This document describes a quantum assembly language (QASM) called OpenQASM that is used to implement experiments with low depth quantum circuits. OpenQASM represents universal physical circuits over the CNOT plus SU(2) basis with straight-line code that includes measurement, reset, fast feedback, and gate subroutines. The simple text language can be written by hand or by higher level tools and may be executed on the IBM Q Experience.

Coherent control, aka quantum control, is a central concept in quantum computing that is attracting increasing attention from both the quantum foundations and quantum software communities. Defining coherent control in the presence of recursion and measurement has long been known to be a major challenge. In particular, no-go results have been established for standard semantical domains like completely positive maps. We address this problem by introducing the first quantum programming language with recursion that allows for the coherent control of arbitrary quantum operations. We equip this language with both an operational and a denotational semantics that we prove to be adequate. To design these semantics, we show that combining coherent control, recursion, and measurement crucially requires describing the evolution of subprograms in the absence of input. To address this, the operational semantics takes into account a default evolution branch, while the denotational semantics uses the concept of coherent quantum operation, based on vacuum extensions. We strengthen the validity of our approach by developing an observational equivalence: two programs are equivalent if their probability of termination is the same in any context. The denotational semantics is shown to be fully abstract with respect to this observational equivalence.

As quantum computing emerges as a promising computational paradigm, quantum programming languages provide the tools that bridge the distance between abstract programming and its hardware implementation. In some cases, restricted programming languages may even provide an avenue for more efficient circuit compilation strategies. In this work, we introduce foq , a first-order quantum programming language which allows for quantum control and recursion, and where a syntactically restricted subset of programs ( pfoq ) is shown to be sound and complete for quantum polytime computation. This is achieved by bounding both the recursion depth and the branching width of programs, which we demonstrate to still be compatible with various interesting applications, such as quantum teleportation and the quantum Fourier transform. pfoq constitutes the first programming-language-based characterization of the quantum complexity class fbqp , and we provide a semantics-preserving compilation algorithm such that any pfoq program can be compiled into a quantum circuit that grows polynomially on its number of input qubits, using an anchoring-and-merging technique to solve the problem of branch sequentialization.

Abstract The two main notions of control in quantum programming languages are often referred to as “quantum” control and “classical” control. With the latter, the control flow is based on classical information, potentially resulting from a quantum measurement, and this paradigm is well-suited to mixed state quantum computation. Whereas with quantum control, we are primarily focused on pure quantum computation and there the “control” is based on superposition. The two paradigms have not mixed well traditionally and they are almost always treated separately. In this work, we show that the paradigms may be combined within the same system. The key ingredients for achieving this are: (1) syntactically: a modality for incorporating pure quantum types into a mixed state quantum type system; (2) operationally: an adaptation of the notion of “quantum configuration” from quantum lambda-calculi, where the quantum data is replaced with pure quantum primitives; (3) denotationally: suitable (sub)categories of Hilbert spaces, for pure computation and von Neumann algebras, for mixed state computation in the Heisenberg picture of quantum mechanics.

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The class of Basic Feasible Functionals BFF is the second-order counterpart of the class of first-order functions computable in polynomial time. We present several implicit characterizations of BFF based on a typed programming language of terms. These terms may perform calls to non-recursive imperative procedures. The type discipline has two layers: the terms follow a standard simply-typed discipline and the procedures follow a standard tier-based type discipline. BFF consists exactly of the second-order functionals that are computed by typable and terminating programs. The completeness of this characterization surprisingly still holds in the absence of lambda-abstraction. Moreover, the termination requirement can be specified as a completeness-preserving instance, which can be decided in time quadratic in the size of the program. As typing is decidable in polynomial time, we obtain the first tractable (i.e., decidable in polynomial time), sound, complete, and implicit characterization of BFF, thus solving a problem opened for more than 20 years.

Abstract In this paper, we study quantitative properties of quantum programs. Properties of interest include (positive) almost-sure termination, expected runtime or expected cost, that is, for example, the expected number of applications of a given quantum gate, etc. After studying the completeness of these problems in the arithmetical hierarchy over the Clifford+T fragment of quantum mechanics, we express these problems using a variation of a quantum pre-expectation transformer, a weakest pre-condition based technique that allows to symbolically compute these quantitative properties. Under a smooth restriction—a restriction to polynomials of bounded degree over a real closed field—we show that the quantitative problem, which consists in finding an upper-bound to the pre-expectation, can be decided in time double-exponential in the size of a program, thus providing, despite its great complexity, one of the first decidable results on the analysis and verification of quantum programs. Finally, we sketch how the latter can be transformed into an efficient synthesis method.

