wikipedia | Adiabatic quantum computation (AQC) relies on the adiabatic theorem to do calculations[1] and is closely related to, and may be regarded as a subclass of, quantum annealing.[2][3][4][5] First, a (potentially complicated) Hamiltonian is found whose ground state describes the solution to the problem of interest. Next, a system with a simple Hamiltonian is prepared and initialized to the ground state. Finally, the simple Hamiltonian is adiabatically evolved to the desired complicated Hamiltonian. By the adiabatic theorem, the system remains in the ground state, so at the end the state of the system describes the solution to the problem. Adiabatic Quantum Computing has been shown to be polynomially equivalent to conventional quantum computing in the circuit model.[6] The time complexity for an adiabatic algorithm is the time taken to complete the adiabatic evolution which is dependent on the gap in the energy eigenvalues (spectral gap) of the Hamiltonian. Specifically, if the system is to be kept in the ground state, the energy gap between the ground state and the first excited state of 
provides an upper bound on the rate at which the Hamiltonian can be evolved at time 
.[7] When the spectral gap is small, the Hamiltonian has to be evolved slowly. The runtime for the entire algorithm can be bounded by 
Where 
is the minimum spectral gap for .
provides an upper bound on the rate at which the Hamiltonian can be evolved at time .[7] When the spectral gap is small, the Hamiltonian has to be evolved slowly. The runtime for the entire algorithm can be bounded by Where is the minimum spectral gap for .
AQC is a possible method to get around the problem of energy relaxation. Since the quantum system is in the ground state, interference with the outside world cannot make it move to a lower state. If the energy of the outside world (that is, the "temperature of the bath") is kept lower than the energy gap between the ground state and the next higher energy state, the system has a proportionally lower probability of going to a higher energy state. Thus the system can stay in a single system eigenstate as long as needed.
The D-Wave One is a device made by a Canadian company D-Wave Systems which describes it as doing quantum annealing.[13] In 2011, Lockheed-Martin purchased one for about US$10 million; in May 2013,Google purchased a D-Wave Two with 512 qubits.[14] As of now, the question of whether the D-Wave processors offer a speedup over a classical processor is still unanswered. Tests performed by researchers atUSC, ETH Zurich, and Google show that as of now, there is no evidence of a quantum advantage.[15][16]
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