Philippe Jorrand

Toward a quantum process algebra

Quantum computations operate in the quantum world. For their results to be useful in any way, there is an intrinsic necessity of cooperation and communication controlled by the classical world. As a consequence, full formal descriptions of algorithms making use of quantum principles must take into account both quantum and classical computing components and assemble them so that they communicate and cooperate. This paper aims at defining a high level language allowing the description of classical and quantum programming, and their cooperation. Since process algebras provide a framework to model cooperating computations and have well defined semantics, they have been chosen as a basis for this language. Starting with a classical process algebra, this paper explains how to transform it for including quantum computation. The result is a quantum process algebra with its operational semantics, which can be used to fully describe quantum algorithms in their classical context.

Unifying quantum computation with projective measurements only and one-way quantum computation

Quantum measurement is universal for quantum computation. Two models for performing measurement-based quantum computation exist: the one-way quantum computater was introduced by Briegel and Raussendorf and quantum computation via projective measurements only by Nielsen. The more recent development of this second model is based on state transfers instead of teleportation. From this development a finite but approximate quantum universal family of observables is exhibited which includes only one two-qubit observable while others are one-qubit observables. In this article an infinite but exact quantum universal family of observables is proposed including also only one two-qubit observable. The rest of the paper is dedicated to compare these two models of measurement-based quantum computation i.e. one-way quantum computation and quantum computation via projective measurements only. From this comparison which was initiated by Cirac and Verstraete closer and more natural connections appear between these two models. These close connections lead to a unified view of measurement-based quantum computation.

Classically-controlled Quantum Computation

It is reasonable to assume that quantum computations take place under the control of the classical world. For modelling this standard situation, we introduce a Classically-controlled Quantum Turing Machine (CQTM) which is a Turing machine with a quantum tape for acting on quantum data, and a classical transition function for a formalized classical control. In CQTM, unitary transformations and quantum measurements are allowed. We show that any classical Turing machine is simulated by a CQTM without loss of efficiency. Furthermore, we show that any k-tape CQTM is simulated by a 2-tape CQTM with a quadratic loss of efficiency. The gap between classical and quantum computations which was already pointed out in the framework of measurement-based quantum computation (see [S. Perdrix, Ph. Jorrand, Measurement-Based Quantum Turing Machines and their Universality, arXiv, quant-ph/0404146, 2004]) is confirmed in the general case of classically-controlled quantum computation. In order to appreciate the similarity between programming classical Turing machines and programming CQTM, some examples of CQTM will be given in the full version of the paper. Proofs of lemmas and theorems are omitted in this extended abstract.

From quantum physics to programming languages: a process algebraic approach

Research in quantum computation is looking for the consequences of having information encoding, processing and communication exploit the laws of quantum physics, i.e. the laws of the ultimate knowledge that we have, today, of the foreign world of elementary particles, as described by quantum mechanics. After an introduction to the principles of quantum information processing and a brief survey of the major breakthroughs brought by the first ten years of research in this domain, this paper concentrates on a typically “computer science” way to reach a deeper understanding of what it means to compute with quantum resources, namely on the design of programming languages for quantum algorithms and protocols, and on the questions raised by the semantics of such languages. Special attention is devoted to the process algebraic approach to such languages, through a presentation of QPAlg, the Quantum Process Algebra which is being designed by the authors.

Classically controlled quantum computation

It is reasonable to assume that quantum computations take place under the control of the classical world. For modelling this standard situation, we introduce a Classically controlled Quantum Turing Machine (CQTM), which is a Turing machine with a quantum tape for acting on quantum data, and a classical transition function for formalised classical control. In a CQTM, unitary transformations and quantum measurements are allowed. We show that any classical Turing machine can be simulated by a CQTM without loss of efficiency. Furthermore, we show that any

SEPARABILITY OF PURE N-QUBIT STATES: TWO CHARACTERIZATIONS

k

Towards a Quantum Calculus

-tape CQTM can be simulated by a 2-tape CQTM with a quadratic loss of efficiency. In order to compare CQTMs with existing models of quantum computation, we prove that any uniform family of quantum circuits (Yao 1993) is efficiently approximated by a CQTM. Moreover, we prove that any semi-uniform family of quantum circuits (Nishimura and Ozawa 2002), and any measurement calculus pattern (Danos et al. 2004) are efficiently simulated by a CQTM. Finally, we introduce a Measurement-based Quantum Turing Machine (MQTM), which is a restriction of CQTMs in which only projective measurements are allowed. We prove that any CQTM is efficiently simulated by a MQTM. In order to appreciate the similarity between programming classical Turing machines and programming CQTMs, some examples of CQTMs are given.

QUANTUM OCTAL GAMES

Given a pure state |ψN>∈ℋN of a quantum system composed of n qubits, where ℋN is the Hilbert space of dimension N=2n, this paper answers two questions: what conditions should the amplitudes in |ψN> satisfy for this state to be separable (i) into a tensor product of n qubit states |ψ2>0⊗ |ψ2>1 ⊗⋯⊗ |ψ2>n-1, and (ii), into a tensor product of two subsystems states |ψP> ⊗ |ψQ> with P=2p and Q=2q such that p+q=n? For both questions, necessary and sufficient conditions are proved, thus characterizing at the same time families of separable and entangled states of n qubit systems. A number of more refined questions about separability in n qubit systems can be studied on the basis of these results.

The quantum challenge to computer science

The aim of this paper is to introduce a general model of quantum computation, the quantum calculus: both unitary transformations and projective measurements are allowed; furthermore a complete classical control, including conditional structures and loops, is available. Complementary to its operational semantics, we introduce a pure denotational semantics for the quantum calculus. Based on probabilistic power domains [Jones, C. and G. D. Plotkin, A probabilistic powerdomain of evaluations, in: LICS, 1989, pp. 186-195. URL http://homepages.inf.ed.ac.uk/gdp/publications/Prob_Powerdomain.pdf], this pure denotational semantics associates with any description of a computation in the quantum calculus its action in a mathematical setting. Adequacy between operational and pure denotational semantics is established. Additionally to this pure denotational semantics, an observable denotational semantics is introduced. Following the work by Selinger, this observable denotational semantics is based on density matrices and super-operators. Finally, we establish an exact abstraction connection between these two semantics.

Minimal sets of conditions for distinguishing among separable and entangled states of multiqubit systems

Quantum games offer situations where quantum information theory may help in solving or improving games with lack of information. For example, one may rely upon properties of entanglement for showin...