In automated complexity analysis, noninterference-based type systems statically guarantee, via soundness, the property that well-typed programs compute functions of a given complexity class, e.g., the class FP of functions computable in polynomial time. These characterizations are also extensionally complete - they capture all functions - but are not intensionally complete as some polytime algorithms are rejected. This impact on expressive power is an unavoidable cost of achieving a tractable characterization. To circumvent this issue, an avenue arising from security applications is to find a relaxation of noninterference based on a declassification mechanism that allows critical data to be released in a safe and controlled manner. Following this path, we present a new and intuitive declassification policy preserving FP-soundness and capturing strictly more programs than existing noninterference-based systems. We show the versatility of the approach: it also provides a new characterization of the class BFF of second-order polynomial time computable functions in a second-order imperative language, with first-order procedure calls. Type inference is tractable: it can be done in polynomial time.

Abstract We introduce a first-order quantum programming language, named foq , whose terminating programs are reversible. We restrict foq to a strict and tractable subset, named pfoq , of terminating programs with bounded width, that provides a first programming language-based characterization of the quantum complexity class fbqp . We finally present a tractable semantics-preserving algorithm compiling a pfoq program to a quantum circuit of size polynomial in the number of input qubits.

We introduce a new noninterference policy to capture the class of functions computable in polynomial time on an object-oriented programming language. This policy makes a clear separation between the standard noninterference techniques for the control flow and the layering properties required to ensure that each “security” level preserves polynomial time soundness, and is thus very powerful as for the class of programs it can capture. This new characterization is a proper extension of existing tractable characterizations of polynomial time based on safe recursion. Despite the fact that this noninterference policy is Π 1 0 -complete, we show that it can be instantiated to some decidable and conservative instance using shape analysis techniques.

Abstract The class of Basic Feasible Functionals $$\mathtt{BFF}$$ BFF is the second-order counterpart of the class of first-order functions computable in polynomial time. We present several implicit characterizations of $$\mathtt{BFF}$$ BFF based on a typed programming language of terms. These terms may perform calls to imperative procedures, which are not recursive. The type discipline has two layers: the terms follow a standard simply-typed discipline and the procedures follow a standard tier-based type discipline. $$\mathtt{BFF}$$ BFF consists exactly of the second-order functionals that are computed by typable and terminating programs. The completeness of this characterization surprisingly still holds in the absence of lambda-abstraction. Moreover, the termination requirement can be specified as a completeness-preserving instance, which can be decided in time quadratic in the size of the program. As typing is decidable in polynomial time, we obtain the first tractable (i.e., decidable in polynomial time), sound, complete, and implicit characterization of $$\mathtt{BFF}$$ BFF , thus solving a problem opened for more than 20 years.

We introduce a new kind of expectation transformer for a mixed classical-quantum programming language. Our semantic approach relies on a new notion of a cost structure, which we introduce and which can be seen as a specialisation of the Kegelspitzen of Keimel and Plotkin. We show that our weakest precondition analysis is both sound and adequate with respect to the operational semantics of the language. Using the induced expectation transformer, we provide formal analysis methods for the expected cost analysis and expected value analysis of classical-quantum programs. We illustrate the usefulness of our techniques by computing the expected cost of several well-known quantum algorithms and protocols, such as coin tossing, repeat until success, entangled state preparation, and quantum walks.

The class of Basic Feasible Functionals BFF$_2$ is the type-2 counterpart of the class FP of type-1 functions computable in polynomial time. Several characterizations have been suggested in the literature, but none of these present a programming language with a type system guaranteeing this complexity bound. We give a characterization of BFF$_2$ based on an imperative language with oracle calls using a tier-based type system whose inference is decidable. Such a characterization should make it possible to link higher-order complexity with programming theory. The low complexity (cubic in the size of the program) of the type inference algorithm contrasts with the intractability of the aforementioned methods and does not overly constrain the expressive power of the language.

