Hybrid quantum-classical algorithms appear to be the most promising approach for near-term quantum applications. An important bottleneck is the classical optimization loop, where the multiple local minima and the emergence of barren plateaux make these approaches less appealing. To improve the optimization the Quantum Natural Gradient (QNG) method [15] was introduced – a method that uses information about the local geometry of the quantum state-space. While the QNG-based optimization is promising, in each step it requires more quantum resources, since to compute the QNG one requires $O(m^2)$ quantum state preparations, where $m$ is the number of parameters in the parameterized circuit. In this work we propose two methods that reduce the resources/state preparations required for QNG, while keeping the advantages and performance of the QNG-based optimization. Specifically, we first introduce the Random Natural Gradient (RNG) that uses random measurements and the classical Fisher information matrix (as opposed to the quantum Fisher information used in QNG). The essential quantum resources reduce to linear $O(m)$ and thus offer a quadratic "speed-up", while in our numerical simulations it matches QNG in terms of accuracy. We give some theoretical arguments for RNG and then benchmark the method with the QNG on both classical and quantum problems. Secondly, inspired by stochastic-coordinate methods, we propose a novel approximation to the QNG which we call Stochastic-Coordinate Quantum Natural Gradient that optimizes only a small (randomly sampled) fraction of the total parameters at each iteration. This method also performs equally well in our benchmarks, while it uses fewer resources than the QNG.

中文翻译：

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 随机自然渐变


混合量子经典算法似乎是近期量子应用最有前途的方法。一个重要的瓶颈是经典优化循环，其中多个局部最小值和贫瘠高原的出现使这些方法的吸引力降低。为了改进优化，引入了量子自然梯度 （QNG） 方法 [15]——一种使用量子状态空间局部几何信息的方法。虽然基于 QNG 的优化很有前途，但在每一步中都需要更多的量子资源，因为要计算 QNG，需要 $O（m^2）$ 量子态准备，其中 $m$ 是参数化电路中的参数数量。在这项工作中，我们提出了两种方法，可以减少 QNG 所需的资源/状态准备，同时保持基于 QNG 的优化的优势和性能。具体来说，我们首先介绍了使用随机测量的随机自然梯度 （RNG） 和经典的 Fisher 信息矩阵（而不是 QNG 中使用的量子 Fisher 信息）。基本量子资源简化为线性 $O（m）$，从而提供二次“加速”，而在我们的数值模拟中，它在准确性方面与 QNG 相匹配。我们给出了 RNG 的一些理论论点，然后在经典和量子问题上用 QNG 对该方法进行了基准测试。其次，受随机坐标方法的启发，我们提出了一种新颖的 QNG 近似值，我们称之为随机坐标量子自然梯度，它在每次迭代时仅优化总参数的一小部分（随机采样）。这种方法在我们的基准测试中也表现同样出色，但使用的资源比 QNG 少。