Formally Verified Quantum Programming

The field of quantum mechanics predates computer science by at least ten years, the time between the publication of the Schrodinger equation and the Church-Turing thesis. It took another fifty years for Feynman to recognize that harnessing quantum mechanics is necessary to efficiently simulate physics and for David Deutsch to propose the quantum Turing machine. After thirty more years, we are finally getting close to the first general-purpose quantum computers based upon prototypes by IBM, Intel, Google and others. While physicists and engineers have worked on building scalable quantum computers, theoretical computer scientists have made their own advances. Complexity theorists introduced quantum complexity classes like BQP and QMA; Shor and Grover developed their famous algorithms for factoring and unstructured search. Programming languages researchers pursued two main research directions: Small-scale languages like QPL and the quantum lambda-calculi for reasoning about quantum computation and large-scale languages like Quipper and Q# for industrial-scale quantum software development. This thesis aims to unify these two threads while adding a third one: formal verification. We argue that quantum programs demand machine-checkable proofs of correctness. We justify this on the basis of the complexity of programs manipulating quantum states, the expense of running quantum programs, and the inapplicability of traditional debugging techniques to programs whose states cannot be examined. We further argue that the existing mathematical models of quantum computation make this an easier task than one could reasonably expect. In light of these observations we introduce QWIRE, a tool for writing verifiable, large scale quantum programs. QWIRE is not merely a language for writing and verifying quantum circuits: it is a verified circuit description language. This means that the semantics of QWIRE circuits are verified in the Coq proof assistant. We also implement verified abstractions, like ancilla management and reversible circuit compilation. Finally, we turn QWIRE and Coq\u27s abilities outwards, towards verifying popular quantum algorithms like quantum teleportation. We argue that this tool provides a solid foundation for research into quantum programming languages and formal verification going forward

Two linearities for quantum computing in the lambda calculus

We propose a way to unify two approaches of non-cloning in quantum lambda-calculi: logical and algebraic linearities. The first approach is to forbid duplicating variables, while the second is to consider all lambda-terms as algebraic-linear functions. We illustrate this idea by defining a quantum extension of first-order simply-typed lambda-calculus, where the type is linear on superposition, while allows cloning base vectors. In addition, we provide an interpretation of the calculus where superposed types are interpreted as vector spaces and non-superposed types as their basis.Comment: Long journal version of TPNC'17 paper (doi:10.1007/978-3-319-71069-3_22) extended with third author's "Licenciatura"'s thesi

Classically Time-Controlled Quantum Automata: Definition and Properties

In this paper we introduce classically time-controlled quantum automata or CTQA, which is a reasonable modification of Moore-Crutchfield quantum finite automata that uses time-dependent evolution and a "scheduler" defining how long each Hamiltonian will run. Surprisingly enough, time-dependent evolution provides a significant change in the computational power of quantum automata with respect to a discrete quantum model. Indeed, we show that if a scheduler is not computationally restricted, then a CTQA can decide the Halting problem. In order to unearth the computational capabilities of CTQAs we study the case of a computationally restricted scheduler. In particular we showed that depending on the type of restriction imposed on the scheduler, a CTQA can (i) recognize non-regular languages with cut-point, even in the presence of Karp-Lipton advice, and (ii) recognize non-regular languages with bounded-error. Furthermore, we study the closure of concatenation and union of languages by introducing a new model of Moore-Crutchfield quantum finite automata with a rotating tape head. CTQA presents itself as a new model of computation that provides a different approach to a formal study of "classical control, quantum data" schemes in quantum computing.Comment: Long revisited version of LNCS 11324:266-278, 2018 (TPNC 2018

Stochastic control liaisons: Richard Sinkhorn meets gaspard monge on a schr\uf6dinger bridge

