Mathematical Methods for a Quantum Annealing Computer

Abstract

This paper describes the logic and creativity needed in order to have high probability of solving discrete optimization problems on a quantum annealing computer. Current features of quantum computing via annealing are discussed. We illustrate the logic at the forefront of this new era of computing, describe some of the work done in this field, and indicate the distinct mindset that is used when programming this type of machine. The traveling salesman problem is formulated for solving on a quantum annealing computer, which illustrates the methods for this computer.

Quantum annealing for microstructure equilibration with long-range elastic interactions

“…During the annealing process, the Hamiltonian is turned into the desired one based on an Ising model 9 H p = ∑ i h i s i + ∑ i< j J i j s i s j with spin states s i = ±1, bias h i and coupling strength J i j between qubits i and j, for which an energetic minimum is sought, min {s i =±1} H p . Both Hamiltonians do not commute 9 , and the time of the initial Hamiltonian to adopt the low energy state is sufficiently large to ensure the validity of the adiabatic theorem of quantum mechanics 35 , which states that a system remains in its eigenstate, if changes occur adiabatically. Notice that the quantum annealing employs tunneling to leave metastable regions, contrary to the simulated annealing 6 .…”

“…In the meantime, a technology known as quantum annealing (QA) [6][7][8] has emerged and is available on several sites worldwide. The use of such machines differs significantly from traditional gate based computers and therefore currently only specific problems can be handled by quantum annealers 9 . The concept of a quantum annealer is that its qubits are initialized in a well defined state which is described by a Hamiltonian with a unique ground state 10 .…”

“…During the operation at cryogenic temperatures, this Hamiltonian is changed adiabatically such that the ground state converts into the one of the final, desired Hamiltonian 10,11 , and therefore allows to perform global energy minimization computations efficiently. The structure of these Hamiltonians is a binary quadratic model, which can be expressed in terms of a quadratic unconstrained binary optimization or equivalently through an Ising model 9 . Due to this specific structure, so far, materials science related applications of this technology are still rare.…”

Efficient low temperature Monte Carlo sampling using quantum annealing

“…a strong transverse magnetic field 46 , 47 . During the annealing process, the Hamiltonian is turned into the desired one based on an Ising model 48 with spin states , bias and couplings between spins and , for which an energetic minimum is sought, .…”

“…The Hamiltonians and do not commute 48 , and the time of the initial Hamiltonian to adopt the low energy state is sufficiently large to ensure the validity of the adiabatic theorem of quantum mechanics 52 , which states that a system remains in its eigenstate, if changes occur adiabatically. Nevertheless, the machines are not perfect and do not always adopt the corresponding low energy state of the system.…”