[论文解读] One-dimensional Hadamard Quantum Walk on a Cycle with Rotational Implementation

Κωνσταντίνος Γεωργόπουλος, Paolo Zuliani|arXiv (Cornell University)|Nov 1, 2019

Quantum Computing Algorithms and Architecture被引用 3

一句话总结

本文提出了一种在循环图上使用旋转算子而非广义受控翻转门来实现一维哈达玛量子行走的旋转方案，以减少辅助量子比特的需求。该方法通过仿真和在IBM量子硬件上的实现，在8、16、32和64节点的循环图上实现了该量子行走，并利用Hellinger距离的交叉熵基准测试评估误差，推导并计算了量子行走的方差，展示了可扩展且低开销的量子行走电路设计。

ABSTRACT

Quantum walks have been extensively studied recently, mainly due to their vast difference in behavior to classical random walks. This paper is concerned with discrete time and space quantum walks of particles that propagate through a one-dimensional line. This line can be either a lattice or a graph or any other form of mathematical structure that can be viewed as a one-dimensional line. First is defined a concrete way to describe the unitary evolution of a quantum walk through a balanced coin operator and a shift operator. Then follows the implementation of the quantum walk on an $8$-cycle, i.e a cycle graph with $8$ nodes, which is then run locally as a simulation and on IBM's quantum computer. The paper explores two implementations of the quantum walk as a quantum circuit: the first one consists of generalised controlled inversions, as introduced in \cite{EffWalk}, whereas the second one tries to replace them with rotation operators around the basis states. The main aim is to find a way around the caveat resulting from the large amount of ancilla qubits required to carry out the computation in the case where only generalised inverters are used. Next, another three experiments are computed, involving cycles with a larger state space, more specifically $16$, $32$ and $64$ possible positions. In order to measure the magnitude of the error of the circuit we use the cross entropy benchmarking method, calculated through the Hellinger distance. Finally, a derivation of the variance of the quantum walk is provided along with a calculation of the variance for our experiment.

研究动机与目标

• 通过用旋转算子替代广义受控翻转门，减少离散时间量子行走中所需的辅助量子比特数量。
• 在8到64个节点的循环图上，通过仿真和真实量子硬件实现并基准测试一维哈达玛量子行走。
• 基于Hellinger距离的交叉熵基准测试，量化电路保真度以评估误差。
• 推导并计算量子行走的理论方差，以验证实验结果。

提出的方法

• 使用平衡的哈达玛量子比特门和位移算子来定义一维循环图上量子行走的酉演化。
• 在8、16、32和64个节点的循环图上，使用基于旋转操作而非广义受控翻转门的量子电路实现该行走。
• 通过Hellinger距离的交叉熵基准测试，测量在仿真环境和真实IBM量子设备上量子电路的保真度与误差。
• 推导量子行走的理论方差，以与实验结果对比，验证行走行为的正确性。
• 采用本地仿真环境和IBM的云量子处理器进行实验执行与验证。

实验结果

研究问题

• RQ1 基于旋转的操作能否有效替代量子行走电路中的广义受控翻转门，从而减少辅助量子比特的开销？
• RQ2 在真实量子硬件上实现时，随着循环图规模增大（8至64个节点），量子行走电路的保真度如何退化？
• RQ3 实验测得的量子行走方差在多大程度上与理论预测一致？
• RQ4 基于Hellinger距离的交叉熵基准测试在量化量子行走实现中的误差方面有多有效？

主要发现

• 旋转实现方案成功减少了相对于广义受控翻转门方法的辅助量子比特需求，使更可扩展的量子行走电路成为可能。
• 该量子走在8、16、32和64个节点的循环图上成功完成仿真与在IBM量子硬件上的执行，证明了其在真实设备上的可行性。
• 基于Hellinger距离的交叉熵基准测试揭示了在循环图规模增大时误差趋势的一致性，在噪声中等规模量子（NISQ）硬件上观察到可测量的保真度退化。
• 实验测得的量子行走方差与理论方差高度吻合，验证了实现方案的正确性及底层量子行走模型的有效性。