Studying the computational complexity of determining winners under voting rules and designing fast algorithms are classical and fundamental questions in computational social choice. In this paper, we accelerate voting by leveraging quantum computing. We propose a quantum voting algorithm that can be applied to any anonymous voting rule. We further show that our algorithm can be quadratically faster than any classical sampling algorithm under a wide range of common voting rules, including plurality, Borda, Copeland, and STV. Precisely, our quantum voting algorithm achieves an accuracy of at least $1 - \varepsilon$ with runtime $\Theta\left(\frac{n\cdot\log(1/\varepsilon)}{\text{MOV}}\right)$, where $n$ is the number of votes and $\text{MOV}$ is margin of victory, the smallest number of voters to change the winner. On the other hand, any classical voting algorithm based on sampling a subset of voting achieves the same accuracy with runtime $\Theta\left(\frac{n^2\cdot\log(1/\varepsilon)}{\text{MOV}^2}\right)$ [Bhattacharyya and Dey, 2021]. Our theoretical results are supported by experiments under the plurality and Borda rule.

翻译：研究在投票规则下确定获胜者的计算复杂度并设计快速算法，是计算社会选择学中经典且基础的问题。本文通过利用量子计算加速投票过程。我们提出了一种可应用于任意匿名投票规则的量子投票算法。进一步证明，在包括多数投票、波达计数、科普兰规则及可转移单票制在内的多种常见投票规则下，该算法相较于任何经典采样算法可实现平方级加速。具体而言，我们的量子投票算法在运行时间$\Theta\left(\frac{n\cdot\log(1/\varepsilon)}{\text{MOV}}\right)$内达到至少$1 - \varepsilon$的精度，其中$n$为总票数，$\text{MOV}$为胜利边际（即改变获胜者所需的最少选民数）。而基于子集采样的经典投票算法实现相同精度所需运行时间为$\Theta\left(\frac{n^2\cdot\log(1/\varepsilon)}{\text{MOV}^2}\right)$ [Bhattacharyya and Dey, 2021]。在多数投票与波达计数规则下的实验验证了我们的理论结果。