Determining properties of ground states of spin Hamiltonians remains a topic of central relevance connecting disciplines of mathematical, theoretical and applied physics. In the last few decades, ground state properties of physical systems have been increasingly considered as computational resources. This thesis develops parts of the mathematical apparatus to create (program) ground states relevant for quantum and classical computation. The core findings presented in this thesis (now over a decade old) including that (i) logic operations (gates) can be embedded into the low-energy sector of Ising spins whereas three (and higher) body Ising interaction terms can be mimicked through the minimisation of 2- and 1-body Ising terms yet require the introduction of slack spins; (ii) Perturbation theory gadgets enable the emulation of interactions not present in a given Hamiltonian, e.g.~$YY$ interactions can be realized from $ZZ$, $XX$, the thesis contains a result from 2007 showing that physically relevant two-body model Hamiltonian's have a QMA-hard ground state energy decision problem. Merged with other results, this established that these models provide a universal resource for ground state quantum computation. More recent findings include the proof that an idealised version of the contemporary variational approach to quantum algorithms enables a universal model of quantum computation. Other related results are also presented as they relate to ground state quantum computation and the minimisation of Hamiltonians by quantum circuits. The topics covered include: Ising model reductions, stochastic versus quantum processes on graphs, quantum gates and circuits as tensor networks, variational quantum algorithms and Hamiltonian gadgets.
翻译:确定旋转汉密尔顿人的地面状态特性仍然是连接数学、理论和应用物理学学科的核心相关主题。 在过去的几十年中,物理系统的地面状态特性被日益视为计算资源。 该论文开发了数学机器中与量子计算和经典计算相关的部分( 程序) 创建( 程序) 地面状态状态。 该论文中的核心结论( 已有十多年的历史) 包括 (一) 逻辑操作( 门) 可以嵌入Ising旋转的低能量部门, 而三个( 更高) 机构是 互动术语可以通过2和1个体的最小化条件被模拟为计算资源。 在过去的几十年中, 物理状态状态状态状态状态特性特性特性属性属性属性属性属性属性属性属性属性属性属性属性属性属性属性属性属性属性属性属性属性属性属性属性属性属性属性属性属性属性属性; 逻辑属性属性属性属性属性属性属性属性属性属性属性属性属性属性属性属性属性属性属性属性属性属性属性属性属性属性属性属性属性属性属性属性属性属性属性属性值值值值值值值的货币值值值值值值值值值值值值值值值值定义。