Implementing Hadamard Matrices in SageMath

Hadamard matrices are $(-1, +1)$ square matrices with mutually orthogonal rows. The Hadamard conjecture states that Hadamard matrices of order $n$ exist whenever $n$ is $1$, $2$, or a multiple of $4$. However, no construction is known that works for all values of $n$, and for some orders no Hadamard matrix has yet been found. Given the many practical applications of these matrices, it would be useful to have a way to easily check if a construction for a Hadamard matrix of order $n$ exists, and in case to create it. This project aimed to address this, by implementing constructions of Hadamard and skew Hadamard matrices to cover all known orders less than or equal to $1000$ in SageMath, an open-source mathematical software. Furthermore, we implemented some additional mathematical objects, such as complementary difference sets and T-sequences, which were not present in SageMath but are needed to construct Hadamard matrices. This also allows to verify the correctness of the results given in the literature; within the $n\leq 1000$ range, just one order, $292$, of a skew Hadamard matrix claimed to have a known construction, required a fix.Comment: pdflatex+biber, 32 page

A survey of base sequences, disjoint complementary sequences and OD(4t; t, t, t, t)

We survey the existence of base sequences, that is four sequences of lengths m + p, m + p, m, m, p odd with zero auto correlation function which can be used with Yang numbers and four disjoint complementary sequences (and matrices) with zero non-periodic (periodic) autocorrelation function to form longer sequences. We survey their application to make orthogonal designs OD(4t; t, t, t, t). We give the method of construction of OD(4t; t, t, t, t) for t = 1,3,..., 41, 45,...,65, 67, 69, 75, 77, 81, 85, 87, 91, 93, 95, 99, 101, 105, 111, 115, 117, 119, 123, 125, 129, 133, 141,..., 147, 153, 155, 159, 161, 165, 169, 171, 175, 177, 183, 185, 189, 195, 201, 203, 205, 209

COMPLEX HADAMARD MATRICES AND APPLICATIONS

A complex Hadamard matrix is a square matrix H ∈ M N (C) whose entries are on the unit circle, |H ij | = 1, and whose rows and pairwise orthogonal. The main example is the Fourier matrix, F N = (w ij) with w = e 2πi/N. We discuss here the basic theory of such matrices, with emphasis on geometric and analytic aspects. CONTENT

Computational Methods for Combinatorial and Number Theoretic Problems

Computational methods have become a valuable tool for studying mathematical problems and for constructing large combinatorial objects. In fact, it is often not possible to find large combinatorial objects using human reasoning alone and the only known way of accessing such objects is to use computational methods. These methods require deriving mathematical properties which the object in question must necessarily satisfy, translating those properties into a format that a computer can process, and then running a search through a space which contains the objects which satisfy those properties. In this thesis, we solve some combinatorial and number theoretic problems which fit into the above framework and present computational strategies which can be used to perform the search and preprocessing. In particular, one strategy we examine uses state-of-the-art tools from the symbolic computation and SAT/SMT solving communities to execute a search more efficiently than would be the case using the techniques from either community in isolation. To this end, we developed the tool MathCheck2, which combines the sophisticated domain-specific knowledge of a computer algebra system (CAS) with the powerful general-purpose search routines of a SAT solver. This fits into the recently proposed SAT+CAS paradigm which is based on the insight that modern SAT solvers (some of the best general-purpose search tools ever developed) do not perform well in all applications but can be made more efficient if supplied with appropriate domain-specific knowledge. To our knowledge, this is the first PhD thesis which studies the SAT+CAS paradigm which we believe has potential to be used in many problems for a long time to come. As case studies for the methods we examine, we study the problem of computing Williamson matrices, the problem of computing complex Golay sequences, and the problem of computing minimal primes. In each case, we provide results which are competitive with or improve on the best known results prior to our work. In the first case study, we provide for the first time an enumeration of all Williamson matrices up to order 45 and show that 35 is the smallest order for which Williamson matrices do not exist. These results were previously known under the restriction that the order was odd but our work also considers even orders, as Williamson did when he defined such matrices in 1944. In the second case study, we provide an independent verification of the 2002 conjecture that complex Golay sequences do not exist in order 23 and enumerate all complex Golay sequences up to order 25. In the third case study, we compute the set of minimal primes for all bases up to 16 as well for all bases up to 30 with possibly a small number of missing elements

