This is a review paper outlining recent progress in the spectral analysis of first order systems. We work on a closed manifold and study an elliptic self-adjoint first order system of linear partial differential equations. The aim is to examine the spectrum and derive asymptotic formulae for the two counting functions. Here the two counting functions are those for the positive and the negative eigenvalues. One has to deal with positive and negative eigenvalues separately because the spectrum is, generically, asymmetric.

Consider a formally self-adjoint first order linear differential operator acting on pairs (2-columns) of complex-valued scalar fields over a 4manifold without boundary. We examine the geometric content of such an operator and show that it implicitly contains a Lorentzian metric, Pauli matrices, connection coefficients for spinor fields and an electromagnetic covector potential. This observation allows us to give a simple representation of the massive Dirac equation as a system of four scalar equations involving an arbitrary two-by-two matrix operator as above and its adjugate. The point of the paper is that in order to write down the Dirac equation in the physically meaningful 4-dimensional hyperbolic setting one does not need any geometric constructs. All the geometry required is contained in a single analytic object -an abstract formally self-adjoint first order linear differential operator acting on pairs of complex-valued scalar fields.

We work on a parallelizable time-orientable Lorentzian 4-manifold and prove that in this case the notion of spin structure can be equivalently defined in a purely analytic fashion. Our analytic definition relies on the use of the concept of a non-degenerate two-by-two formally self-adjoint first order linear differential operator and gauge transformations of such operators. We also give an analytic definition of spin structure for the 3-dimensional Riemannian case.

A significant challenge in quantum annealing is to map a real-world problem onto a hardware graph of limited connectivity. When the problem graph is not a subgraph of the hardware graph, one might employ minor embedding in which each logical qubit is mapped to a tree of physical qubits. Pairwise interactions between physical qubits in the tree are set to be ferromagnetic with some coupling strength F < 0. Here we address the theoretical question of what the best value F should be in order to achieve unbroken trees in the pre-quantum-processing. The sum of |F| for each logical qubit is defined as minor embedding energy, and the best value F is obtained when the minor embedding energy is minimized. We also show that our new analytical lower bound on |F| is a tighter bound than that previously derived by Choi (Quantum Inf Process 7:193-209, 2008). In contrast to Choi's work, our new method depends more delicately on minor embedding parameters, which leads to a higher computational cost.