Peter Selinger

Towards a quantum programming language

We propose the design of a programming language for quantum computing. Traditionally, quantum algorithms are frequently expressed at the hardware level, for instance in terms of the quantum circuit model or quantum Turing machines. These approaches do not encourage structured programming or abstractions such as data types. In this paper, we describe the syntax and semantics of a simple quantum programming language with high-level features such as loops, recursive procedures, and structured data types. The language is functional in nature, statically typed, free of run-time errors, and has an interesting denotational semantics in terms of complete partial orders of superoperators.

A Survey of Graphical Languages for Monoidal Categories

This article is intended as a reference guide to various notions of monoidal categories and their associated string diagrams. It is hoped that this will be useful not just to mathematicians, but also to physicists, computer scientists, and others who use diagrammatic reasoning. We have opted for a somewhat informal treatment of topological notions, and have omitted most proofs. Nevertheless, the exposition is sufficiently detailed to make it clear what is presently known, and to serve as a starting place for more in-depth study. Where possible, we provide pointers to more rigorous treatments in the literature. Where we include results that have only been proved in special cases, we indicate this in the form of caveats.

Control categories and duality: on the categorical semantics of the lambda-mu calculus

We give a categorical semantics to the call-by-name and call-by-value versions of Parigots λμ-calculus with disjunction types. We introduce the class of control categories, which combine a cartesian-closed structure with a premonoidal structure in the sense of Power and Robinson. We prove, via a categorical structure theorem, that the categorical semantics is equivalent to a CPS semantics in the style of Hofmann and Streicher. We show that the call-by-name λμ-calculus forms an internal language for control categories, and that the call-by-value λμ-calculus forms an internal language for the dual co-control categories. As a corollary, we obtain a syntactic duality result: there exist syntactic translations between call-by-name and call-by-value that are mutually inverse and preserve the operational semantics. This answers a question of Streicher and Reus.

A lambda calculus for quantum computation with classical control

In this paper we develop a functional programming language for quantum computers by extending the simply-typed lambda calculus with quantum types and operations. The design of this language adheres to the ‘quantum data, classical control’ paradigm, following the first authors work on quantum flow-charts. We define a call-by-value operational semantics, and give a type system using affine intuitionistic linear logic. The main results of this paper are the safety properties of the language and the development of a type inference algorithm.

Quipper: a scalable quantum programming language

The field of quantum algorithms is vibrant. Still, there is currently a lack of programming languages for describing quantum computation on a practical scale, i.e., not just at the level of toy problems. We address this issue by introducing Quipper, a scalable, expressive, functional, higher-order quantum programming language. Quipper has been used to program a diverse set of non-trivial quantum algorithms, and can generate quantum gate representations using trillions of gates. It is geared towards a model of computation that uses a classical computer to control a quantum device, but is not dependent on any particular model of quantum hardware. Quipper has proven effective and easy to use, and opens the door towards using formal methods to analyze quantum algorithms.

A Brief Survey of Quantum Programming Languages

This article is a brief and subjective survey of quantum programming language research. 1 Quantum Computation Quantum computing is a relatively young subject. It has its beginnings in 1982, when Paul Benioff and Richard Feynman independently pointed out that a quantum mechanical system can be used to perform computations [11, p.12]. Feynman’s interest in quantum computation was motivated by the fact that it is computationally very expensive to simulate quantum physical systems on classical computers. This is due to the fact that such simulation involves the manipulation is extremely large matrices (whose dimension is exponential in the size of the quantum system being simulated). Feynman conceived of quantum computers as a means of simulating nature much more efficiently. The evidence to this day is that quantum computers can indeed perform certain tasks more efficiently than classical computers. Perhaps the best-known example is Shor’s factoring algorithm, by which a quantum computer can find the prime factors of any integer in probabilistic polynomial time [15]. There is no known classical probabilistic algorithm which can solve this problem in polynomial time. In the ten years since the publication of Shor’s result, there has been an enormous surge of research in quantum algorithms and quantum complexity theory. 2 Quantum Programming Languages Quantum physics involves phenomena, such as superposition and entanglement, whose properties are not always intuitive. These same phenomena give quantum computation its power, and are often at the heart of an interesting quantum algorithm. However, there does not yet seem to be a unifying set of principles by which quantum algorithms are developed; each new algorithm seems to rely on a unique set of “tricks” to achieve its particular goal. One of the goals of programming language design is to identify and promote useful “high-level” concepts — abstractions or paradigms which allow humans 2 to think about a problem in a conceptual way, rather than focusing on the details of its implementation. With respect to quantum programming, it is not yet clear what a useful set of abstractions would be. But the study of quantum programming languages provides a setting in which one can explore possible language features and test their usefulness and expressivity. Moreover, the definition of prototypical programming languages creates a unifying formal framework in which to view and analyze existing quantum algorithm. 2.1 Virtual Hardware Models Advances in programming languages are often driven by advances in compiler design, and vice versa. In the case of quantum computation, the situation is complicated by the fact that no practical quantum hardware exists yet, and not much is known about the detailed architecture of any future quantum hardware. To be able to speak of “implementations”, it is therefore necessary to fix some particular, “virtual” hardware model to work with. Here, it is understood that future quantum hardware may differ considerably, but the differences should ideally be transparent to programmers and should be handled automatically by the compiler or operating system. There are several possible virtual hardware models to work with, but fortunately all of them are equivalent, at least in theory. Thus, one may pick the model which fits one’s computational intuitions most closely. Perhaps the most popular virtual hardware model, and one of the easiest to explain, is the quantum circuit model. Here, a quantum circuit is made up from quantum gates in much the same way as a classical logic circuit is made up from logic gates. The difference is that quantum gates are always reversible, and they correspond to unitary transformations over a complex vector space. See e.g. [3] for a succinct introduction to quantum circuits. Of the two basic quantum operations, unitary transformations and measurements, the quantum circuit model emphasizes the former, with measurements always carried out as the very last step in a computation. Another virtual hardware model, and one which is perhaps even better suited for the interpretation of quantum programming languages, is the QRAM model of Knill [9]. Unlike the quantum circuit model, the QRAM models allows unitary transformations and measurements to be freely interleaved. In the QRAMmodel, a quantum device is controlled by a universal classical computer. The quantum device contains a large, but finite number of individually addressable quantum bits, much like a RAM memory chip contains a multitude of classical bits. The classical controller sends a sequence of instructions, which are either of the form “apply unitary transformation U to qubits i and j” or “measure qubit i”. The quantum device carries out these instruction, and responds by making the results of the measurements available. A third virtual hardware model, which is sometimes used in complexity theory, is the quantum Turing machine. Here, measurements are never performed, and the entire operation of the machine, which consists of a tape, head, and finite control, is assumed to be unitary. While this model is theoretically equivalent

Exact synthesis of multiqubit Clifford+Tcircuits

We prove that a unitary matrix has an exact representation over the Clifford+

Quantum circuits of T-depth one

T

Quantum lambda calculus

gate set with local ancillas if and only if its entries are in the ring

First-Order Axioms for Asynchrony

\mathbb{Z}[\frac{1}{\sqrt{2}},i]