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arXiv · 2010.03461

Verification of Group Non-membership by Shallow Quantum Circuits

Kai Sun·Zi-Jian Zhang·Fei Meng·Bin Cheng·Zhu Cao·Jin-Shi Xu·Man-Hong Yung·Chuan-Feng Li·Guang-Can Guo
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Abstract

Decision problems are the problems whose answer is either YES or NO. As the quantum analogue of $\mathsf{NP}$ (nondeterministic polynomial time), the class $\mathsf{QMA}$ (quantum Merlin-Arthur) contains the decision problems whose YES instance can be verified efficiently with a quantum computer. The problem of deciding the group non-membership (GNM) of a group element is known to be in $\mathsf{QMA}$. Previous works on the verification of GNM required a quantum circuit with $O(n^5)$ group oracle calls. Here we propose an efficient way to verify GNM problems, reducing the circuit depth to $O(1)$ and the number of qubits by half. We further experimentally demonstrate the scheme, in which two-element subgroups in a four-element group are employed for the verification task. A significant completeness-soundness gap is observed in the experiment.

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Kai Sun, Zi-Jian Zhang, Fei Meng, Bin Cheng, Zhu Cao, Jin-Shi Xu, Man-Hong Yung, Chuan-Feng Li, Guang-Can Guo. 2020-10-07. Verification of Group Non-membership by Shallow Quantum Circuits. https://arxiv.org/abs/2010.03461

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