基于量子力学和深度强化学习的魔方求解
Solving Rubik's Cube via Quantum Mechanics and Deep Reinforcement Learning
摘要：
魔方是最著名的组合谜题之一，涉及几乎$4.3 \times 10^{19}$可能的配置。它的数学描述由魔方组表示，其元素定义了它的层如何旋转。我们发展了这样的群的酉表示和量子形式主义来描述立方体的几何约束。立方体是用单粒子状态来描述的，它们的角和边分别表现得像玻色子和费米子。当在其解决的配置，立方体，作为一个几何对象，显示对称性时，被打破，当驱动远离这个配置。对于每一个这样的对称，我们建立一个哈密顿算子。当一个哈密顿量处于它的基态时，立方体的对称性保持不变。当所有这些对称性都被保留时，立方体的配置与博弈的解相匹配。为了达到所有哈密顿算子的基态，我们使用基于哈密顿奖励的深度强化学习算法。立方体分四个阶段求解，所有阶段都基于基于其频谱的相应哈密顿奖励，灵感来自伊辛模型。将组合问题嵌入到量子力学形式主义中，提出了新的可能算法和未来在量子硬件上的实现。
英文摘要：
Rubik's Cube is one of the most famous combinatorial puzzles involving nearly $4.3 \times 10^{19}$ possible configurations. Its mathematical description is expressed by the Rubik's group, whose elements define how its layers rotate. We develop a unitary representation of such group and a quantum formalism to describe the Cube from its geometrical constraints. Cubies are describedby single particle states which turn out to behave like bosons for corners and fermions for edges, respectively. When in its solved configuration, the Cube, as a geometrical object, shows symmetrieswhich are broken when driven away from this configuration. For each of such symmetries, we build a Hamiltonian operator. When a Hamiltonian lies in its ground state, the respective symmetry of the Cube is preserved. When all such symmetries are preserved, the configuration of the Cube matches the solution of the game. To reach the ground state of all the Hamiltonian operators, we make use of a Deep Reinforcement Learning algorithm based on a Hamiltonian reward. The Cube is solved in four phases, all based on a respective Hamiltonian reward based on its spectrum, inspired by the Ising model. Embedding combinatorial problems into the quantum mechanics formalism suggests new possible algorithms and future implementations on quantum hardware.
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