Abstract The rapid progress of quantum technologies, fostered by the efforts of both academia and industry, has stimulated the design of quantum programming languages and the development of methods to support their verification and optimization. As in the classical setting, static analysis plays a fundamental role in such an endeavour. In this paper, we provide a survey on static analysis approaches for quantum programs, which have been proposed in the literature, distinguishing between dataflow-oriented approaches, which are based on a graph representation of the program information flow, and domain-oriented approaches, which essentially consist of the definition of some appropriate abstract domains representing the program property to be analysed. To illustrate these two perspectives concretely, we also present in detail two specific analyses: a dataflow analysis for managing quantum variables and uncomputation, and a static analysis based on abstract interpretation for detecting state entanglement.

Semantics-based program analysis uses an abstract semantics of programs/systems to statically determine run-time properties. Classic examples from compiler technology include analyses to support constant propagation and constant folding transformations and estimation of pointer values to prevent buffer overruns. More recent examples include the estimation of information flows (to enforce security constraints) and estimation of non-functional properties such as timing (to determine worst case execution times in hard real-time applications). The classical approaches are based on semantics involving discrete mathematics. Paralleling trends in model-checking, there have been recent moves towards using probabilistic and quantitative methods in program analysis. In this paper we will start by reviewing both classical and probabilistic/quantitative approaches to program analysis. We will provide a comparison of the two approaches. We will use a simple information flow analysis to exemplify the classical approach. The existence of covert information flows through timing channels are difficult to detect using classical techniques; we show how such problems can be addressed using probabilistic techniques.

The -calculus that maintains irreversibility. Recently, reversible computational models have been studied mainly in the context of quantum computation, as (without measurements) quantum physics is inherently reversible. However, reversibility also fundamentally changes the semantical framework in which classical computation has to be investigated. We describe an implementation of classical combinatory logic in a reversible calculus for which we present an algebraic model based on a generalisation of the notion of a group.

We introduce a quantitative approach to the analysis of distributed systems which relies on a linear operator based network semantics. A typical problem in a distributed setting is how information propagates through a network, and a typical qualitative analysis is concerned with establishing whether some information will eventually be transmitted from one node to another node in the network. The quantitative approach we present allows us to obtain additional information such as an estimation of the probability that some data is transmitted within a given interval of time. We formalise situations like this using a probabilistic version of a process calculus which is the core of KLAIM, a language for distributed and mobile computing based on interactions through distributed tuple spaces. The analysis we present exploits techniques based on Probabilistic Abstract Interpretation and is characterised by compositional aspects which greatly simplify the inspection of the nodes interaction and the detection of the information propagation through a computer network.

We show how the framework of probabilistic abstract interpretation can be applied to statically analyse a probabilistic version of the λ-calculus. The resulting analysis allows for a more speculative use of its outcomes based on the consideration of statistically defined quantities. After introducing a linear operator based semantics for our probabilistic λ-calculus Λp, and reviewing the framework of abstract interpretation and strictness analysis, we demonstrate our technique by constructing a probabilistic (first-order) strictness analysis for Λp.

We address the problem of characterising the security of a program against unauthorised information flows. Classical approaches are based on non-interference models which depend ultimately on the notion of process equivalence. In these models confidentiality is an absolute property stating the abse nce of any illegal information flow. We present a model in which the notion of non-interference is approximated in the sense that it allows for some exactly quantified leakage of information. This is characterised via a notion of process similarity which replaces the indistinguishability of processes by a quantitative measure of their behavioural difference. Such a quantity is related to the number of statistical tests needed to distinguish two behaviours. We also present two semantics-based analyses of approximate non-interference and we show that one is a correct abstraction of the other.

We address the problem of characterising the security of a program against unauthorised information flows. Classical approaches are based on non-interference models which depend ultimately on the notion of process equivalence. In these models confidentiality is an absolute property stating the absence of any illegal information flow. We present a model in which the notion of non-interference is approximated in the sense that it allows for some exactly quantified leakage of information. This is characterised via a notion of process similarity which replaces the indistinguishability of processes by a quantitative measure of their behavioural difference. Such a quantity is related to the number of statistical tests needed to distinguish two behaviours. We also present two semantics-based analyses of approximate noninterference and we show that one is a correct abstraction of the other.

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We develop a denotational, fully abstract semantics for constraint logic programming (clp) with respect to successful and failed observables. The denotational approach turns out very useful for the definition of new operators on the language as the counterpart of some abstract operations on the denotational domain. In particular, by defining our domain as a cylindric Heyting algebra, we can exploit, to this aim, operations of both cylindric algebras (such as cylindrification), and Heyting algebras (such as implication and negation). The former allows us to generalize the clp language by introducing an explicit hiding operator, the latter allows us to define a notion of negation which extends the classical negation used in logic programming. In particular, we show that our notion subsumes both negation as failure and negation as instantiation.