Public-Key Quantum Money and Fast Real Transforms

Abstract: 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.