绝热量子计算

绝热量子计算是先把初态制备到product state，然后利用绝热路径演化初始哈密顿量到对应待解决问题的哈密顿量，从而使基态演化到包含我们需要的解的态。
学习一下何为绝热量子计算，以及他如何与circuit 量子计算等价。

Definition: (Adiabatic Quantum Computation) A k-local adiabatic quantum computation is specified by two k-local Hamiltonians, and , acting on -state particles, .
The ground state of is unique and is a product state.
The output of the algorithm is a state that is -close to the ground state of in 2-norm.
Let and let the smallest time for an adiabatic evolution to get the -close gs of .

Comments:

• 的基态可以简并，虽然一般来说要求不简并。
• 一般来说AQC都是用基态，但有时候也会利用激发态。
• 可以引入“中间哈密顿量”，在 和 时消失。

Cost of running an AQC algorithm:

• Runtime ：
  • 最差情况：, 是最小能隙。
  • 好一些的情况： up to a polylogarithmic factor in .
• Full cost (dimensionless)：. 这里的 代表 operator norm。
• Related to gap analysis in condenced matter.
• Cost is strongly influenced by the type of quantum phase transition.