Verification of Group Non-membership by Shallow Quantum Circuits

TL;DR

This paper introduces a more efficient quantum verification method for group non-membership problems, reducing circuit depth and qubit usage, and demonstrates its effectiveness through experimental validation.

Contribution

It presents a shallow quantum circuit approach for GNM verification, significantly improving efficiency over previous methods and experimentally validating the scheme.

Findings

Reduced circuit depth to O(1)

Halved the number of qubits needed

Observed a significant completeness-soundness gap in experiments

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.