In 1931-1932, Erwin Schr\uf6dinger studied a hot gas Gedankenexperiment (an instance of large deviations of the empirical distribution). Schr\uf6dinger's problem represents an early example of a fundamental inference method, the so-called maximum entropy method, having roots in Boltzmann's work and being developed in subsequent years by Jaynes, Burg, Dempster, and Csisz\ue1r. The problem, known as the Schr\uf6dinger bridge problem (SBP) with "uniform"prior, was more recently recognized as a regularization of the Monge-Kantorovich optimal mass transport (OMT) problem, leading to effective computational schemes for the latter. Specifically, OMT with quadratic cost may be viewed as a zerotemperature limit of the problem posed by Schr\uf6dinger in the early 1930s. The latter amounts to minimization of Helmholtz's free energy over probability distributions that are constrained to possess two given marginals. The problem features a delicate compromise, mediated by a temperature parameter, between minimizing the internal energy and maximizing the entropy. These concepts are central to a rapidly expanding area of modern science dealing with the so-called Sinkhorn algorithm, which appears as a special case of an algorithm first studied in the more challenging continuous space setting by the French analyst Robert Fortet in 1938-1940 specifically for Schr\uf6dinger bridges. Due to the constraint on end-point distributions, dynamic programming is not a suitable tool to attack these problems. Instead, Fortet's iterative algorithm and its discrete counterpart, the Sinkhorn iteration, permit computation of the optimal solution by iteratively solving the so-called Schr\uf6dinger system. Convergence of the iteration is guaranteed by contraction along the steps in suitable metrics, such as Hilbert's projective metric. In both the continuous as well as the discrete time and space settings, stochastic control provides a reformulation of and a context for the dynamic versions of general Schr\uf6dinger bridge problems and of their zero-temperature limit, the OMT problem. These problems, in turn, naturally lead to steering problems for flows of one-time marginals which represent a new paradigm for controlling uncertainty. The zero-temperature problem in the continuous-time and space setting turns out to be the celebrated Benamou-Brenier characterization of the McCann displacement interpolation flow in OMT. The formalism and techniques behind these control problems on flows of probability distributions have attracted significant attention in recent years as they lead to a variety of new applications in spacecraft guidance, control of robot or biological swarms, sensing, active cooling, and network routing as well as in computer and data science. This multifaceted and versatile framework, intertwining SBP and OMT, provides the substrate for the historical and technical overview of the field given in this paper. A key motivation has been to highlight links between the classical early work in both topics and the more recent stochastic control viewpoint, which naturally lends itself to efficient computational schemes and interesting generalizations

Extending scientific computing system with structural quantum programming capabilities

We present a basic high-level structures used for developing quantum programming languages. The presented structures are commonly used in many existing quantum programming languages and we use quantum pseudo-code based on QCL quantum programming language to describe them. We also present the implementation of introduced structures in GNU Octave language for scientific computing. Procedures used in the implementation are available as a package quantum-octave, providing a library of functions, which facilitates the simulation of quantum computing. This package allows also to incorporate high-level programming concepts into the simulation in GNU Octave and Matlab. As such it connects features unique for high-level quantum programming languages, with the full palette of efficient computational routines commonly available in modern scientific computing systems. To present the major features of the described package we provide the implementation of selected quantum algorithms. We also show how quantum errors can be taken into account during the simulation of quantum algorithms using quantum-octave package. This is possible thanks to the ability to operate on density matrices

Foundations of Software Science and Computation Structures

This open access book constitutes the proceedings of the 23rd International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2020, which took place in Dublin, Ireland, in April 2020, and was held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2020. The 31 regular papers presented in this volume were carefully reviewed and selected from 98 submissions. The papers cover topics such as categorical models and logics; language theory, automata, and games; modal, spatial, and temporal logics; type theory and proof theory; concurrency theory and process calculi; rewriting theory; semantics of programming languages; program analysis, correctness, transformation, and verification; logics of programming; software specification and refinement; models of concurrent, reactive, stochastic, distributed, hybrid, and mobile systems; emerging models of computation; logical aspects of computational complexity; models of software security; and logical foundations of data bases.