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This article reports an experimental work that unveils some interesting yet unknown phenomena underneath all smooth nonlinear maps. The findings are based on the fact that, generalizing the conventional gradient dynamics, the right singular vectors of the Jacobian matrix of any differentiable map point in directions that are most pertinent to the infinitesimal deformation of the underlying function and that the singular values measure the rate of deformation in the corresponding directions. A continuous adaption of these singular vectors, therefore, constitutes a natural moving frame that carries indwelling information of the variation. This structure exists in any dimensional space, but the development of the fundamental theory and algorithm for surface exploration is an important first step for immediate application and further generalization. In this case, trajectories of these singular vectors, referred to as singular curves, unveil some intriguing patterns per the given function. At points where singular values coalesce, curious and complex behaviors occur, manifesting specific landmarks for the function. Upon analyzing the dynamics, it is discovered that there is a remarkably simple and universal structure underneath all smooth two-parameter maps. This work delineates graphs with this interesting dynamical system and the possible new discovery that, analogous to the double helix with two base parings in DNA, two strands of critical curves and eight base pairings could encode properties of a generic and arbitrary surface. This innate structure suggests that this approach could provide a unifying paradigm in functional genetics, where all smooth surfaces could be genome-sequenced and classified accordingly. Such a concept has sparked curiosity and warrants further investigation. 

Simulating the time evolution of a Hamiltonian system on a classical computer is hard—The computational power required to even describe a quantum system scales exponentially with the number of its constituents, let alone integrate its equations of motion. Hamiltonian simulation on a quantum machine is a possible solution to this challenge—Assuming that a quantum system composing of spin-½ particles can be manipulated at will, then it is tenable to engineer the interaction between those particles according to the one that is to be simulated, and thus predict the value of physical quantities by simply performing the appropriate measurements on the system. Establishing a linkage between the unitary operators described mathematically as a logic solution and the unitary operators recognizable as quantum circuits for execution, is therefore essential for algorithm design and circuit implementation. Most current techniques are fallible because of truncation errors or the stagnation at local solutions. This work offers an innovative avenue by tackling the Cartan decomposition with the notion of Lax dynamics. Within the integration errors that is controllable, this approach gives rise to a genuine unitary synthesis that not only is numerically feasible, but also can be utilized to gauge the quality of results produced by other means, and extend the knowledge to a wide range of applications. This paper aims at establishing the theoretic and algorithmic foundations by exploiting the geometric properties of Hamiltonian subalgebras and describing a common mechanism for deriving the Lax dynamics.