Abstract Inductive datatypes in programming languages allow users to define useful data structures such as natural numbers, lists, trees, and others. In this paper we show how inductive datatypes may be added to the quantum programming language QPL. We construct a sound categorical model for the language and by doing so we provide the first detailed semantic treatment of user-defined inductive datatypes in quantum programming. We also show our denotational interpretation is invariant with respect to big-step reduction, thereby establishing another novel result for quantum programming. Compared to classical programming, this property is considerably more difficult to prove and we demonstrate its usefulness by showing how it immediately implies computational adequacy at all types. To further cement our results, our semantics is entirely based on a physically natural model of von Neumann algebras, which are mathematical structures used by physicists to study quantum mechanics.

The class of Basic Feasible Functionals BFF2 is the type-2 counterpart of the class FP of type-1 functions computable in polynomial time. Several characterizations have been suggested in the literature, but none of these present a programming language with a type system guaranteeing this complexity bound. We give a characterization of BFF2 based on an imperative language with oracle calls using a tier-based type system whose inference is decidable. Such a characterization should make it possible to link higher-order complexity with programming theory. The low complexity (cubic in the size of the program) of the type inference algorithm contrasts with the intractability of the aforementioned methods and does not restrain strongly the expressive power of the language.

We design an interpretation-based theory of higher-order functions that is well-suited for the complexity analysis of a standard higher- order functional language a` la ml. We manage to express the interpretation of a given program in terms of a least fixpoint and we show that when restricted to functions bounded by higher-order polynomials, they characterize exactly classes of tractable functions known as Basic Feasible Functions at any order.

Algebra and coalgebra are widely used to model data types in functional programming languages and proof assistants. Their use permits to better structure the computations and also to enhance the expressivity of a language or of a proof system. Interestingly, parametric polymorphism à la System F provides a way to encode algebras and coalgebras in strongly normalizing languages without losing the good logical properties of the calculus. Even if these encodings are sometimes unsatisfying because they provide only limited forms of algebras and coalgebras, they give insights on the expressivity of System F in terms of functions that we can program in it. With the goal of contributing to a better understanding of the expressivity of Implicit Computational Complexity systems, we study the problem of defining algebras and coalgebras in the Light Affine Lambda Calculus, a system characterizing the complexity class FPTIME. This system limits the computational complexity of programs but it also limits the ways we can use parametric polymorphism, and in general the way we can write our programs. We show here that while the restrictions imposed by the Light Affine Lambda Calculus pose some issues to the standard System F encodings, they still permit to encode some form of algebra and coalgebra. Using the algebra encoding one can define in the Light Affine Lambda Calculus the traditional inductive types. Unfortunately, the corresponding coalgebra encoding permits only a very limited form of coinductive data types. To extend this class we study an extension of the Light Affine Lambda Calculus by distributive laws for the modality §. This extension has been discussed but not studied before.

The sup-interpretation method is proposed as a new tool to control memory resources of first order functional programs with pattern matching by static analysis. It has been introduced in order to increase the intensionality, that is the number of captured algorithms, of a previous method, the quasi-interpretations. Basically, a sup-interpretation provides an upper bound on the size of function outputs. A criterion, which can be applied to terminating as well as nonterminating programs, is developed in order to bound the stack frame size polynomially. Since this work is related to quasi-interpretation, dependency pairs, and size-change principle methods, we compare these notions obtaining several results. The first result is that, given any program, we have heuristics for finding a sup-interpretation when we consider polynomials of bounded degree. Another result consists in the characterizations of the sets of functions computable in polynomial time and in polynomial space. A last result consists in applications of sup-interpretations to the dependency pair and the size-change principle methods.

A sup-interpretation is a tool which provides upper bounds on the size of the values computed by the function symbols of a program. Sup-interpretations have shown their interest to deal with the complexity of first order functional programs. This paper is an attempt to adapt the framework of sup-interpretations to a fragment of object-oriented programs, including loop and while constructs and methods with side effects. We give a criterion, called brotherly criterion, which uses the notion of sup-interpretation to ensure that each brotherly program computes objects whose size is polynomially bounded by the inputs sizes. Moreover we give some heuristics in order to compute the sup-interpretation of a given method.

In this paper, we study characterizations of polynomial complexity classes using first order functional programs and we try to improve their intensionality, that is the number of natural algorithms captured. We use polynomial assignments over the reals. The polynomial assignments used are inspired by the notions of quasiinterpretation and sup-interpretation, and are decidable when considering polynomials of bounded degree ranging over real numbers. Contrarily to quasi-interpretations, the considered assignments are not required to have the subterm property. Consequently, they capture a strictly larger number of natural algorithms (including quotient, gcd, duplicate elimination from a list) than previous characterizations using quasi-interpretations