Theory Seminar: On the mathematical structure of quantum models of computation based on Hamiltonian minimisation

Jacob Biamonte (Skoltech)
Determining properties of ground states of spin Hamiltonians remains a topic of central relevance connecting disciplines of mathematical, theoretical and applied physics.

July 21, 2021

Jacob Biamonte (Skoltech)
Group Seminar via Zoom
Wed, 21. July 2021, 10:00 am (MEZ)

Abstract:

Determining properties of ground states of spin Hamiltonians remains a topic of central relevance connecting disciplines of mathematical, theoretical and applied physics.  In the last few decades, ground state properties of physical systems have been increasingly considered as computational resources.  This thesis develops parts of the mathematical apparatus to create (program) ground states relevant for quantum and classical computation.  The core results presented in this thesis include several findings (some over a decade old) including that logic operations (gates) can be embedded into the low-energy sector of Ising spins whereas three (and higher) body Ising interaction terms can be mimicked through the minimisation of 2- and 1-body Ising terms yet require the introduction of strongly coupled slack spins; moreover perturbation theory gadgets enable the emulation of interactions not present in a given Hamiltonian, e.g.~$YY$ interactions can be realized from $ZZ$, $XX$, the thesis contains a result showing that physically relevant two-body model Hamiltonian's have a {\sf QMA}-complete ground state energy decision problem. Merged with other results, this established that these models provide a universal resource for ground state quantum computation.  More recent findings include the proof that an idealised version of the contemporary variational approach to quantum algorithms enables a universal model of quantum computation.  Other related results are also presented as they relate to ground state quantum computation and the minimisation of Hamiltonians by variational quantum circuits.

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