There is a sizable literature on investigating the minimum and maximum
numbers of cycles in a class of graphs. However, the answer is known only for
special classes. This paper presents a result on the smallest number of cycles
in hamiltonian 3-connected cubic graphs. Further, it describes a proof
technique that could improve an upper bound of the largest number of cycles in
a hamiltonian graph.

本文介绍了 3 连通立方图哈密顿图中最小正常周期数的结果，并描述了一种证明技术，可以改进哈密顿图中最大周期数的上限。

图中的环数

On the Number of Cycles in a Graph

A quantum system will stay near its instantaneous ground state if the
Hamiltonian that governs its evolution varies slowly enough. This quantum
adiabatic behavior is the basis of a new class of algorithms for quantum
computing. We test one such algorithm by applying it to randomly generated,
hard, instances of an NP-complete problem. For the small examples that we can
simulate, the quantum adiabatic algorithm works well, and provides evidence
that quantum computers (if large ones can be built) may be able to outperform
ordinary computers on hard sets of instances of NP-complete problems.

通过应用量子绝热算法来解决 NP 完全问题，证明了如果构建大型计算机，则量子计算机可能能够在困难的 NP 完全问题上胜过普通计算机。

一种应用于 NP 完全问题随机实例的量子绝热进化算法

A Quantum Adiabatic Evolution Algorithm Applied to Random Instances of  an NP-Complete Problem

We give a quantum algorithm for solving instances of the satisfiability
problem, based on adiabatic evolution. The evolution of the quantum state is
governed by a time-dependent Hamiltonian that interpolates between an initial
Hamiltonian, whose ground state is easy to construct, and a final Hamiltonian,
whose ground state encodes the satisfying assignment. To ensure that the system
evolves to the desired final ground state, the evolution time must be big
enough. The time required depends on the minimum energy difference between the
two lowest states of the interpolating Hamiltonian. We are unable to estimate
this gap in general. We give some special symmetric cases of the satisfiability
problem where the symmetry allows us to estimate the gap and we show that, in
these cases, our algorithm runs in polynomial time.

本文提出了一种基于绝热演化的量子算法，用于解决可满足性问题，其中 Hamiltonian 函数随时间变化，插入了初始 Hamiltonian 和最终 Hamiltonian，通过保证时间足够长，来确保系统演化到预期的最终基态，算法时间复杂度在某些可满足性问题的对称情况下为多项式时间。