To harness the potential of noisy intermediate-scale quantum devices, it is paramount to find the best type of circuits to run hybrid quantum-classical algorithms. Key candidates are parametrized quantum circuits that can be effectively implemented on current devices. Here, we evaluate the capacity and trainability of these circuits using the geometric structure of the parameter space via the effective quantum dimension, which reveals the expressive power of circuits in general as well as of particular initialization strategies. We assess the expressive power of various popular circuit types and find striking differences depending on the type of entangling gates used. Particular circuits are characterized by scaling laws in their expressiveness. We identify a transition in the quantum geometry of the parameter space, which leads to a decay of the quantum natural gradient for deep circuits. For shallow circuits, the quantum natural gradient can be orders of magnitude larger in value compared to the regular gradient; however, both of them can suffer from vanishing gradients. By tuning a fixed set of circuit parameters to randomized ones, we find a region where the circuit is expressive, but does not suffer from barren plateaus, hinting at a good way to initialize circuits. We show an algorithm that prunes redundant parameters of a circuit without affecting its effective dimension. Our results enhance the understanding of parametrized quantum circuits and can be immediately applied to improve variational quantum algorithms.
翻译:为了利用杂乱的中间级量子装置的潜力,至关重要的是要找到最佳类型的电路来运行混合量子古典算法。 关键候选人是可在当前装置上有效运行的准美量子电路。 在这里, 我们通过有效的量子维度, 利用参数空间的几何结构来评估这些电路的能力和可训练性。 这揭示了一般电路以及特定初始化战略的表达力。 我们评估了各种流行电路类型的表达力,并发现取决于所使用电门类型的惊人差异。 特定电路的特征是缩放法。 我们确定参数空间的量度几何测量方法的过渡, 从而导致深度电路的量自然梯度衰减。 对于浅线电路来说, 量自然梯度的量可比通常的梯度值大得多; 但是, 这两种电路的表达力都可能因渐渐变的梯度而受损。 我们通过调整固定的电路参数来显示电路的清晰度区域, 但不会因其直观而受到影响。 我们确定参数的偏差度, 我们确定参数的偏差度的偏差值的偏差值的偏向, 直径将显示我们的直流的直径值的直径值的直径直到直径值的直径直到直到直到直至。