Geometric quantum gates in liquid-state NMR based on a cancellation of dynamical phases

A proposal for applying nonadiabatic geometric phases to quantum computing, called double-loop method [S.-L. Zhu and Z. D. Wang, Phys. Rev. A 67, 022319 (2003)], is demonstrated in a liquid-state nuclear magnetic-resonance quantum computer. Using a spin-echo technique, the original method is modified so that quantum gates are implemented in a standard high-precision nuclear magnetic-resonance system for chemical analysis. We show that a dynamical phase is successfully eliminated and a one-qubit quantum gate is realized although the gate fidelity is not high.

Figure 1

Example of dynamics of a single-qubit cyclic vector. (a) A time-dependent external field $\mathbf{\Omega }\left(t\right)$ and (b) a closed trajectory on the Bloch sphere corresponding to a cyclic vector $|{\psi }_{+}\left(t\right)⟩$, $0\le t\le \tau =2\pi /\left|{\omega }_{\text{rf}}\right|$. The end point of each arrow represents the initial value. We set the parameters ${\omega }_0=2\pi$, ${\omega }_1=0.5\times 2\pi$, ${\omega }_{\text{rf}}=0.8\times 2\pi$, and $\varphi =0$ in Eq. (4).

Figure 2

Schematic diagram of double-loop method for dynamical phase cancellation according to the proposal by Zhu and Wang [8]. Two time-dependent magnetic fields are applied sequentially. The first magnetic field (loop 1) rotates counterclockwise, while the second one (loop 2) rotates clockwise in order to eliminate a dynamical phase.

Figure 3

(Color online) Example of Zhu-Wang’s double-loop method. The time-dependent external fields ${\mathbf{\Omega }}_1\left(t\right)$ and ${\mathbf{\Omega }}_2\left(t\right)$ are shown in (a), while the closed trajectory on the Bloch sphere corresponding to the cyclic vectors $|{\psi }_{1,+}⟩$ and $|{\psi }_{2,+}⟩$ in (b). We note that these are connected and thus form one closed trajectory. We set the loop parameters ${\omega }_{1,1}={\omega }_{2,1}=2\pi$, ${\omega }_{\text{rf}}=0.7\times 2\pi$, ${\omega }_{1,0}=0.27\times 2\pi$, and ${\omega }_{2,0}=1.5\times 2\pi$ in Eqs. (7, 8). We note that these parameters are calculated on the basis of a condition for nulling dynamical phases in Ref. [8]. In this example, $\Gamma =\frac{1}{2}$ in Eq. (13).

Figure 4

Schematic diagram of double-loop method for dynamical phase cancellation on the basis of a spin-echo approach. Two (four) soft (hard) square pulses are applied. We note that $R_y\left(\theta \right)=e^{-i\theta {\sigma }_y/2}$, in which $\theta ={\chi }_2-{\chi }_1$.

Figure 5

Gate operations visualized. The Bloch sphere in (a) is mapped to the surfaces in (b)–(d) under the gates with $ϵ=0.5$, 0.3 and 0.1, respectively. The right surface in (a) is an expected Bloch sphere when $\Theta =-\pi /4$, which corresponds to the Hadamard gate. Each left panel in (b)–(d) corresponds to the theoretical final state. The middle panels are the results for the single gate operation $U_{\text{echo}}\left(\Theta \right)$. The right panels are for the two-successive (double) gate operations.