Experimental observation of the effect of global phase on optimal times of $\mathrm{SU}\left(2\right)$ quantum operations

We study the role that the global phase plays in quantum operations. Previous theoretical works suggested that the optimal (minimum) times to realize $\mathrm{SU}\left(n\right)$ quantum operations with different global phases are generally different. Here, we experimentally constructed two SU(2) operations with different global phases, i.e., $U$ and $-U$, in optimal times. Then, utilizing nuclear magnetic resonance interferometry, we measured these phases and observed the global phase's effect on the optimal times of these two operations. Our result further clarifies that the effect that the global phase has on unitary operations is not only mathematical, but also physical. In addition, this work has potential applications in many areas, such as designing time-optimal controls in quantum systems.

Figure 1

Quantum circuit for detecting the global phases of different unitary operations.

Figure 2

(a) The molecular structure of the chloroform sample. (b) Pulse sequence in NMR experiments. Note that the $R_{\vec{n}}\left(\theta \right)$ pulse (on ${}^1\mathrm{H}$) is applied to create $|\epsilon \rangle$, $R_{1y}(\pm \pi )$ pulses (on ${}^{13}\mathrm{C}$) are only applied for $C_{-U_{\mathrm{id}}}$. ${\alpha }^{\pm }$ and ${\tau }^{\pm }$ are given in the main text.

Figure 3

The experimental ${}^{13}\mathrm{C}$ spectrum of the final states for (a) $|\epsilon \rangle =|0\rangle$ and (b) $|\epsilon \rangle =|1\rangle$, respectively. The red solid (blue dashed) curve denotes the spectra after the control operation $C_{+U_{\mathrm{id}}}$ ($C_{-U_{\mathrm{id}}}$).

Figure 4

Experimental ${}^{13}\mathrm{C}$ and ${}^1\mathrm{H}$ spectra after the implementation of the controlled operations on the states, namely, (a) $C_{+U_{\mathrm{id}}}|00\rangle$, (b) $C_{+U_{\mathrm{id}}}|10\rangle$, (c) $C_{-U_{\mathrm{id}}}|00\rangle$, and (d) $C_{-U_{\mathrm{id}}}|10\rangle$, respectively.

Figure 5

Theoretical and experimental process matrix (real part) ${\chi }_{+}$ of $C_{+U_{\mathrm{id}}}$ and ${\chi }_{-}$ of $C_{-U_{\mathrm{id}}}$. The numbers 1 to 16 in the horizontal and vertical axes refer to the operators in the operator basis set $S$. $\left|\mathrm{Imag}\right({\chi }_{\pm }\left)\right|=0$ in theory, and $\left|\mathrm{Imag}\right({\chi }_{\pm }\left)\right|<0.025$ in the actual experiment.