Zijian Song and Isaac H. Kim

1. Analyzing the paper

Summary

The paper is about a new method of recycling qudits in quantum circuits that have a convolutional structure. The authors show that some of the qudits can be reset to the initial state without measurement, by applying a rewinding circuit that is the inverse of a subcircuit acting on the idle qudits. The authors analyze the fidelity of this protocol for a family of quantum circuits that interpolate between convolutional circuits and local quantum circuits, by averaging over Haar-random gates. They find that the fidelity depends on the eigenvalue gap of a matrix related to the shape of the circuit, and they derive an exact expression for the fidelity in terms of a path counting problem on a square lattice. They also study the effect of noise on the fidelity and show that it can be expressed in terms of the entanglement fidelity of the channel. The paper demonstrates that the rewinding protocol can work well for typical instances of convolutional circuits with constant-sized windows, which are useful for variational quantum algorithms.

这篇论文介绍了一种新的方法，用于在具有卷积结构的量子电路中回收量子比特。作者表明，通过应用一个反转电路，可以将一些量子比特重置到初始状态而无需测量，该反转电路是作用于空闲量子比特的子电路的反向电路。作者通过对哈尔随机门进行平均，分析了该协议在一族量子电路上的保真度，该族电路在卷积电路和局部量子电路之间插值。他们发现，保真度取决于与电路形状相关的矩阵的本征值间隔，并推导出一个关于正方形格子上路径计数问题的保真度的精确表达式。他们还研究了噪声对保真度的影响，并表明可以用通道的纠缠保真度来表达。该论文证明，反演协议可以在具有恒定大小窗口的典型卷积电路实例中很好地工作，这对于变分量子算法非常有用。

The logic of the proposed method is to use a unitary reset protocol to recycle some of the qudits that are idle in a quantum circuit, and to analyze the fidelity of this protocol by relating it to a path counting problem on a graph. The main steps are:

• The authors consider a family of quantum circuits that interpolate between convolutional circuits and local circuits, which are both relevant for variational quantum algorithms.
• The authors show that some of the qudits in these circuits can be reset to |0> state by applying the inverse of a subcircuit that only acts on the idle qudits. This can reduce the number of qudits needed for the computation.
• The authors compute the average fidelity of the reset protocol over Haar-random gates, which is a measure of how well the protocol works for typical instances of the circuit. The fidelity is defined as the overlap between the reset state and the ideal |0> state.
• The authors map the fidelity to a partition function of a statistical mechanics model on a square lattice, using a folding technique and diagrammatic rules. The partition function can be expressed as a sum over contributions from domain walls or paths on the lattice.
• The authors simplify the calculation by only considering configurations with a single domain wall or path, which are determined by the shape of the circuit and some boundary conditions. They use existing results from combinatorics to count the number of paths and their weights.
• The authors obtain exact expressions for the fidelity of the reset protocol for different cases of circuit shape, window size, qudit dimension, noise model, and number of recycled qudits. They also derive expressions for the correlations between reset qudits and the fidelity in the presence of noise.

所提出的方法的逻辑是使用单一重置协议回收量子电路中闲置的一些四能级系统，并通过将其与图上的路径计数问题相联系来分析该协议的保真度。主要步骤包括：

• 作者考虑了一族量子电路，这些电路介于卷积电路和局部电路之间，这两种电路都与变分量子算法相关。
• 作者证明这些电路中的一些四能级系统可以通过应用仅作用于闲置四能级系统的子电路的逆来重置为|0>状态。这可以减少计算所需的四能级系统数量。
• 作者计算了逐个应用哈尔随机门时重置协议的平均保真度，这是协议对电路的典型实例的有效性的度量，保真度定义为重置状态与理想|0>状态之间的重叠。
• 作者使用折叠技术和图形规则将保真度映射到方格网格上统计力学模型的配分函数，该配分函数可以表示为来自网格上的域壁或路径的贡献之和。
• 作者通过仅考虑具有单个域壁或路径的构型来简化计算，这些构型由电路的形状和一些边界条件确定。他们使用组合数学中的现有结果来计数路径数和它们的权重。
• 作者获得了重置协议的保真度的精确表达式，这些表达式适用于电路形状、窗口大小、四能级系统维度、噪声模型和回收的四能级系统数量的不同情况。他们还推导了在存在噪声的情况下，重置四能级系统和保真度之间的相关性的表达式。

The authors use rigorous mathematical derivations and numerical simulations to support their results. They also provide clear explanations and illustrations for their main concepts and techniques. They acknowledge the limitations and assumptions of their method, and suggest possible extensions and improvements for future work.