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 integrating its equations of motion. Hamiltonian simulation on a quantum machine is a possible solution to this challenge. Assuming that a quantum machine composed of spin-½ particles that we could manipulate at will, 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 link 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. The approach employs an eclectic mix of techniques from Lie theory, dynamical systems, linear algebra, combinatorics, and numerical ODEs. Not only the process is numerically feasible, but also it produces a genuine unitary synthesis that is optimal in both the precision with controllable integration errors and the usage of only minimally required synthesis components. This talk aims at providing a glimpse into the theoretic and algorithmic foundations by exploiting the geometric properties of Hamiltonian subalgebras and describing a common mechanism for deriving the Lax dynamics. (Lots of examples will be given. This talk does not assume a priori knowledge on quantum computing.)

Moody Chu joined NCSU in 1982. He was formally trained as a numerical analyst with specialty in stiff ordinary differential systems, but he has maintained a vivid interest in the broader spectrum of mathematics. His curiosity in action has been to search for the interconnection across various fields of knowledge and show that often one part can help advance the other. His work has been supported by the NSF, DOE, and ARO, with the most recent two on mathematical aspects of quantum computing. Among his publications, he is most proud of three SIAM Review articles and two Acta Numerica treatises. In addition to research, he loves teaching. He has been awarded six times on this campus for his teaching.