Families of quantum fingerprinting protocols

Abstract

We introduce several families of quantum fingerprinting protocols to evaluate the equality function on two $n$-bit strings in the simultaneous message passing model. The original quantum fingerprinting protocol uses a tensor product of a small number of $O(logn)$-qubit high-dimensional signals [H. Buhrman et al., Phys. Rev. Lett. 87, 167902 (2001)], whereas a recently proposed optical protocol uses a tensor product of $O(n)$ single-qubit signals, while maintaining the $O(logn)$ information leakage of the original protocol [J. M. Arazola and N. L\"utkenhaus, Phys. Rev. A 89, 062305 (2014)]. We find a family of protocols which interpolate between the original and optical protocols while maintaining the $O(logn)$ information leakage, thus demonstrating a tradeoff between the number of signals sent and the dimension of each signal. There has been interest in experimental realization of the recently proposed optical protocol using coherent states [F. Xu et al., Nat. Commun. 6, 8735 (2015); J.-Y. Guan et al., Phys. Rev. Lett. 116, 240502 (2016)], but as the required number of laser pulses grows linearly with the input size $n$, eventual challenges for the long-time stability of experimental setups arise. We find a coherent state protocol which reduces the number of signals by a factor $\frac{1}{2}$ while also reducing the information leakage. Our reduction makes use of a simple modulation scheme in optical phase space, and we find that more complex modulation schemes are not advantageous. Using a similar technique, we improve a recently proposed coherent state protocol for evaluating the Euclidean distance between two real unit vectors [N. Kumar et al., Phys. Rev. A 95, 032337 (2017)] by reducing the number of signals by a factor $\frac{1}{2}$ and also reducing the information leakage.