Experimental demonstration of the Deutsch-Jozsa algorithm in homonuclear multispin systems

Despite early experimental tests of the Deutsch-Jozsa (DJ) algorithm, there have been only a very few nontrivial balanced functions tested for register number $n>3$. In this paper, we experimentally demonstrate the DJ algorithm in four- and five-qubit homonuclear spin systems by the nuclear-magnetic-resonance technique, by which we encode the one function evaluation into a long shaped pulse with the application of the gradient ascent algorithm. Our work, dramatically reducing the accumulated errors due to gate imperfections and relaxation, demonstrates a better implementation of the DJ algorithm.

Figure 1

Molecular structure of crotonic acid together with a table of the chemical shifts (on the diagonal) and $J$-coupling constants (below the diagonal) for four ${}^{13}$C spins. The chemical shifts are given with respect to a reference frequency of ${\nu }_{rf}=100.632251$ MHz. All the values are in Hz units. The relaxation times $T_2$ for carbons Nos. 1, 2, 3, and 4 are about 1.075, 0.910, 0.932, and 1.175 s, respectively.

Figure 2

Molecular structure of arginine together with a table of the chemical shifts (on the diagonal) and $J$-coupling constants (below the diagonal) for five ${}^{13}$C spins (from C${}_1$ to C${}_5$). ${\text{C}}_6$ has negligible $J$ couplings with other carbons [23]. The chemical shifts are given with respect to a reference frequency of ${\nu }_{rf}=100.63029$ MHz. All the values are in Hz units. The relaxation times $T_2$ for carbons Nos. 1, 2, 3, 4, and 5 are about 0.822, 0.312, 0.227, 0.243, and 0.278 s, respectively.

Figure 3

Experimental spectra of the final states ${\rho }_{f_i}(i=0,1,...4)$ after the implementation of the DJ algorithm in crotonic acid. From the phase information of these spectra, we can determine that $f_0$ is a constant function and $f_1$, $f_2$, $f_3$, and $f_4$ are balanced functions. All the spectra are drawn in the same vertical scales.

Figure 4

Experimental spectra of the final states ${\rho }_{f_i}(i=0,1,...5)$ after the implementation of the DJ algorithm in arginine. From the phase information of these spectra, we can determine that $f_0$ is a constant function and $f_1$, $f_2$, $f_3$, $f_4$, and $f_5$ are balanced functions. All the spectra are drawn in the same vertical scales.

Figure 5

Comparison of the simulated spectra with the experimental spectra for ${\rho }_{f_4}$ and ${\rho }_{f_5}$ in arginine. For convenience of comparison, we scaled the stimulated spectra signals almost as the same intensities as the experimental spectra. The right-hand three columns are the peaks of partial spins theoretically expected to appear with expanded horizontal scales. The signals of other spins are relatively small.