Programming a quantum computer, i.e., implementing quantum algorithms on a quantum processor-based copmputer architecture, is a task that can be addressed (just as for classical computers) at different levels of abstraction. This paper proposes a denotational semantics for high-level quantum programming constructs, focusing on the conceptual meaning of quantum-controlled branching and iteration. We introduce a denotational domain where a mathematical meaning of a quantum control flow with loops can be defined, which reflects the coherent evolution of the quantum system implementing the program.
Programming a quantum computer, i.e., implementing quantum algorithms on a quantum processor-based copmputer architecture, is a task that can be addressed (just as for classical computers) at different levels of abstraction. This paper proposes a denotational semantics for high-level quantum programming constructs, focusing on the conceptual meaning of quantum-controlled branching and iteration. We introduce a denotational domain where a mathematical meaning of a quantum control flow with loops can be defined, which reflects the coherent evolution of the quantum system implementing the program.
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.
Quantum algorithms for tasks such as factorization, search, and simulation rely on control flow such as branching and iteration that depends on the value of data in superposition. High-level programming abstractions for control flow, such as switches, loops, higher-order functions, and continuations, are ubiquitous in classical languages. By contrast, many quantum languages do not provide high-level abstractions for control flow in superposition, and instead require the use of hardware-level logic gates to implement such control flow. The reason for this gap is that whereas a classical computer supports control flow abstractions using a program counter that can depend on data, the typical architecture of a quantum computer does not analogously provide a program counter that can depend on data in superposition. As a result, the complete set of control flow abstractions that can be correctly realized on a quantum computer has not yet been established. In this work, we provide a complete characterization of the properties of control flow abstractions that are correctly realizable on a quantum computer. First, we prove that even on a quantum computer whose program counter exists in superposition, one cannot correctly realize control flow in quantum algorithms by lifting the classical conditional jump instruction to work in superposition. This theorem denies the ability to directly lift general abstractions for control flow such as the λ-calculus from classical to quantum programming. In response, we present the necessary and sufficient conditions for control flow to be correctly realizable on a quantum computer. We introduce the quantum control machine, an instruction set architecture featuring a conditional jump that is restricted to satisfy these conditions. We show how this design enables a developer to correctly express control flow in quantum algorithms using a program counter in place of logic gates.
Quantum algorithms for tasks such as factorization, search, and simulation rely on control flow such as branching and iteration that depends on the value of data in superposition. High-level programming abstractions for control flow, such as switches, loops, and higher-order functions, are ubiquitous in classical languages. By contrast, many quantum languages do not provide high-level abstractions for control flow in superposition, and instead require the use of hardware-level logic gates to implement such control flow. The reason for this gap is that whereas a classical computer supports control flow using a program counter that can depend on data, the typical architecture of a quantum computer does not provide a program counter that can depend on data in superposition. As a result, the complete set of control flow abstractions that can be correctly realized on a quantum computer has not yet been established. In this work, we provide a complete characterization of the properties of control flow abstractions that are correctly realizable on a quantum computer. First, we prove that even on a quantum computer whose program counter exists in superposition, one cannot correctly realize control flow in quantum algorithms by lifting the classical conditional jump instruction to work in superposition. This theorem denies the ability to directly lift general abstractions for control flow such as the λ-calculus from classical to quantum programming. In response, we present the necessary and sufficient conditions for control flow to be correctly realizable on a quantum computer. We introduce the quantum control machine, an instruction set architecture featuring a conditional jump that is restricted to satisfy these conditions. We show how this design enables a developer to correctly express control flow in quantum algorithms using a program counter in place of logic gates.
In this thesis, we study quantum programming languages, and focus on the quantum lambda-calculus, a paradigmatic language defined by Selinger and Valiron, which mixes a rich classical control flow (higher order functions, etc.) and quantum data (qubit creation, measure, and unitary operations). In this thesis, we search for a denotational semantics of this language, in other words, we extract the meaning and behaviour of programs by interpreting them in an adequately chosen mathematical universe. In order to model the quantum lambda-calculus, one must adapt the traditional denotational models to the mathematics of quantum computation (Hilbert spaces, density matrices, etc.). We have three main contributions:Firstly, we extend game semantics to the quantum case. Game semantics is a dynamic approach to denotational semantics, which focuses on the sequence of interactions between a program and its environment. We obtain the first denotational model of the full quantum lambda-calculus which is both compositional and interactive, meaning that it represents the dynamic of the execution. Our model generalise the recently developed model for the probabilistic case.Secondly, we formally link our model to the only pre-existing denotational model of the full lambda-calculus, the quantum relational model of Pagani, Selinger and Valiron.Lastly, we prove that our game model is fully abstract, the gold standard of denotational semantics, which ensures a total correspondence between the programming language and the model. We deduce that the quantum relational model was also fully abstract, which was an open problem.This thesis is constituted of three parts. The first part details some preliminaries on category theory, used for defining a formal framework including both the quantum game model and the quantum relational model, some preliminaries on the mathematics of quantum computation, and then some preliminaries on quantum programming and its models. The second part starts with some preliminaries on game semantics and defines our quantum game model in the simplified case of the linear quantum lambda-calculus. The third part builds on the second part to define the full quantum game model, prove its full abstraction, and deduce that the quantum relational model is fully abstract too.
Current quantum programs are mainly designed at the level of quantum gates acting on individual qubits; on a large scale and for complex problems this may involve a high cognitive load on the programmer, making the program specification nontrivial and error-prone. In this context, providing quantum programming with higher abstraction mechanisms will assist in making this task more manageable and robust against design errors. In this work, a conceptual framework is addressed following the notion of the whole quantum computation as a structure composed of quantum registers representing each an undivided entity. Thus, computation progresses through semantically well-defined transformations that act on, or entangle, quantum registers, thereby modifying the global state. Ultimately, the program reaches the desired state by following a specific composition strategy. With this in mind, high-level syntax is presented through an algebraic formalism that bridges them with their low-level semantics. Proposed syntax is based on certain well-know operations used on quantum algorithms that apply phase shifts upon logical condition satisfaction or leverage on parallel evaluation. Based solely on the formalized operations, a quantum satisfiability modulo theories (SMT) solver can be designed. At its core, this work contributes to establishing some methodological principles towards realizing a high-level quantum structured programming.
We propose a notion of quantum control in a quantum programming language which permits the superposition of finitely many quantum operations without performing a measurement. This notion takes the form of a conditional construct similar to the "if" statement in classical programming languages. We show that adding such a quantum "if" statement to the QPLprogramming language simplifies the presentation of several quantum algorithms. This motivates the possibility of extending the denotational semantics of QPL to include this form of quantum alternation. We give a denotational semantics for this extension of QPL based on Kraus decompositions rather than on superoperators. Finally, we clarify the relation between quantum alternation and recursion, and discuss the possibility of lifting the semantics defined by Kraus operators to the superoperator semantics defined by Selinger.