of quantum maps of increasingly higher order. In this work, we develop a formalism for quantum computation with indefinite causal structures; namely we characterize the computational structure of higher order quantum maps. Taking an axiomatic approach, the rules of this computation are identified as the most general compositions of higher order maps which are compatible with the mathematical structure of quantum theory. We provide a mathematical characterization of the admissible composition for arbitrary higher order quantum maps. We prove that these...

arXiv preprint arXiv:0912.0195, 2009

Abstract: The manuscript poses and addresses a new very fundamental issue in Quantum Computer Science, which is going to have a radical impact on the way we currently conceive quantum computation. We show that there exists a new kind of" higher-order" quantum computation, potentially much more powerful than the usual quantum processing, which is feasible, but cannot be realized by a usual quantum circuit. In order to implement this new kind of computations one needs to change the rules of quantum circuits, also considering ...

Physical Review A, 2013

We show that quantum theory allows for transformations of black boxes that cannot be realized by inserting the input black boxes within a circuit in a pre-defined causal order. The simplest example of such a transformation is the classical switch of black boxes, where two input black boxes are arranged in two different orders conditionally on the value of a classical bit. The quantum version of this transformation-the quantum switch-produces an output circuit where the order of the connections is controlled by a quantum bit, which becomes entangled with the circuit structure. Simulating these transformations in a circuit with fixed causal structure requires either postselection, or an extra query to the input black boxes.

Electronic Notes in Theoretical Computer Science, 2008

Several domains can be used to define the semantics of quantum programs. Among them Abramsky [1] has introduced a semantics based on probabilistic power domains, whereas the one by Selinger associates with every program a completely positive map. In this paper, we mainly introduce a semantical domain based on admissible transformations, i.e. multisets of linear operators. In order to establish a comparison with existing domains, we introduce a simple quantum imperative language (QIL), equipped with three different denotational semantics, called pure, observable, and admissible respectively. The pure semantics is a natural extension of probabilistic (classical) semantics and is similar to the semantics proposed by Abramsky [1]. The observable semantics,à la Selinger [16], associates with any program a superoperator over density matrices. Finally, we introduce an admissible semantics which associates with any program an admissible transformation. These semantics are not equivalent, but exact abstraction or interpretation relations are established between them, leading to a hierarchy of quantum semantics. 2 Trace decreasing, completely positive maps. 3 Multisets of linear operators satisfying a completeness condition, see section 2.

Entropy

In this work we advance a generalization of quantum computational logics capable of dealing with some important examples of quantum algorithms. We outline an algebraic axiomatization of these structures.

2018

In this work we advance a generalization of quantum computational logics capable of dealing with some important examples of quantum algorithms. We outline an algebraic axiomatization of these structures. 1Universidad Nacional de La Plata, Instituto de F́ısica (IFLP-CCT-CONICET), C.C. 727, 1900 La Plata, Argentina 2Universitá di Cagliari, Via Is Mirrionis 1, 09123, Cagliari-Italia 3Center Leo Apostel for Interdisciplinary Studies and, Department of Mathematics, Brussels Free University Krijgskundestraat 33, 1160 Brussels, Belgium

We present an axiomatic and operational theory of quantum mechan- ics. The theory is founded on the axiomatic and operational approach started in Geneva mainly by Constantin Piron and his students and collaborators and developed further in Brussels by myself and differ- ent students and collaborators. A physical entity, which a priori can be a classical entity or a quantum entity or a combination of both, is described by means of its set of states, its set of properties and a phys- ical notion of ‘actuality of a property the entity being in a state’. This leads to the mathematical structure of a state property space (Σ, L, ξ), where Σ is the set of states of the entity, L the set of properties, and ξ : Σ → P(L) a function mapping each state on the set of properties that are actual if the entity is in that state. We introduce seven axioms such that if satisfied the state property space can be represented by the direct union over a classical state space of irreducible state prop- erty spaces, where each one of the irreducible state property spaces is a Hilbert space state property space of standard quantum mechanics, over the real, complex or quaternionic numbers. The axioms are in- troduced in an as much as possible operational way, such that we can analyze their physical meaning.

This thesis studies the categorical formalisation of quantum computing, through the prism of type theory, in a three-tier process. The first stage of our investigation involves the creation of the dagger lambda calculus, a lambda calculus for dagger compact categories. Our second contribution lifts the expressive power of the dagger lambda calculus, to that of a quantum programming language, by adding classical control in the form of complementary classical structures and dualisers. Finally, our third contribution demonstrates how our lambda calculus can be applied to various well known problems in quantum computation.