Quantum Factorization of 143 on a Dipolar-Coupling Nuclear Magnetic Resonance System

Quantum algorithms could be much faster than classical ones in solving the factoring problem. Adiabatic quantum computation for this is an alternative approach other than Shor’s algorithm. Here we report an improved adiabatic factoring algorithm and its experimental realization to factor the number 143 on a liquid-crystal NMR quantum processor with dipole-dipole couplings. We believe this to be the largest number factored in quantum-computation realizations, which shows the practical importance of adiabatic quantum algorithms.

Figure 1

Process of the adiabatic factorization of 143. (a) the lowest three energy levels of the time-dependent Hamiltonian in Eq. (1), The parameter $g$ in the initial Hamiltonian is 0.6. (b) $k=1–5$ shows the populations on computational basis of the system during the adiabatic evolution at different times marked in a); $k=6$ shows the result got from our experiment. The experimental result agrees well with the theoretical expectation. The system finally stays on a superposition of $|6⟩$ and $|9⟩$, which denotes that the answer is {$p=11$, $q=13$} or {$p=13$, $q=11$}.

Figure 2

Quantum register in our experiment. (a) The structure of the 1-Bromo-2-Chlorobenzene molecule. The four ${}^1\mathrm{H}$ nuclei in ovals forms the qubits in our experiment. (b) Spectrum of ${}^1\mathrm{H}$ of the thermal state ${\rho }_{th}=\sum _{i=1}^4{\sigma }_z^i$ applying a $[\pi /2{]}_y$ pulse. Transitions are labeled according to descending order of their frequencies. (c) Labeling scheme for the states of the four-qubit system.

Figure 3

NMR spectra for the small-angle-flip observation of the PPS and the output state ${\rho }_{\mathrm{out}}$, respectively. The blue spectra (thick) are the experimental results, and the red spectra (thin) are the simulated ones. (a) (c) Spectra corresponding to the PPS ${\rho }_{0000}$ and ${\rho }_{1111}$ by applying a small-angle-flip (3°) pulse. The main peaks are No. 33 and No. 3 labeled in the thermal equilibrium spectrum. (b) Spectrum corresponding to the output state ${\rho }_{\mathrm{out}}$ after applying a small-angle-flip (3°) pulse, which just consists of the peaks of No. 33 and No. 3.