  Prime factorization (P = M*N) is considered to be a promising application in
quantum computations. We perform 4-bit factorization in experiments using a
superconducting flux qubit toward quantum annealing. Our proposed method uses a
superconducting quantum circuit implementing a multiplier Hamiltonian, which
provides combinations of M and N as a factorization solution after quantum
annealing when the integer P is initially set. The circuit comprises multiple
multiplier units combined with connection qubits. The key points are a native
implementation of the multiplier Hamiltonian to the superconducting quantum
circuit and its fabrication using a Nb multilayer process with a Josephson
junction dedicated to the qubit. The 4-bit factorization circuit comprises 32
superconducting flux qubits. Our method has superior scalability because the
Hamiltonian is implemented with fewer qubits than in conventional methods using
a chimera graph architecture. We perform experiments at 10 mK to clarify the
validity of interconnections of a multiplier unit using qubits. We demonstrate
experiments at 4.2 K and simulations for the factorization of integers 4, 6,
and 9.
