Non-hermitian Adiabatic Quantum Optimization

Many physical and combinatorial problems associated with complex networks of interacting degrees of freedom can be mapped to equivalent problems of finding the ground (or minimum cost) state of a corresponding quantum Hamiltonian H0 1,2,3,4,5,6,7,8,9. One of the approaches to finding the ground state of H0 is adiabatic quantum computation which can be formulated as follows. Consider the time dependent Hamiltonian H(t) = (t/τ)H0 + (1 − t/τ)H1, where H0 is the Hamiltonian whose ground state is to be found, τ is the given time-interval of quantum computation, H1 is an auxiliary “initial” Hamiltonian and [H0,H1] 6= 0. As time varies from t = 0 to t = τ , the Hamiltonian interpolates between H1 and H0. If the system is initially close to the ground state of H1, and if τ is sufficiently large (slow variation), then the system will remain close to the instantaneous ground state (i.e., that of Hτ (t)) for all t ∈ [0, τ ]. In particular, at t = τ the ground state of the total Hamiltonian, Hτ , will be close to the ground state of H0, which is the state we seek. In practice, H1 is chosen such that its ground state is known, then the dynamics is allowed to evolve and the state of the system evolves into the final state which is the solution to the problem.