Diabatic Quantum Annealing using XX-Catalysts

我提议的研究将着眼于一种被称为量子退火法的量子算法。该算法用于解决组合优化问题，目的是以比经典方法更快的速度获得解。量子退火法的工作原理是将此类问题的最优解编码为哈密顿量的基态，哈密顿量本质上是定义系统能级的数学对象。通过这种方式，解决优化问题就变成了让量子系统进入这个基态，以便可以测量它。然而，对于任何非平凡的问题，这个哈密顿量都将是令人难以置信的复杂的，所以让一个量子系统进入它的基态并不是一项简单的任务。在传统的量子退火法中，系统在一个简单的哈密顿量的基态中被初始化，然后慢慢过渡到定义这个问题的哈密顿量。根据所谓的绝热定理，系统在整个转变过程中将保持在基态，只要它发生得足够慢，因此系统结束于问题哈密顿量的基态。然而，对于一些问题，这必须发生得很慢，以避免远离基态的跃迁-破坏任何量子加速的机会。我的博士将研究这个问题的潜在解决方案，其中允许发生到更高能级的跃迁，但以确保系统在算法结束时返回到基态的方式。这是通过引入第三哈密顿量来实现的，该哈密顿量有助于这些转变回到基态。这种所谓的非绝热量子退火法有可能导致其绝热退火法甚至超过经典算法的显著加速。

My proposed research will look at a quantum algorithm known as Quantum Annealing. This algorithm is used for solving combinatorial optimisation problems with the aim of reaching a solution faster than is classically achievable.Quantum Annealing works by encoding the optimal solution for such a problem into the ground state of what is called a Hamiltonian - essentially a mathematical object that defines the energy levels of a system. In this way, solving the optimisation problem becomes a matter of getting a quantum system into this ground state so that it can be measured. However, for any non-trivial problem this Hamiltonian will be incredibly complicated and so getting a quantum system into its ground state is not a simple task.In traditional Quantum Annealing, the system is initialised in the ground state of a simple Hamiltonian before slowly transitioning to the Hamiltonian defining the problem. By what is known as the Adiabatic theorem, the system will remain in the ground state throughout this transition provided it happens slowly enough and so the system ends in the ground state of the problem Hamiltonian. However, for some problems, this must occur intractably slowly to avoid transitions away from the ground state - destroying any chance of quantum speedup.My PhD will examine a potential solution to this problem in which transitions to higher energy states are allowed to occur but in such a way that ensures the system has returned to the ground state by the end of the algorithm. This is achieved through the introduction of a third Hamiltonian that facilitates these transitions back into the ground state. This so-called diabatic approach to quantum annealing has the potential to lead to dramatic speedup over its adiabatic counterpart and indeed over classical algorithms.