We propose a public-key quantum money scheme based on group actions and the Hartley transform. Our scheme adapts the quantum money scheme of Zhandry (2024), replacing the Fourier transform with the Hartley transform. This substitution ensures the banknotes have real amplitudes rather than complex amplitudes, which could offer both computational and theoretical advantages. To support this new construction, we propose a new verification algorithm that uses group action twists to address verification failures caused by the switch to real amplitudes. We also show how to efficiently compute the serial number associated with a money state using a new algorithm based on continuous-time quantum walks. Finally, we present a recursive algorithm for the quantum Hartley transform, achieving lower gate complexity than prior work and demonstrate how to compute other real quantum transforms, such as the quantum sine transform, using the quantum Hartley transform as a subroutine.

翻译：我们提出了一种基于群作用和哈特利变换的公钥量子货币方案。该方案改编自Zhandry（2024）的量子货币方案，用哈特利变换替代了傅里叶变换。这一替换确保了银行票据具有实振幅而非复振幅，这可能带来计算和理论上的双重优势。为支持这一新构造，我们提出了一种新的验证算法，该算法利用群作用扭曲来解决因转向实振幅而导致的验证失败问题。我们还展示了如何基于连续时间量子行走的新算法高效计算与货币态相关联的序列号。最后，我们提出了一种用于量子哈特利变换的递归算法，其实现了比先前工作更低的门复杂度，并演示了如何以量子哈特利变换为子程序计算其他实数量子变换，例如量子正弦变换。