Quantum Adiabatic Evolution

Wei Wang, S. C. Hou, X. X. Yi

Mária Kieferová, Nathan Wiebe

A major challenge facing adiabatic quantum computing is that algorithm design and error correction can be difficult for adiabatic quantum computing. Recent work has considered addressing his challenge by using coherently controlled adiabatic evolutions in the place of classically controlled evolution. An important question remains: what is the relative power of controlled adiabatic evolution to traditional adiabatic evolutions? We address this by showing that coherent control and measurement provides a way to average different adiabatic evolutions in ways that cause their diabatic errors to cancel, allowing for adiabatic evolutions to combine the best characteristics of existing adiabatic optimizations strategies that are mutually exclusive in conventional adiabatic QIP. This result shows that coherent control and measurement can provide advantages for adiabatic state preparation. We also provide upper bounds on the complexity of simulating such evolutions on a circuit based quantum computer and provide sufficiency conditions for the equivalence of controlled adiabatic evolutions to adiabatic quantum computing.

Jonas Larson, Erik Sjöqvist, Patrik Öhberg

Zhaohui Wei, Mingsheng Ying

Quantum adiabatic computation is a novel paradigm for the design of quantum algorithms, which is usually used to find the minimum of a classical function. In this paper, we show that if the initial hamiltonian of a quantum adiabatic evolution with a interpolation path is too simple, the minimal gap between the ground state and the first excited state of this quantum adiabatic evolution is an inverse exponential distance. Thus quantum adiabatic evolutions of this kind can't be used to design efficient quantum algorithms. Similarly, we show that a quantum adiabatic evolution with a simple final hamiltonian also has a long running time, which suggests that some functions can't be minimized efficiently by any quantum adiabatic evolution with a interpolation path.

Jie Sun, Songfeng Lu, Fang Liu

Zhenyu Wang

We decompose the quantum adiabatic evolution as the products of gauge invariant unitary operators and obtain the exact nonadiabatic correction in the adiabatic approximation. A necessary and sufficient condition that leads to adiabatic evolution with geometric phases is provided and we determine that in the adiabatic evolution, while the eigenstates are slowly varying, the eigenenergies and degeneracy of the Hamiltonian can change rapidly. We exemplify this result by the example of the adiabatic evolution driven by parametrized pulse sequences. For driving fields that are rotating slowly with the same average energy and evolution path, fast modulation fields can have smaller nonadiabatic errors than obtained under the traditional approach with a constant amplitude.