Efficient synthesis of quantum gates on indirectly coupled spins

Experiments in coherent nuclear and electron magnetic resonance and quantum computing in general correspond to control of quantum-mechanical systems, guiding them from initial to final target states by unitary transformations. The control inputs (pulse sequences) that accomplish these unitary transformations should take as little time as possible so as to minimize the effects of relaxation and decoherence and to optimize the sensitivity of the experiments. Here, we derive a time-optimal sequences as fundamental building blocks to synthesize unitary transformations. Such sequences can be widely implemented on various physical systems, including the simulation of effective Hamiltonians for topological quantum computing on spin lattices. Experimental demonstrations are provided for a system consisting of three nuclear spins.

Figure 1

Three linearly coupled spins.

Figure 2

Rotation corresponds to $e^{\alpha {\Omega }_x}$ that moves $a=(0,sin\alpha ,cos\alpha )$ to $d=(0,0,1)$ via the point $b=(cos\alpha sin{t}_1,sin\alpha ,cos\alpha cos{t}_1)$ and $c=(\sqrt{{cos}^2\alpha {sin}^2{t}_1+{sin}^2\alpha },0,cos\alpha cos{t}_1)$.

Figure 3

The solid black line represents the time cost of generating $e^{-i4\theta [I_{1x}{I}_{2z}{I}_{3y}+I_{1y}{I}_{2z}{I}_{3x}]}$ using the optimal sequence; the star red line represents $T_C$ of Eq. (5), the total time cost by concatenating the sequence described in Sec. 2; the dotted blue line represents the total time cost using BCH formula as in Eq. (8).

Figure 4

Time needed to generate $e^{-2i\theta [I_{1x}{I}_{3x}+I_{1y}{I}_{3y}]}$.

Figure 5

The spins live on the edges of the square lattice. The spins adjacent to a star operator $A_s$ and a plaquette operator $B_p$ are shown.

Figure 6

Pulse sequence diagram for the unitary transformation $e^{2\pi i(I_{1x}{I}_{2z}{I}_{3y}+I_{1y}{I}_{2z}{I}_{3x})}$: the pulses of the first line are on spins 1 and 3, pulses of the second line are in spin 2; thin vertical lines denote 90${}^{\circ }$ pulses and wide vertical lines denote 180${}^{\circ }$ pulses which are inserted for refocusing of frequency offset effects; the durations $t_1$, $t_2$, and $\delta t$ can be calculated according to the above theory, which is $t_1=t_2=\text{arccos}\frac{1}{sin\frac{\theta }{2}+cos\frac{\theta }{2}},\delta t=\text{arccos}(cos\frac{\theta }{2}-sin\frac{\theta }{2})$.

Figure 7

(a), (c), (e), (g), (i), and (k) correspond to the spectra of six different initial states $I_{1x}$, $I_{1y}$, $I_{3x}$, $I_{3y}$, $I_{1z}$, $I_{3z}$; (b), (d), (f), (h), (j), and (l) correspond to the spectra of respective final states $I_{1z}{I}_{2z}{I}_{3x}$, $-I_{1z}{I}_{2z}{I}_{3y}$, $I_{1x}{I}_{2z}{I}_{3z}$, $-I_{1y}{I}_{2z}{I}_{3z}$, $-I_{3z}$, and $-I_{1z}$.

Figure 8

The evolution trajectory of $m_x,m_y,m_z$.

Figure 9

Rotations that move $a=(0,sin\alpha ,cos\alpha )$ to $d=(0,0,1)$ via the point $b=(cos\alpha sin{t}_1,sin\alpha ,cos\alpha cos{t}_1)$ and $c=(\sqrt{{cos}^2\alpha {sin}^2{t}_1+{sin}^2\alpha },0,cos\alpha cos{t}_1)$.