Ground State Spin Calculus

Abstract

We present an intuitive compositional theory from which one is able to predict and also to control the ground state manifold (and higher energy excitations) of interacting spin systems governed by variants of tunable Ising models, hence giving precise control over the apriori additive structure of Hamiltonian composition. This compositional theory is given in terms of string diagrams: these results were made possible by mapping a variant of the Boolean F2-calculus onto spins and synthesizing modern ideas appearing in Category Theory, Coalgebras, Classical Network Theory and Graphical Calculus. Specifically, we present an algebraic method which allows one to explicitly engineer several energy levels including the low-energy subspace of interacting spin systems. We call this new framework: Ground State Spin Calculus, and in the first instance, the theory requires interactions of up to third order (3- body). By introducing ancillary qubits, we present a novel approach allowing k-body interactions to be captured exactly using only two-body Hamiltonians [Biamonte, Phys. Rev. A 77(5), 052331 (2008)]. Our reduction method has no dependence on perturbation theory or the associated large spectral gap and allows for problem instance solutions to be embedded into the ground energy state of Ising spin systems. This could have important applications for future technology as adiabatic quantum evolution might be used to place such a computational system into it’s ground state.