Speedup of quantum-state transfer by three-qubit interactions: Implementation by nuclear magnetic resonance

Universal quantum information processing requires single-qubit rotations and two-qubit interactions as minimal resources. A possible step beyond this minimal scheme is the use of three-qubit interactions. We consider such three-qubit interactions and show how they can reduce the time required for a quantum state transfer in an $XY$ spin chain. For the experimental implementation, we use liquid-state nuclear magnetic resonance, where three-qubit interactions can be implemented by sequences of radio-frequency pulses.

Figure 1

(Color online) The duration of the QST $t_{QS{T}_{1\to 3}}$ (dot dashed), $t_{QS{T}_{3\to 1}}$ (dashed), and $t_{QS{T}_{3\to 1\to 3}}$ (solid) vs $\lambda$. The unit of the vertical axes is chosen as $t_0=\frac{\pi }{2\sqrt{2}}$, normalized to the duration for $\lambda =0$.

Figure 2

The frame for operations $U_C\left(t\right)$ and $U_D\left(t\right)$. The vectors ${\mathbf{n}}_{\mathbf{c}}$ and ${\mathbf{n}}_{\mathbf{d}}$ denote the directions of the rotation axes for the two operations, respectively. They are tilted from the $z$ axis by the angles ${\theta }_c$ and ${\theta }_d=\pi -{\theta }_c$. The projections of ${\mathbf{n}}_{\mathbf{c}}$ and ${\mathbf{n}}_{\mathbf{d}}$ into the $xy$ plane are identical and indicated by a black line. The angles between the projection and the $x$ and $y$ axes are $\pi ∕4$.

Figure 3

The parameters of carbon-13 labeled trichloroethylene (TCE). The diagonal terms in the table are the shifts (in Hz) of the carbons and protons with respect to the reference frequencies 500.13 and $125.76\mathrm{MHz}$, respectively. The nondiagonal terms are the coupling constants, also in Hz.

Figure 4

The carbon spectrum (a) and proton spectrum (b) obtained by applying selective readout pulses to the system in its thermal equilibrium state. Each qubit gives rise to four resonance lines, which correspond to specific states of the other qubits. The highest frequency lines always correspond to the other qubits being in the $\mid 00⟩$ state, the lowest frequency lines to the $\mid 11⟩$ state.

Figure 5

(Color online) Pulse sequences for the implementation of $U_C\left(t\right)$ (a) and $U_D\left(t\right)$ (b). Steps 1–10 correspond to the ten compound operations separated by “$\times$” in $U_C\left(t\right)$ and $U_D\left(t\right)$, respectively. The unfilled rectangles denote $\pi ∕2$ pulses, and the filled rectangles denote $\pi$ pulses. $X$, $\overline{X}$, $Y$, and $\overline{Y}$ below the pulses denote the $x$, $-x$, $y$, and $-y$ directions along which the pulses are applied. Those $\pi$ pulses for which directions are not denoted are refocusing pulses. They are applied in pairs in which the two pulses take opposite directions to reduce experimental errors. The durations of the pulses applied to H2 and the nonselective pulses applied to C1 and C3 are so short that they can be ignored. The selective $\pi$ pulses for C1 and C3, denoted by the green and blue rectangles marked by the tilted and horizontal short lines, are implemented as RE-BURP [29] and Gauss shaped pulses with 6.2649 and $2.8252\mathrm{ms}$ durations, respectively. The delays are $d_1=\frac{9}{8J_{12}}$, $d_2=\frac{1}{16J_{23}}$, $d_3=\frac{1}{4\pi J_{23}}[\frac{\pi }{2}-\arctan \left(\frac{2\sqrt{2}}{\lambda }\right)]$, $d_4=\frac{1}{2\pi J_{12}}(t\sqrt{2+\frac{{\lambda }^2}{4}}+2\pi )$, $d_5=\frac{1}{2\pi J_{12}}\{\pi -\frac{1}{2}[\frac{\pi }{2}-\arctan \left(\frac{2\sqrt{2}}{\lambda }\right)]\}$, $d_6=\frac{t}{2\pi J_{23}}\sqrt{2+\frac{{\lambda }^2}{4}}$. For the case of $\lambda =0$, steps 2, 4, 7, and 9 are omitted.

Figure 6

The experimental results demonstrating the QST. The initial state is ${\sigma }_x^3$; the corresponding spectrum is shown in the left-hand column. The results of the QST C3$\to$ C1 are shown as (d)–(f), and the results of the cyclic transfer C3 $\to$ C1 $\to$ C3 are shown as (g)–(i). The three rows correspond to increasing three-qubit coupling strength, $\lambda =0$, $1.5$, and $4$. The time required for each transfer is shown in the figures. At $t=t_{QS{T}_{3\to 1}}$, the three-spin system is in state $-{\sigma }_x^1{\sigma }_z^2{\sigma }_z^3$, and at $t=t_{QS{T}_{3\to 1\to 3}}$, the system is in state ${\sigma }_x^3$.

Figure 7

(Color online) Progress of the QST, starting from ${\sigma }_x^3$, for different strengths of the three-body coupling. The upper part of the figure shows the overlap of the density operator with the target state ${\sigma }_x^1{\sigma }_z^2{\sigma }_z^3$ as a function of time. The unit $t_0$ of the time axis corresponds to the transfer time in the absence of the three-body interaction. The data for $\lambda =0$, $1.5$, and $4$ are marked by *, +, and $\times$, respectively. The solid lines represent the theoretical results, the individual points correspond to the simulated data by setting TCE as the weak-coupling system without decoherence. Points A, B, and C indicate the maxima corresponding to the transfer times C3$\to$ C1 and the points D, E, and F to the transfer times C3 $\to$ C1 $\to$ C3. This clearly demonstrates the speedup of the transfer by the three-body interaction.

Figure 8

(Color online) Progress of the QST, starting from ${\sigma }_y^3$, for different strengths of the three-body coupling. The data for $\lambda =0$, $1.5$, and $4$ are marked by *, +, and $\times$, respectively. The graphs are the theoretical results, used to fit the corresponding data. Points A, B, and C indicate the maxima corresponding to the transfer times C$3$ $\to$ C$1$ and points D, E, and F to the transfer times C$3$ $\to$ C$1$ $\to$ C$3$.

Figure 9

(Color online) Progress of the QST, starting from ${\sigma }_z^3$, for different strengths of the three-body coupling. The data for $\lambda =0$, $1.5$, and $4$ are marked by *, +, and $\times$, respectively. The graphs are the theoretical results, used to fit the corresponding data. Points A, B, and C indicate the maxima corresponding to the transfer times C$3$ $\to$ C$1$ and points D, E, and F to the transfer times C$3$ $\to$ C$1$ $\to$ C$3$.