qubits) is better than that of the conventional scheme. The run-time of adiabatic algorithms is not just determined by the minimum value of the fundamental energy gap (between the ground state and the exited states), but also by its curvature at the critical point. The proposed symmetry-restoring adiabatic quantum algorithm only contains contributions linear and quadratic in the Pauli matrices and can be generalized to other problem Hamiltonians which are decomposed of terms involving one and two qubits. We show how the factoring problem can be cast into such a quadratic form. These findings suggest that adiabatic quantum algorithms can solve a large class of NP problems much faster than the Grover search routine (which corresponds to a first-order transition and yields a quadratic enhancement only).

DOI: 10.48550/arxiv.0708.1882

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{\cal O}\{20\}

We analyze the ground-state entanglement in a quantum adiabatic evolution algorithm designed to solve the

NP

-complete Exact Cover problem. The entropy of entanglement seems to obey linear and universal scaling at the point where the energy gap becomes small, suggesting that the system passes near a quantum phase transition. Such a large scaling of entanglement suggests that the effective connectivity of the system diverges as the number of qubits goes to infinity and that this algorithm cannot be efficiently simulated by classical means. On the other hand, entanglement in Grover's algorithm is bounded by a constant.