Figure 1

The Zeeman energy levels of a spin-$\frac{15}{2}$ nucleus along with one of the optimum labeling schemes for the logical operation given in Table . The magnetic quantum number $\left(m\right)$ corresponding to each eigenstate is given on the left-hand side, and the qubit labeling is given on the right-hand side. The maximal sets of Table are shown in the energy-level diagram along with the transition selective $\pi$ pulses required to implement the truth table of Table using this labeling scheme.Reuse & Permissions

Figure 2

(a) Conventional labeling scheme and the pulses required for implementing the truth table of Table in a four-qubit spin-$\frac{1}{2}$ system. (b) Relabeled energy levels to implement the truth table of Table with optimum pulses. (c) The conventional scheme of (a) by rearranging the sequence of pulses to implement the logic of Table with the minimum number of pulses.Reuse & Permissions

Figure 3

Circuit of a quantum full adder. The two bits $A$ and $B$ are added with a carry “$C_0$” from the previous operation. An ancillary bit $L$ is included in the input to make the operation reversible. After the full-adder operation, the sum gets stored in $S$ $(S=C_0\oplus A\oplus B)$, and the carry is stored in $C_1$ $\left[C_1=L\oplus (AB\oplus A{C}_0\oplus B{C}_0)\right]$.Reuse & Permissions

Figure 4

The four-qubit system of 2–3 diflouro 6-nitrophenol. The two protons ($I_1$ and $I_3$) and the two fluorine nuclei ($I_2$ and $I_4$) constitute the four-qubit system. The equilibrium spectrum of each nucleus is individually shown. The assignment of the transitions is given above each line, which has been determined by transition tickling and HET-Z-COSY experiments [29, 30]. Each assignment identifies a transition of the spin corresponding to the states of other spins. In a $500\mathrm{MHz}$ spectrometer, the chemical shift difference between the two fluorine spins is $16\mathrm{kHz}$ while that between the two protons is $250\mathrm{Hz}$. The couplings range from $19.1\mathrm{Hz}$ to $-2.3\mathrm{Hz}$.Reuse & Permissions

Figure 5

Relabeled energy-level diagram for implementing a quantum full adder in the four-qubit system of 2–3 diflouro 6-nitrophenol. The maximal sets were created using the scheme outlines in Sec. 2. The chains of elements in the maximal sets were mapped on to chains of states in this four-qubit system. To simplify the experimental protocol, it was noted that all the pulses that commute (which also means that they are not connected to a common energy level) can be applied simultaneously. The maximal sets are mutually exclusive and so the pulses of different sets commute with each other. This means that ${\pi }_7$ and ${\pi }_8$ of sets $S_7$ and $S_8$ can be applied simultaneously with the pulses of $S_5$ and $S_6$. In the set $S_5$, the labels of states $\mid \alpha \beta \beta \alpha ⟩$ and $\mid \alpha \beta \beta \beta ⟩$ were interchanged so that the pulses ${\pi }_1$ and ${\pi }_2$ can be applied simultaneously. Similarly, in $S_6$, by interchanging the states $\mid \beta \beta \beta \alpha ⟩$ and $\mid \beta \beta \beta \beta ⟩$, ${\pi }_4$ and ${\pi }_5$ can be applied simultaneously. Hence, by this labeling, ${\pi }_1{\pi }_2{\pi }_4{\pi }_5{\pi }_7{\pi }_8$ followed by ${\pi }_3{\pi }_6$ would implement the quantum full adder. The initial (equilibrium) populations and final (after implementation of the full adder) populations are given beside each energy level as initial population (final population). The intensity of various transitions changes after the full adder. For example, we note that the intensity of the $\alpha \beta \beta$ transition of $I_4$ changes from $+1$ to $-2$, after implementation of the full adder.Reuse & Permissions

Figure 6

Implementation of the quantum full adder in the four-qubit system of 2–3 diflouro 6-nitrophenol. Starting from equilibrium, three selective Gaussian-shaped pulses of lengths $50\mu \mathrm{s}$, $350\mathrm{ms}$, and $450\mathrm{ms}$ were applied on selected transitions (as explained in text). Gradients were applied after each pulse to destroy any unwanted coherence created by the imperfection of pulses. A nonselective small flip-angle $(5°)$ pulse was used to map the final populations. The experimental spectra are shown above with the expected spectra shown (as a stick diagram) below each spin. The intensities in the stick diagram are $0$, $\pm 1$, and $\pm 2$ corresponding to the final populations of Fig. 5. All the spectra are Fourier-transformed after multiplication of the signal with a Gaussian window function. The longitudinal relaxation rates for the four spins are $T_1^1=7\mathrm{s}$, $T_1^2=3.5\mathrm{s}$, $T_1^3=10\mathrm{s}$, and $T_1^4=4\mathrm{s}$. The observed experimental intensities are within $18\%$ of the expected intensities.Reuse & Permissions