Quantum lambda calculus has been studied mainly as an idealized programming language - the evaluation essentially corresponds to a deterministic abstract machine. Very little work has been done to develop a rewriting theory for quantum lambda calculus. Recent advances in the theory of probabilistic rewriting give us a way to tackle this task with tools unavailable a decade ago. Our primary focus are standardization and normalization results.

In this paper, we present a linear and reversible programming language with inductives types and recursion. The semantics of the languages is based on pattern-matching; we show how ensuring syntactical exhaustivity and non-overlapping of clauses is enough to ensure reversibility. The language allows to represent any Primitive Recursive Function. We then give a Curry-Howard correspondence with the logic $μ$MALL: linear logic extended with least fixed points allowing inductive statements. The critical part of our work is to show how primitive recursion yields circular proofs that satisfy $μ$MALL validity criterion and how the language simulates the cut-elimination procedure of $μ$MALL.

In this paper, we explore the interaction between two monoidal structures: a multiplicative one, for the encoding of pairing, and an additive one, for the encoding of choice. We propose a colored PROP to model computation in this framework, where the choice is parameterized by an algebraic side effect: the model can support regular tests, probabilistic and non-deterministic branching, as well as quantum branching, i.e. superposition. The graphical language comes equipped with a denotational semantics based on linear applications, and an equational theory. We prove the language to be universal, and the equational theory to be complete with respect to this semantics.

This paper is concerned with the expressivity and denotational semantics of a functional higher-order reversible programming language based on Theseus. In this language, pattern-matching is used to ensure the reversibility of functions. We show how one can encode any Reversible Turing Machine in said language. We then build a sound and adequate categorical semantics based on join inverse categories, with additional structures to capture pattern-matching and to interpret inductive types and recursion. We then derive a notion of completeness in the sense that any computable, partial, first-order injective function is the image of a term in the language.

In this paper, we present a linear and reversible programming language with inductives types and recursion. The semantics of the languages is based on pattern-matching; we show how ensuring syntactical exhaustivity and non-overlapping of clauses is enough to ensure reversibility. The language allows to represent any Primitive Recursive Function. We then give a Curry-Howard correspondence with the logic μMALL: linear logic extended with least fixed points allowing inductive statements. The critical part of our work is to show how primitive recursion yields circular proofs that satisfy μMALL validity criterion and how the language simulates the cut-elimination procedure of μMALL.

We introduce the first complete equational theory for quantum circuits. More precisely, we introduce a set of circuit equations that we prove to be sound and complete: two circuits represent the same unitary map if and only if they can be transformed one into the other using the equations. The proof is based on the properties of multi-controlled gates – that are defined using elementary gates – together with an encoding of quantum circuits into linear optical circuits, which have been proved to have a complete axiomatisation.

We extend the circuit model of quantum computation so that the wiring between gates is soft-coded within registers inside the gates. The addresses in these registers can be manipulated and put into superpositions. This aims at capturing indefinite causal orders and making their geometrical layout explicit: we express the quantum switch and the polarizing beam-splitter within the model. In this context, our main contribution is a full characterization of the anonymity constraints. Indeed, the names used as addresses should not matter beyond the wiring they describe; i.e., quantum evolutions should commute with “renamings.” We show that these quantum evolutions can still act non-trivially upon the names. We specify the structure of “nameblind” matrices.

We introduce the LO_v-calculus, a graphical language for reasoning about linear optical quantum circuits with so-called vacuum state auxiliary inputs. We present the axiomatics of the language and prove its soundness and completeness: two LO_v-circuits represent the same quantum process if and only if one can be transformed into the other with the rules of the LO_v-calculus. We give a confluent and terminating rewrite system to rewrite any polarisation-preserving LO_v-circuit into a unique triangular normal form, inspired by the universal decomposition of Reck et al. (1994) for linear optical quantum circuits.

Current proposals for quantum compilers require the synthesis and optimization of linear reversible circuits and among them CNOT circuits. Since these circuits represent a significant part of the cost of running an entire quantum circuit, we aim at reducing their size. In this paper we present a new algorithm for the synthesis of CNOT circuits based on the solution of the syndrome decoding problem. Our method addresses the case of ideal hardware with an all-to-all qubit connectivity and the case of near-term quantum devices with restricted connectivity. For both cases, we present benchmarks showing that our algorithm outperforms existing algorithms.

Abstract While recent progress in quantum hardware open the door for significant speedup in certain key areas, quantum algorithms are still hard to implement right, and the validation of such quantum programs is a challenge. In this paper we propose Qbricks, a formal verification environment for circuit-building quantum programs, featuring both parametric specifications and a high degree of proof automation. We propose a logical framework based on first-order logic, and develop the main tool we rely upon for achieving the automation of proofs of quantum specification: PPS, a parametric extension of the recently developed path sum semantics. To back-up our claims, we implement and verify parametric versions of several famous and non-trivial quantum algorithms, including the quantum parts of Shor’s integer factoring, quantum phase estimation (QPE) and Grover’s search.

