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Coalgebraic Quantum Computation

R. Duncan, P. Panangaden (eds, Frank Roumen

Published Aug 17, 2016• 2 citations
OpenAlex

Abstract

Coalgebras generalize various kinds of dynamical systems occuring in mathematics and computer science. Examples of systems that can be modeled as coalgebras include automata and Markov chains. We will present a coalgebraic representation of systems occuring in the field of quantum computation, using convex sets of density matrices as state spaces. This will allow us to derive a method to convert quantum mechanical systems into simpler probabilistic systems with the same probabilistic behaviour. 1

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Coalgebraic Quantum Computation

Frank Roumen

Coalgebras generalize various kinds of dynamical systems occuring in mathematics and computer science. Examples of systems that can be modeled as coalgebras include automata and Markov chains. We will present a coalgebraic representation of systems occuring in the field of quantum computation, using convex sets of density matrices as state spaces. This will allow us to derive a method to convert quantum mechanical systems into simpler probabilistic systems with the same probabilistic behaviour.

arXiv:1408.0050•Electronic Proceedings in Theoretical Computer Science•Jul 28, 2014•2 citations

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As a quantum counterpart of labeled transition system (LTS), quantum labeled transition system (QLTS) is a powerful formalism for modeling quantum programs or protocols, and gives a categorical understanding for quantum computation. With the help of quantum branching monad, QLTS provides a framework extending some ideas in non-deterministic or probabilistic systems to quantum systems. On the other hand, quantum finite automata (QFA) emerged as a very elegant and simple model for resolving some quantum computational problems. In this paper, we propose the notion of reactive quantum system (RQS), a variant of QLTS capturing reactive system behavior, and develop a coalgebraic semantics for QLTS, RQS and QFA by an endofunctor on the category of convex sets, which has a final coalgebra. Such a coalgebraic semantics provides a unifying abstract interpretation for QLTS, RQS and QFA. The notions of bisimulation and simulation can be employed to compare the behavior of different types of quantum systems and judge whether a coalgebra can be behaviorally simulated by another.

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Recent works have shown that defining a behavioural equivalence that matches the observational properties of a quantum-capable, concurrent, non-deterministic system is a surprisingly difficult task.We explore coalgebras over distributions taking weights from a generic effect algebra, which subsumes probabilities and quantum effects, a physical formalism that represents the probabilistic behaviour of an open quantum system.To abide by the properties of quantum theory, we introduce monads graded on a partial commutative monoid, intuitively allowing composition of two processes only if they use different quantum resources, as prescribed by the no-cloning theorem.We investigate the relation between an open quantum system and its probabilistic counterparts obtained when instantiating the input with a specific quantum state.We consider Aczel-Mendler and kernel bisimilarities, advocating for the latter as it characterizes quantum systems that exhibit the same probabilistic behaviour for all input states.Finally, we propose operators on quantum effect labelled transition systems, paving the way for a process calculi semantics that is parametric over the quantum input.

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Ulrich Faigle, Alexander Schönhuth

A framework for finite-dimensional quantum Markov chains on Hilbert spaces is introduced. Quantum Markov chains generalize both classical Markov chains with possibly hidden states and existing models of quantum walks on finite graphs. Quantum Markov chains are based on Markov operations that may be applied to quantum systems and include quantum measurements, for example. It is proved that quantum Markov chains are asymptotically stationary and hence possess ergodic and entropic properties. With a quantum Markov chain one may associate a quantum Markov process, which is a stochastic process in the classical sense. Generalized Markov chains allow a representation with respect to a generalized Markov source model with definite (but possibly hidden) states relative to which observables give rise to classical stochastic processes. It is demonstrated that this model allows for observables to violate Bell’s inequality.

