Demonstration of multiparty quantum clock synchronization

Abstract

Quantum clock synchronization (QCS) is a kind of method to measure the time difference among spatially separated clocks with the principle of quantum mechanics. The first QCS algorithm proposed by Chuang and Jozsa is merely based on two parties, which is further extended and generalized to multiparty situation by Krco and Paul. They present a multiparty QCS protocol based upon W-state, utilizing shared prior entanglement and broadcasting the classical information to synchronize spatially separated clocks. Shortly afterward, Ben-Av and Exman came up with an optimized multiparty QCS based on Z-state. In this work, we firstly report the demonstrations of these two multiparty QCS protocols in a four-qubit liquid-state nuclear magnetic resonance system. The experimental results show a great agreement with the theoretical predictions and also prove that Ben-AV’s multiparty QCS algorithm is more accurate than Krco’s.

Appendices

Appendix A: GRAPE robustness profiles

As we all know the GRAPE technical is to realize the optimal control of the coupled spin dynamics [31]. In this method, the chosen transfer time T is discretized in N equal steps of duration $\Delta t=T/N$ and during each step, the control amplitudes are constant. The time evolution of the spin system during a time step j is given by the propagator

$$\begin{aligned} {U_j=\exp ^{-i\Delta t(H_0+X_j\cdot \sigma _x+Y_j\cdot \sigma _y)}=\exp ^{-i\Delta t(H_0+A_j(\cos (\phi _j) \sigma _x+\sin (\phi _j) \sigma _y))}} \end{aligned}$$

(24)

where the $H_0$ is the internal Hamiltonian in Eq. 14, the $X_j$ and $Y_j$ are amplitudes of the radiofrequency (RF) pulses applied on the x and y axis, the $A_j$ and $\phi _j$ are the amplitudes and phases of RF pulses. The final density operator at time $t=T$ is

$$\begin{aligned} {\rho (T)=U_N\ldots U_1\rho _0 U_1^{\dagger }\ldots U_N^{\dagger }} \end{aligned}$$

(25)

In our experiments to realize U in Eq. 17, we choose the $T=20$ ms, $N=4000$, $\Delta t=5\upmu$s, the $A_j$ and $\phi _j$ are calculated by the GRAPE method [31]. The pulses to realize U are designed to be robust to the static field distributions, RF inhomogeneity and decoherence. Then, we will prove the robustness in detail.

1.1 Static field distributions

In Ref. [20], we know that the static field distributions $B_0$ only have the influences on $\nu _j$ in the internal Hamiltonian $H_0$. So we add some noises on $B_0$ and the $\nu _j$ in the Eq. 17 can be rewritten as $\nu _j+\Delta \nu _j$. We assume that $\Delta \nu _j$ is changing from $-10$ Hz to 10 Hz for the real experimental noise is far less than 10 Hz. We compare the final density matrix with the theoretical value to obtain the fidelity. As shown in Fig. 8, the fidelities are all more than 0.99.

Fig. 8

The robustness of static field distributions. The x axis is $\Delta \nu _j$, changing from $-10$ Hz to 10 Hz. The y axis represents the fidelities (in the normalization units)

Fig. 9

The robuseness of RF inhomogenezity. The x axis in the left figure is $\Delta A_j$, changing from $-10$ Hz to 10 Hz and the x axis in the right figure is $\Delta \phi _j$, changing from $-5^{\circ }$ to $5^{\circ }$. The y axis represents the fidelities (in the normalization units)

1.2 RF inhomogenezity

The radiofrequency (RF) pulses are applied on the x and y axis, the $A_j$ and $\phi _j$ are the amplitudes and phases of RF pulses. Similarly, we assume that the noise $\Delta A_j$ is changing from $-10$ Hz to 10Hz and the $\Delta \phi _j$ is changing from $-5^{\circ }$ to $5^{\circ }$. As shown in Fig. 9, the fidelities are all more than 0.996.

1.3 Decoherence

The final density at time $t=T$ is given by the Eq. 25. Considering the influence of decoherence, the time evolution of the spin system during a time step j is approximately given by the propagator:

$$\begin{aligned} U_j=\exp ^{-i\Delta t(H_0+A_j(\cos (\phi _j) \sigma _x+\sin (\phi _j) \sigma _y))}\cdot \exp ^{ -\frac{\Delta t}{T_2^1}\sigma _x}\otimes \exp ^{\frac{\Delta t}{T_2^2}\sigma _x}\otimes \exp ^{\frac{\Delta t}{T_2^3}\sigma _x}\otimes \exp ^{\frac{\Delta t}{T_2^4}\sigma _x} \end{aligned}$$

(26)

where $\sigma _x$ is $2\times 2$ Pauli operator, $T_2^i$ are the decoherence time of ith carbon atom as shown in Fig. 2. Finally, we can obtain the fidelities with the influence of decoherence in the Table 2 and the result proves that our method are robust with decoherence with a length of 20 ms.

Appendix B: Fitting of the experimental spectrum

See Figs. 8, 9, 10 and Table 2.

Fig. 10

The fitting of the spectrum of spin C4 in the final state. As shown in the Fig. 5f, we can not obtain the values of peaks directly. Through the method of fitting, we can obtain the values of peaks. The fitting results of $|000\rangle$, $|001\rangle$, $|010\rangle$ and $|011\rangle$ are 0.2824, 0.0123, $-\,0.0143$ and $-\,0.0118$

Table 2 The influence of decoherence