On Quantum and Probabilistic Linear Lambda-calculi (Extended Abstract)

AbstractIn this paper we give a fully complete model for a linear probabilistic lambda-calculus. The model is a Kripke semantics based on the category of stochastic relations. We sketch how this relates to quantum computation

The Vectorial λ\lambda-Calculus

We describe a type system for the linear-algebraic $\lambda$-calculus. The type system accounts for the linear-algebraic aspects of this extension of $\lambda$-calculus: it is able to statically describe the linear combinations of terms that will be obtained when reducing the programs. This gives rise to an original type theory where types, in the same way as terms, can be superposed into linear combinations. We prove that the resulting typed $\lambda$-calculus is strongly normalising and features weak subject reduction. Finally, we show how to naturally encode matrices and vectors in this typed calculus.Comment: Long and corrected version of arXiv:1012.4032 (EPTCS 88:1-15), to appear in Information and Computatio

Concrete resource analysis of the quantum linear system algorithm used to compute the electromagnetic scattering cross section of a 2D target

We provide a detailed estimate for the logical resource requirements of the quantum linear system algorithm (QLSA) [Phys. Rev. Lett. 103, 150502 (2009)] including the recently described elaborations [Phys. Rev. Lett. 110, 250504 (2013)]. Our resource estimates are based on the standard quantum-circuit model of quantum computation; they comprise circuit width, circuit depth, the number of qubits and ancilla qubits employed, and the overall number of elementary quantum gate operations as well as more specific gate counts for each elementary fault-tolerant gate from the standard set {X, Y, Z, H, S, T, CNOT}. To perform these estimates, we used an approach that combines manual analysis with automated estimates generated via the Quipper quantum programming language and compiler. Our estimates pertain to the example problem size N=332,020,680 beyond which, according to a crude big-O complexity comparison, QLSA is expected to run faster than the best known classical linear-system solving algorithm. For this problem size, a desired calculation accuracy 0.01 requires an approximate circuit width 340 and circuit depth of order $10^{25}$ if oracle costs are excluded, and a circuit width and depth of order $10^8$ and $10^{29}$, respectively, if oracle costs are included, indicating that the commonly ignored oracle resources are considerable. In addition to providing detailed logical resource estimates, it is also the purpose of this paper to demonstrate explicitly how these impressively large numbers arise with an actual circuit implementation of a quantum algorithm. While our estimates may prove to be conservative as more efficient advanced quantum-computation techniques are developed, they nevertheless provide a valid baseline for research targeting a reduction of the resource requirements, implying that a reduction by many orders of magnitude is necessary for the algorithm to become practical.Comment: 37 pages, 40 figure

Call-by-value, call-by-name and the vectorial behaviour of the algebraic \lambda-calculus

We examine the relationship between the algebraic lambda-calculus, a fragment of the differential lambda-calculus and the linear-algebraic lambda-calculus, a candidate lambda-calculus for quantum computation. Both calculi are algebraic: each one is equipped with an additive and a scalar-multiplicative structure, and their set of terms is closed under linear combinations. However, the two languages were built using different approaches: the former is a call-by-name language whereas the latter is call-by-value; the former considers algebraic equalities whereas the latter approaches them through rewrite rules. In this paper, we analyse how these different approaches relate to one another. To this end, we propose four canonical languages based on each of the possible choices: call-by-name versus call-by-value, algebraic equality versus algebraic rewriting. We show that the various languages simulate one another. Due to subtle interaction between beta-reduction and algebraic rewriting, to make the languages consistent some additional hypotheses such as confluence or normalisation might be required. We carefully devise the required properties for each proof, making them general enough to be valid for any sub-language satisfying the corresponding properties

Completeness of algebraic CPS simulations

The algebraic lambda calculus and the linear algebraic lambda calculus are two extensions of the classical lambda calculus with linear combinations of terms. They arise independently in distinct contexts: the former is a fragment of the differential lambda calculus, the latter is a candidate lambda calculus for quantum computation. They differ in the handling of application arguments and algebraic rules. The two languages can simulate each other using an algebraic extension of the well-known call-by-value and call-by-name CPS translations. These simulations are sound, in that they preserve reductions. In this paper, we prove that the simulations are actually complete, strengthening the connection between the two languages.Comment: In Proceedings DCM 2011, arXiv:1207.682

Semantics of a Typed Algebraic Lambda-Calculus

Algebraic lambda-calculi have been studied in various ways, but their semantics remain mostly untouched. In this paper we propose a semantic analysis of a general simply-typed lambda-calculus endowed with a structure of vector space. We sketch the relation with two established vectorial lambda-calculi. Then we study the problems arising from the addition of a fixed point combinator and how to modify the equational theory to solve them. We sketch an algebraic vectorial PCF and its possible denotational interpretations

Perceval: A Software Platform for Discrete Variable Photonic Quantum Computing

We introduce Perceval, an evolutive open-source software platform for simulating and interfacing with discrete variable photonic quantum computers, and describe its main features and components. Its Python front-end allows photonic circuits to be composed from basic photonic building blocks like photon sources, beam splitters, phase shifters and detectors. A variety of computational back-ends are available and optimised for different use-cases. These use state-of-the-art simulation techniques covering both weak simulation, or sampling, and strong simulation. We give examples of Perceval in action by reproducing a variety of photonic experiments and simulating photonic implementations of a range of quantum algorithms, from Grover's and Shor's to examples of quantum machine learning. Perceval is intended to be a useful toolkit both for experimentalists wishing to easily model, design, simulate, or optimise a discrete variable photonic experiment, and for theoreticians wishing to design algorithms and applications for discrete-variable photonic quantum computing platforms

Programmer un ordinateur quantique

MathsInfos Hors-Série Numéro 3, Fondation des Sciences Mathématiques de Pari