•Nov 4, 2010•3 citations
Coalgebras generalize various kinds of dynamical systems occuring in mathematics and computer science. Examples of systems that can be modeled as coalgebras include automata and Markov chains. We will present a coalgebraic representation of systems occuring in the field of quantum computation, using convex sets of density matrices as state spaces. This will allow us to derive a method to convert quantum mechanical systems into simpler probabilistic systems with the same probabilistic behaviour. 1
As a quantum counterpart of labeled transition system (LTS), quantum labeled transition system (QLTS) is a powerful formalism for modeling quantum programs or protocols, and gives a categorical understanding for quantum computation. With the help of quantum branching monad, QLTS provides a framework extending some ideas in non-deterministic or probabilistic systems to quantum systems. On the other hand, quantum finite automata (QFA) emerged as a very elegant and simple model for resolving some quantum computational problems. In this paper, we propose the notion of reactive quantum system (RQS), a variant of QLTS capturing reactive system behavior, and develop a coalgebraic semantics for QLTS, RQS and QFA by an endofunctor on the category of convex sets, which has a final coalgebra. Such a coalgebraic semantics provides a unifying abstract interpretation for QLTS, RQS and QFA. The notions of bisimulation and simulation can be employed to compare the behavior of different types of quantum systems and judge whether a coalgebra can be behaviorally simulated by another.
The topic of this thesis is a sequence of results which re-situate exchangeability theorems from probability theory in categories of probabilistic and quantum processes. These theorems take sequences of events, say random variables, random matrices, or quantum states, and study what behaviours are possible after imposing invariance of the action of finite permutations of the elements of the sequence. These are often philosophically motivated, in the classical case by subjectivist foundations of probability, in the quantum case by the school of quantum Bayesianism. In chapters 3 and 4, quantum and classical de Finetti theorems are generalised to limits of diagrams in appropriate categories, moving from statements about measures and quantum states to theorems about parameterised random measures and quantum states. The latter also considers the use of multisets for encoding exchangeability. In chapter 5, an approach using coalgebras of functors as theories of systems is taken. Instead of looking at sequences of events, we consider processes, coalgebras, that take a parameter and probabilistically return both an output and an updated parameter. Such coalgebras can be exchangeable, and it is shown that these coalgebras have a universal object similar to in the limit objects of the previous chapters, namely a final exchangeable coalgebra. Both the multiset and coalgebraic theorems are given quantum analogues.
Recent works have shown that defining a behavioural equivalence that matches the observational properties of a quantum-capable, concurrent, non-deterministic system is a surprisingly difficult task.We explore coalgebras over distributions taking weights from a generic effect algebra, which subsumes probabilities and quantum effects, a physical formalism that represents the probabilistic behaviour of an open quantum system.To abide by the properties of quantum theory, we introduce monads graded on a partial commutative monoid, intuitively allowing composition of two processes only if they use different quantum resources, as prescribed by the no-cloning theorem.We investigate the relation between an open quantum system and its probabilistic counterparts obtained when instantiating the input with a specific quantum state.We consider Aczel-Mendler and kernel bisimilarities, advocating for the latter as it characterizes quantum systems that exhibit the same probabilistic behaviour for all input states.Finally, we propose operators on quantum effect labelled transition systems, paving the way for a process calculi semantics that is parametric over the quantum input.
We give an introduction to research associated with the generalised probabilistic theories framework, also known as the convex framework. States are real vectors representing lists of probabilities of measurement outcomes. Convex combinations of the vectors represent probabilistic combinations of different state preparations. Transformations are real matrices. Measurement outcomes are represented by functionals of the states, inner products of the state with a real vector, whose values are the probability of the measurement outcome in question. The framework generalises quantum theory. We describe the operational meaning of the framework, and how the concepts can be defined in terms of cones of states and measurement outcome vectors. We describe how the classical and quantum probability theories are represented in the framework. We describe Bell non-locality and the theory with super-quantum non-locality known as box world. We discuss generalised Hamiltonian mechanics in the discrete case and in continuous phase space, including the role of negativity of the phase space density in contextuality and tunnelling.
A framework for finite-dimensional quantum Markov chains on Hilbert spaces is introduced. Quantum Markov chains generalize both classical Markov chains with possibly hidden states and existing models of quantum walks on finite graphs. Quantum Markov chains are based on Markov operations that may be applied to quantum systems and include quantum measurements, for example. It is proved that quantum Markov chains are asymptotically stationary and hence possess ergodic and entropic properties. With a quantum Markov chain one may associate a quantum Markov process, which is a stochastic process in the classical sense. Generalized Markov chains allow a representation with respect to a generalized Markov source model with definite (but possibly hidden) states relative to which observables give rise to classical stochastic processes. It is demonstrated that this model allows for observables to violate Bell’s inequality.