In the circuit model of quantum computing, amplitude amplification techniques can be used to find solutions to NP-hard problems defined on $n$-bits in time $\text{poly}(n) 2^{n/2}$. In this work, we investigate whether such general statements can be made for adiabatic quantum optimization, as provable results regarding its performance are mostly unknown. Although a lower bound of $\Omega(2^{n/2})$ has existed in such a setting for over a decade, a purely adiabatic algorithm with this running time has been absent. We show that adiabatic quantum optimization using an unstructured search approach results in a running time that matches this lower bound (up to a polylogarithmic factor) for a broad class of classical local spin Hamiltonians. For this, it is necessary to bound the spectral gap throughout the adiabatic evolution and compute beforehand the position of the avoided crossing with sufficient precision so as to adapt the adiabatic schedule accordingly. However, we show that the position of the avoided crossing is approximately given by a quantity that depends on the degeneracies and inverse gaps of the problem Hamiltonian and is NP-hard to compute even within a low additive precision. Furthermore, computing it exactly (or nearly exactly) is \#P-hard. Our work indicates a possible limitation of adiabatic quantum optimization algorithms, leaving open the question of whether provable Grover-like speed-ups can be obtained for any optimization problem using this approach.

翻译：在量子计算的电路模型中，振幅放大技术可用于在$\text{poly}(n) 2^{n/2}$时间内求解定义在$n$比特上的NP难问题。本研究探讨此类一般性结论是否适用于绝热量子优化，因为关于其性能的可证明结果大多未知。尽管在此类设定下$\Omega(2^{n/2})$的下界已存在十余年，但具有该运行时间的纯绝热算法始终缺失。我们证明，对于广泛类型的经典局域自旋哈密顿量，采用非结构化搜索方法的绝热量子优化可实现与该下界匹配的运行时间（相差多对数因子）。为此，需要在绝热演化过程中界定谱隙，并以足够精度预先计算避免交叉点的位置，从而相应调整绝热调度方案。然而，我们证明避免交叉点的位置近似取决于问题哈密顿量的简并度和逆谱隙，且即使要求较低加性精度，其计算也是NP难的。此外，精确（或接近精确）计算该量是\#P难的。我们的工作揭示了绝热量子优化算法可能存在的局限性，悬而未决的问题是：采用此方法是否能在任何优化问题上获得可证明的类Grover加速。