In this paper, we introduce dynamic lifting to a quantum circuit-description language, following the Proto-Quipper language approach. Dynamic lifting allows programs to transfer the result of measuring quantum data - qubits - into classical data - booleans -. We propose a type system and an operational semantics for the language and we state safety properties. Next, we introduce a concrete categorical semantics for the proposed language, basing our approach on a recent model from Rios&Selinger for Proto-Quipper-M. Our approach is to construct on top of a concrete category of circuits with measurements a Kleisli category, capturing as a side effect the action of retrieving classical content out of a quantum memory. We then show a soundness result for this semantics.

Linear reversible circuits represent a subclass of reversible circuits with many applications in quantum computing. These circuits can be efficiently simulated by classical computers and their size is polynomially bounded by the number of qubits, making them a good candidate to deploy efficient methods to reduce computational costs. We propose a new algorithm for synthesizing any linear reversible operator by using an optimized version of the Gaussian elimination algorithm coupled with a tuned LU factorization. We also improve the scalability of purely greedy methods. Overall, on random operators, our algorithms improve the state-of-the-art methods for specific ranges of problem sizes: The custom Gaussian elimination algorithm provides the best results for large problem sizes (n > 150), while the purely greedy methods provide quasi optimal results when n < 30. On a benchmark of reversible functions, we manage to significantly reduce the CNOT count and the depth of the circuit while keeping other metrics of importance (T-count, T-depth) as low as possible.

In this paper we present a semantics for a linear algebraic lambda-calculus based on realizability. This semantics characterizes a notion of unitarity in the system, answering a long standing issue. We derive from the semantics a set of typing rules for a simply-typed linear algebraic lambda-calculus, and show how it extends both to classical and quantum lambda-calculi.

One perspective on quantum algorithms is that they are classical algorithms having access to a special kind of memory with exotic properties. This perspective suggests that, even in the case of quantum algorithms, the control flow notions of sequencing, conditionals, loops, and recursion are entirely classical. There is however, another notion of control flow, that is itself quantum. The notion of quantum conditional expression is reasonably well-understood: the execution of the two expressions becomes itself a superposition of executions. The quantum counterpart of loops and recursion is however not believed to be meaningful in its most general form. In this paper, we argue that, under the right circumstances, a reasonable notion of quantum loops and recursion is possible. To this aim, we first propose a classical, typed, reversible language with lists and fixpoints. We then extend this language to the closed quantum domain (without measurements) by allowing linear combinations of terms and restricting fixpoints to structurally recursive fixpoints whose termination proofs match the proofs of convergence of sequences in infinite-dimensional Hilbert spaces. We additionally give an operational semantics for the quantum language in the spirit of algebraic lambda-calculi and illustrate its expressiveness by modeling several common unitary operations.

We introduce a Geometry of Interaction model for higher-order quantum computation, and prove its adequacy for a fully fledged quantum programming language in which entanglement, duplication, and recursion are all available.

We study multitoken interaction machines in the context of a very expressive linear logical system with exponentials, fix points and synchronization. The advantage of such machines is to provide models in the style of the Geometry of Interaction, i.e., An interactive semantics which is close to low-level implementation. On the one hand, we prove that despite the inherent complexity of the framework, interaction is guaranteed to be deadlock-free. On the other hand, the resulting logical system is powerful enough to embed PCF and to adequately model its behaviour, both when call-by-name and when call-by-value evaluation are considered. This is not the case for single-token stateless interactive machines.

Finding a denotational semantics for higher order quantum computation is a long-standing problem in the semantics of quantum programming languages. Most past approaches to this problem fell short in one way or another, either limiting the language to an unusably small finitary fragment, or giving up important features of quantum physics such as entanglement. In this paper, we propose a denotational semantics for a quantum lambda calculus with recursion and an infinite data type, using constructions from quantitative semantics of linear logic.

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.

Lambda-calculi with vectorial structures have been studied in various ways, but their semantics remain largely uninvestigated. The main contribution of this paper is to provide a categorical framework for the semantics of such algebraic lambda-calculi. We first develop a categorical analysis of a general simply typed lambda-calculus endowed with the structure of a module. We study the problems arising from the addition of a fixed-point combinator and show how to modify the equational theory to solve them. The categorical analysis carries nicely over to the modified language. We provide various concrete models for both the case without fixpoints and for the case with them.

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 author's 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.