Adipose tissue FABP deficiency promotes metabolic reprogramming and positively impacts healthspan

The adipose tissue lipid chaperones aP2 and mal1, also known as fatty acid binding proteins (FABPs), are significant molecules contributing to metabolic homeostasis, whereby their absence promotes physiological changes that improve systemic metabolism. Identification of palmitoleate as a lipokine generated in aP2-mal1 deficiency--originating from adipose and directing the lipogenic program in liver, established a role for these chaperones in linking adipocyte and hepatic function. We have recently demonstrated a functional role for secreted aP2 in the activation of gluconeogenesis and hepatic glucose output, further designating this molecule as an adipocyte-derived regulatory factor that influences liver metabolism. Key molecules linking the metabolism of nutrients in energy generating pathways are the nucleotide cofactors NAD and NADH. Together, these molecules function to coordinate the maintenance of redox reactions during normal cellular metabolism and act as required substrates for enzymes such as sirtuins and poly ADP-ribose polymerases. Using global metabolite profiling, we show that combined deficiency of the adipose tissue lipid chaperones aP2 and mal1 leads to a hepatic nucleotide imbalance resulting from metabolic reprogramming in liver. We demonstrate that this reprogramming of metabolite flux is accompanied by significant alterations in liver NAD metabolism and establish a role for aP2 in directing substrate utilization through inhibition of the rate-limiting enzyme for NAD synthesis, nicotinamide phosphoribosyltransferase. Several models for the proposed regulatory pathways that link nutrient metabolism to aging include mechanisms that are NAD dependent. Accordingly, we found that long-term FABP deficiency confers a strong resistance to aging related metabolic deterioration. Together, the findings presented in this thesis support a considerable role for FABPs in the regulation of NAD metabolism and healthspan

Methods for parallel quantum circuit synthesis, fault-tolerant quantum RAM, and quantum state tomography

The pace of innovation in quantum information science has recently exploded due to the hope that a quantum computer will be able to solve a multitude of problems that are intractable using classical hardware. Current quantum devices are in what has been termed the ``noisy intermediate-scale quantum'', or NISQ stage. Quantum hardware available today with 50-100 physical qubits may be among the first to demonstrate a quantum advantage. However, there are many challenges to overcome, such as dealing with noise, lowering error rates, improving coherence times, and scalability. We are at a time in the field where minimization of resources is critical so that we can run our algorithms sooner rather than later. Running quantum algorithms ``at scale'' incurs a massive amount of resources, from the number of qubits required to the circuit depth. A large amount of this is due to the need to implement operations fault-tolerantly using error-correcting codes. For one, to run an algorithm we must be able to efficiently read in and output data. Fault-tolerantly implementing quantum memories may become an input bottleneck for quantum algorithms, including many which would otherwise yield massive improvements in algorithm complexity. We will also need efficient methods for tomography to characterize and verify our processes and outputs. Researchers will require tools to automate the design of large quantum algorithms, to compile, optimize, and verify their circuits, and to do so in a way that minimizes operations that are expensive in a fault-tolerant setting. Finally, we will also need overarching frameworks to characterize the resource requirements themselves. Such tools must be easily adaptable to new developments in the field, and allow users to explore tradeoffs between their parameters of interest. This thesis contains three contributions to this effort: improving circuit synthesis using large-scale parallelization; designing circuits for quantum random-access memories and analyzing various time/space tradeoffs; using the mathematical structure of discrete phase space to select subsets of tomographic measurements. For each topic the theoretical work is supplemented by a software package intended to allow others researchers to easily verify, use, and expand upon the techniques herein