Figure 1

(a) Energy-level diagram for the adiabatic factorization of $N=21$ when $s(t)=(t/T{)}^2$ and $T=0.168$. (b) Occupation probabilities for the computational basis states $|z_1{z}_2{z}_3⟩$ for the theoretical simulation [denoted by blue (light gray) bars] and experimentally reconstructed populations of the computational basis states after $m$ evolution steps [denoted by red (dark gray) bars]. (c) Measured spectra of ${}^{13}\mathrm{C}$ for each adiabatic step. The four resonance lines of ${}^{13}\mathrm{C}$ are labeled by the corresponding states of the two other qubits. The spectra were adjusted as absorption spectra by 180° phase correction, which leads to positive amplitude indicating the $|1⟩$ subspace of the ${}^{13}\mathrm{C}$ qubit. A color scale indicates peak intensities, which are in arbitrary units. The system starts in an equal weight superposition and evolves to the desired final state $|111⟩$, which encodes the solution $p=3$, $q=7$.Reuse & Permissions

Figure 2

(a) Molecular structure of Diethyl-fluoromalonate, (b) schematic representation of the experiment and (c) the pulse sequence that implements the adiabatic evolution for factorizing 21. The oval in (a) marks the three spins used as qubits. The rectangles in (c) labeled with ${\theta }_m$ represent rotations by an angle $g[1-(\frac{m}{M}{)}^r]\tau$, while the narrow empty rectangles denote 90° rotations and the wide ones (labeled by $\pi$) denote the refocusing 180° pulses. The delays are $d_3^{(m)}=10(\frac{m}{M}{)}^r\tau (\frac{1}{\pi J_{\mathrm{HF}}}+\frac{2}{\pi J_{\mathrm{HC}}})$, $d_4^{(m)}=10(\frac{m}{M}{)}^r\tau (\frac{1}{\pi J_{\mathrm{HF}}}-\frac{2}{\pi J_{\mathrm{HC}}})$, $d_5^{(m)}=\frac{1}{2|J_{\mathrm{CF}}|}-20(\frac{m}{M}{)}^r\tau (\frac{1}{\pi J_{\mathrm{HC}}}+\frac{1}{\pi |J_{\mathrm{CF}}|})$, and $d_7^{(m)}=32(\frac{m}{M}{)}^r\tau \frac{1}{\pi J_{\mathrm{HC}}}$. Negative durations $d_5^{(m)}<0$ according to this formula were implemented as positive ones, by omitting the dashed $\pi$ pulse preceding $d_5^{(m)}$ and replacing the gray $-x$ pulse immediately after the period by an $x$ pulse.Reuse & Permissions

Figure 3

Average evolution time for achieving the probability $1/8$ for 50 instances as a function of the number of input bits $n$. The circles represent the simulation data for the bit length $7\le n\le 16$, while the solid line shows the quadratic fit to the data.Reuse & Permissions