While scalable error correction schemes and fault tolerant quantum computing seem not to be universally accessible in the near sight, the efforts of many researchers have been directed to the exploration of the contemporary available quantum hardware. Due to these limitations, the depth and dimension of the possible quantum circuits are restricted. This motivates the study of circuits with parameterized operations that can be classically optimized in hybrid methods as variational quantum algorithms, enabling the reduction of circuit depth and size. The characteristics of these Parameterized Quantum Circuits (PQCs) are still not fully understood outside the scope of their principal application, motivating the study of their intrinsic properties. In this work, we analyse the generation of random states in PQCs under restrictions on the qubits connectivities, justified by different quantum computer architectures. We apply the expressibility quantifier and the average entanglement as diagnostics for the characteristics of the generated states and classify the circuits depending on the topology of the quantum computer where they can be implemented. As a function of the number of layers and qubits, circuits following a Ring topology will have the highest entanglement and expressibility values, followed by Linear/All-to-all almost together and the Star topology. In addition to the characterization of the differences between the entanglement and expressibility of these circuits, we also place a connection between how steep is the increase on the uniformity of the distribution of the generated states and the generation of entanglement. Circuits generating average and standard deviation for entanglement closer to values obtained with the truly uniformly random ensemble of unitaries present a steeper evolution when compared to others.

中文翻译：

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通过可表达性和平均纠缠来表征参数化量子电路中的随机性


虽然可扩展的纠错方案和容错量子计算似乎并非在近处普遍可用，但许多研究人员的努力已转向探索当代可用的量子硬件。由于这些限制，可能的量子电路的深度和维度受到限制。这激发了对具有参数化运算的电路的研究，这些电路可以在混合方法中作为变分量子算法进行经典优化，从而能够减小电路深度和尺寸。这些参数化量子电路 （PQC） 的特性在其主要应用范围之外仍未完全理解，这激发了对其本征性质的研究。在这项工作中，我们分析了在量子比特连接性限制下 PQC 中随机状态的生成，由不同的量子计算机架构证明是合理的。我们应用可表达性量词和平均纠缠作为对生成状态特征的诊断，并根据可以实现它们的量子计算机的拓扑对电路进行分类。作为层数和量子比特数的函数，遵循环形拓扑的电路将具有最高的纠缠和可表示性值，其次是线性/几乎一起到位和星形拓扑。除了描述这些电路的纠缠和可表达性之间的差异外，我们还将生成状态分布的均匀性的增加程度与纠缠的产生联系起来。 与其他电路相比，为纠缠生成平均值和标准差的电路更接近于使用真正均匀随机的酉系综获得的值，其演变更加陡峭。