Quantum synchronization on the IBM Q system

We report an experimental demonstration of quantum synchronization. This is achieved by performing a digital simulation of a single spin-1 limit-cycle oscillator on the quantum computers of the IBM Q system. Applying an external signal to the oscillator, we verify typical features of quantum synchronization and demonstrate an interference-based quantum synchronization blockade. Our results show that state-of-the-art noisy intermediate-scale quantum computers are powerful enough to implement realistic dissipative quantum systems. Finally, we discuss limitations of current quantum hardware, and we define the requirements necessary to investigate more complex problems.

Figure 1

(a) Energy level diagram of a spin-1 system hosting a limit-cycle oscillator. The limit cycle is stabilized by dissipative transitions toward the state $\left|0\right.〉$ at rates ${\mathrm{\Gamma }}_{\pm 1,0}$ and is subjected to an external signal that drives transitions $j_{k,l}$ between the spin-1 states. (b) Quantum-circuit implementation of the synchronization dynamics for a time step $dt$, obtained by a Suzuki-Trotter decomposition. The gates shown in white correspond to the free evolution of the oscillator while the other circuit components correspond to the transitions of the same color in (a). Here, $R_z\left(\theta \right)=U_3(0,0,\theta )$ is the phase gate, and the signals $j_{0,\pm 1}$ are mapped onto controlled gates $U_{\pm 1,0}\left(t\right)=U_3(-2\varepsilon \left|j_{0,\pm 1}\right|t,arg\left(j_{0,\pm 1}\right)-\frac{3\pi }{2},-arg\left(j_{0,\pm 1}\right)-\frac{\pi }{2})$. The $U_3(\theta ,\phi ,\lambda )$ gate is a basis gate of the IBM quantum computer, defined in Eq. (3). Open (solid) circles indicate a controlled gate conditioned on the control qubit $q_k$ being in ${\left|0\right.〉}_{q_k}$ (${\left|1\right.〉}_{q_k}$); see Eq. (2). (c) Trotter step of the $j_{1,-1}$ signal using three controlled $U_3(\theta ,\phi ,\lambda )$ gates, where $\tau =arg\left(j_{-1,1}\right)$. (d) Implementation of relaxation dynamics with ${\theta }_k\left(t\right)=2arcsin\left(\sqrt{{\mathrm{\Gamma }}_{k,0}t}\right)$. Note that the two dissipative steps in (b) could also be applied sequentially to a single ancillary qubit.

Figure 2

(a) Time evolution of the state $\left|0\right.〉$ under the semiclassical signal components $j_{0,\pm 1}$ (markers) on a noisy intermediate-scale quantum device (NISQ) and ideal noise-free time evolution (lines). The upper plot is obtained for controlled $U_{\pm 1,0}$ gates as shown in the circuit diagram in Fig. 1 for $\varepsilon dt=0.1$, $j_{-1,0}=0.5\times e^{-\pi i/6}$, $j_{1,0}=1\times e^{5\pi i/6}$, $j_{-1,1}=0$, and $\mathrm{\Delta }/\varepsilon =0$. In the lower plot, uncontrolled single-qubit $U_{\pm 1,0}$ gates were used with the same parameters. The data have been collected on the ibmqx2 computer on qubits $q_0=4$ and $q_1=2$. (b) Dissipative stabilization of the limit-cycle state $\left|0\right.〉$ if no signal is applied, $j_{\pm 1,0}=j_{-1,1}=0$, on a NISQ device (markers) and theoretical expectation taking into account noise (lines). The dynamics of the coherences in the inset is illustrated by the thin connecting lines. The gray circle defines the level of the noise due to the dissipative limit-cycle stabilization. Parameters: ${\mathrm{\Gamma }}_{1,0}dt=0.2$, $\mathrm{\Delta }/{\mathrm{\Gamma }}_{1,0}=0$, $\varepsilon /{\mathrm{\Gamma }}_{1,0}=0.25$, and ${\mathrm{\Gamma }}_{-1,0}/{\mathrm{\Gamma }}_{1,0}=1$. Data have been collected on ibmqx2 on qubits $q_0=2$ and $q_1=2$ in sequential runs. (c) Demonstration of the onset of synchronization if both the signal and the dissipative stabilization of the limit-cycle state are switched on, $j_{-1,0}=1\times e^{2\pi i/6}$, $j_{1,0}=2\times e^{-\pi i/6}$, and $j_{-1,1}=0$. The signal builds up coherences beyond the noise level of the limit cycle. The data have been averaged over three runs, each having 8192 repetitions per circuit. The corresponding error bars are smaller than the plot markers.

Figure 3

(a) Phase distribution $S\left(\phi \right)$ of the spin-1 limit-cycle oscillator as a function of the detuning $\mathrm{\Delta }$ between its natural frequency and the signal frequency after $N=3$ time steps. The dashed black line indicates the theoretical expectation of the position of the maximum of $S\left(\phi \right)$, obtained by combining Eqs. (1) and (4). Parameters are ${\mathrm{\Gamma }}_{-1,0}/{\mathrm{\Gamma }}_{1,0}=1$, ${\mathrm{\Gamma }}_{1,0}dt=0.2$, $\varepsilon /{\mathrm{\Gamma }}_{1,0}=0.25$, $j_{-1,0}=2\times e^{2\pi i/6}$, $j_{1,0}=2\times e^{-\pi i/6}$, and $j_{-1,1}=0$. (b) Populations and coherences as a function of the signal strength $\varepsilon$ for $\mathrm{\Delta }/{\mathrm{\Gamma }}_{1,0}=0$. The gray background indicates the noise level of the coherences introduced in Fig. 2. (c) Upper panel: Phase of the coherences if the overall phase $\chi$ of the signals, $j_{\pm 1,0}{e}^{i\chi }$, is varied for $\mathrm{\Delta }/{\mathrm{\Gamma }}_{1,0}=0$ and $\varepsilon /{\mathrm{\Gamma }}_{1,0}=0.25$. Lower panel: Demonstration of an interference-based quantum synchronization blockade if the phase of only one of the signals is varied, $j_{-1,0}=e^{i\chi }\times 2\times e^{-2\pi i/6}$ and $j_{1,0}=2\times e^{-2\pi i/6}=\mathrm{const}$. Data points are the result obtained on a NISQ device, the solid line corresponds to a simulation taking into account noise, and the dashed line describes the theory result. Parameters are ${\mathrm{\Gamma }}_{-1,0}/{\mathrm{\Gamma }}_{1,0}=1.25$, ${\mathrm{\Gamma }}_{1,0}dt=0.2$, $\varepsilon /{\mathrm{\Gamma }}_{1,0}=0.25$, and $j_{-1,1}=0$. All data of this figure have been collected on the ibmqx2 computer on qubits $q_0=2$ and $q_1=2$ in sequential runs.

Figure 4

Time evolution of the coherences ${\widehat{\rho }}_{1,X}$, ${\widehat{\rho }}_{-1,X}$, and ${\widehat{\rho }}_{0,X}$ for the parameters of Fig. 2. (a) Only dissipative stabilization of the limit cycle is switched on, corresponding to Fig. 2. (b) An additional external signal is applied, corresponding to Fig. 2. The gray circle denotes the noise level of the limit-cycle state.

Figure 5

Time evolution of the populations (top row) and the coherences (bottom row) if only the dissipative stabilization mechanism of the limit cycle is switched on. The initial state is (a) $\left|+1\right.〉$, (b) $\left|-1\right.〉$, and (c) $\left|X\right.〉$. Parameters are the same as in Fig. 2.