论文标题
使用自由费米子来表征变分量子算法
Characterization of variational quantum algorithms using free fermions
论文作者
论文摘要
我们从自由费米子的角度研究变异量子算法。通过得出相关的LIE代数的显式结构，我们表明量子近似优化算法（QAOA）在一维晶格上（具有和没有解耦角）能够准备所有尊重电路符号的费米斯高斯状态。利用这些结果，我们从数值上研究了这些对称性与目标状态的位置之间的相互作用，并发现缺乏对称性使非局部状态更容易准备。对高斯州的有效经典模拟，系统大小高达$ 80 $和深层电路，用于研究电路过度参数化时的行为。在这种优化方案中，我们发现要收敛到解决方案的迭代次数与系统大小线性缩放。此外，我们观察到，收敛到溶液的迭代次数随电路的深度呈指数减小，直到其在系统大小上是二次的深度饱和。最后，我们得出的结论是，可以用梯度提供的更好的局部线性近似值来解释优化的改进。
We study variational quantum algorithms from the perspective of free fermions. By deriving the explicit structure of the associated Lie algebras, we show that the Quantum Approximate Optimization Algorithm (QAOA) on a one-dimensional lattice -- with and without decoupled angles -- is able to prepare all fermionic Gaussian states respecting the symmetries of the circuit. Leveraging these results, we numerically study the interplay between these symmetries and the locality of the target state, and find that an absence of symmetries makes nonlocal states easier to prepare. An efficient classical simulation of Gaussian states, with system sizes up to $80$ and deep circuits, is employed to study the behavior of the circuit when it is overparameterized. In this regime of optimization, we find that the number of iterations to converge to the solution scales linearly with system size. Moreover, we observe that the number of iterations to converge to the solution decreases exponentially with the depth of the circuit, until it saturates at a depth which is quadratic in system size. Finally, we conclude that the improvement in the optimization can be explained in terms of better local linear approximations provided by the gradients.
