Formal Proofs for Nonlinear Optimization

We present a formally verified global optimization framework. Given a semialgebraic or transcendental function $f$ and a compact semialgebraic domain $K$, we use the nonlinear maxplus template approximation algorithm to provide a certified lower bound of $f$ over $K$. This method allows to bound in a modular way some of the constituents of $f$ by suprema of quadratic forms with a well chosen curvature. Thus, we reduce the initial goal to a hierarchy of semialgebraic optimization problems, solved by sums of squares relaxations. Our implementation tool interleaves semialgebraic approximations with sums of squares witnesses to form certificates. It is interfaced with Coq and thus benefits from the trusted arithmetic available inside the proof assistant. This feature is used to produce, from the certificates, both valid underestimators and lower bounds for each approximated constituent. The application range for such a tool is widespread; for instance Hales' proof of Kepler's conjecture yields thousands of multivariate transcendental inequalities. We illustrate the performance of our formal framework on some of these inequalities as well as on examples from the global optimization literature.Comment: 24 pages, 2 figures, 3 table

COLAB : a hybrid knowledge representation and compilation laboratory

Knowledge bases for real-world domains such as mechanical engineering require expressive and efficient representation and processing tools. We pursue a declarative-compilative approach to knowledge engineering. While Horn logic (as implemented in PROLOG) is well-suited for representing relational clauses, other kinds of declarative knowledge call for hybrid extensions: functional dependencies and higher-order knowledge should be modeled directly. Forward (bottom-up) reasoning should be integrated with backward (top-down) reasoning. Constraint propagation should be used wherever possible instead of search-intensive resolution. Taxonomic knowledge should be classified into an intuitive subsumption hierarchy. Our LISP-based tools provide direct translators of these declarative representations into abstract machines such as an extended Warren Abstract Machine (WAM) and specialized inference engines that are interfaced to each other. More importantly, we provide source-to-source transformers between various knowledge types, both for user convenience and machine efficiency. These formalisms with their translators and transformers have been developed as part of COLAB, a compilation laboratory for studying what we call, respectively, "vertical\u27; and "horizontal\u27; compilation of knowledge, as well as for exploring the synergetic collaboration of the knowledge representation formalisms. A case study in the realm of mechanical engineering has been an important driving force behind the development of COLAB. It will be used as the source of examples throughout the paper when discussing the enhanced formalisms, the hybrid representation architecture, and the compilers

Quantum-assisted quantum compiling

Compiling quantum algorithms for near-term quantum computers (accounting for connectivity and native gate alphabets) is a major challenge that has received significant attention both by industry and academia. Avoiding the exponential overhead of classical simulation of quantum dynamics will allow compilation of larger algorithms, and a strategy for this is to evaluate an algorithm's cost on a quantum computer. To this end, we propose a variational hybrid quantum-classical algorithm called quantum-assisted quantum compiling (QAQC). In QAQC, we use the overlap between a target unitary $U$ and a trainable unitary $V$ as the cost function to be evaluated on the quantum computer. More precisely, to ensure that QAQC scales well with problem size, our cost involves not only the global overlap ${\rm Tr} (V^\dagger U)$ but also the local overlaps with respect to individual qubits. We introduce novel short-depth quantum circuits to quantify the terms in our cost function, and we prove that our cost cannot be efficiently approximated with a classical algorithm under reasonable complexity assumptions. We present both gradient-free and gradient-based approaches to minimizing this cost. As a demonstration of QAQC, we compile various one-qubit gates on IBM's and Rigetti's quantum computers into their respective native gate alphabets. Furthermore, we successfully simulate QAQC up to a problem size of 9 qubits, and these simulations highlight both the scalability of our cost function as well as the noise resilience of QAQC. Future applications of QAQC include algorithm depth compression, black-box compiling, noise mitigation, and benchmarking.Comment: 19 + 10 pages, 14 figures. Added larger scale implementations and proof that cost function is DQC1-har