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A quantum shortcut to adiabaticity scheme based on physics-informed neural networks is proposed in this work. Compared with traditional shortcut to adiabaticity technology, our method innovatively integrates machine learning methods by employing parameterized physics-informed neural networks to solve parameterized differential equations. The neural networks serve as an approximating function for quantum adiabatic evolution processes, while incorporating parameter-dependent differential equations and various physical constraints as components of the loss function. Through networks training, we effectively simulate quantum system dynamics and derive the driving control field for population inversion. Numerical simulations show that the quantum system can achieve rapid population inversion within significantly reduced time while maintaining high fidelity and exceptional robustness against parameter fluctuations. The neural networks exhibit remarkable computational capabilities, particularly suitable for generating control functions in complex quantum systems. Compared with conventional counter-diabatic driving and transitionless quantum driving methods, this PINN-based framework not only achieves better control performance but also provides the improved practicality for experimental implementations. The success of this method demonstrates its promising applications in quantum control tasks, including but not limited to quantum state preparation, quantum gate optimization, and adiabatic quantum computing acceleration.
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Keywords:
- shortcuts to adiabaticity /
- deep learning /
- physics-Informed neural networks /
- differential equation
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图 1 STAPINN原理设计图
Figure 1. STAPINN principle design diagram.
图 2 PINN方法仿真和Mathematica软件仿真无驱动布居转移情况对比图 (a) PINN仿真无驱动布居转移; (b) Mathematica软件中无驱动布居转移的仿真. 参数选取为: $ {t_0} = $$ 5 \times {10^{ - 12}} {\text{ s}}, \;{t_{\text{f}}} = 2 \times {10^{ - 10}} {\text{ s}}, \;{\varOmega _0} = 10 {\text{ MHz}}, \;\delta = 600 {\text{ }} {\text{MHz}} $
Figure 2. Comparison between PINN method simulation and Mathematica software simulation of undriven population transfer: (a) PINN simulated undriven population transfer; (b) simulation of undriven population transfer in Mathematica software. The parameters are selected as: $ {t_0} = 5 \times $$ {10^{ - 12}} {\text{ s}}, \;{t_{\text{f}}} = 2 \times {10^{ - 10}} {\text{ s}}, \;{\varOmega _0} = 10 {\text{ MHz}}, \;\delta = 600 {\text{ MHz}} $.
图 3 不同训练次数下的损失函数曲线图. 参数选取为$ {t_0} = 5 \times {10^{ - 12}} {\text{ s}}$, $ {t_{\text{f}}} = 2 \times {10^{ - 10}} {\text{ s}} $, $ {\varOmega _0} = 10 {\text{ MHz}}$, $\delta = $$ 600 {\text{ MHz}} $
Figure 3. Loss function curves under different training times. The parameters are selected as: $ {t_0} = 5 \times {10^{ - 12}} {\text{ s}}, \;{t_{\text{f}}} = 2 \times $$ {10^{ - 10}} {\text{ s}}$, $ {\varOmega _0} = 10 {\text{ MHz}}, \;\delta = 600 {\text{ MHz}} $.
图 4 STAPINN方法和Mathematica软件仿真加驱动的布居转移情况对比 (a) STAPINN仿真加驱动的布居转移情况; (b) Mathematica仿真加驱动的布居转移情况. 参数选取为: ${t_0} = 5 \times {10^{ - 12}} {\text{ s}} $, $ {t_{\text{f}}} = 2 \times {10^{ - 10}} {\text{ s}} $, ${\varOmega _0} = 10 {\text{ MHz}} $, $\delta = 600 {\text{ MHz}} $
Figure 4. Comparison of driver population transfer after STAPINN method and Mathematica software simulation and optimization: (a) STAPINN simulation optimization drives population transfer; (b) population transfer driven by Mathematica simulation optimization. The parameters are selected as: $ {t_0} = 5 \times {10^{ - 12}} {\text{ s}}$, ${t_{\text{f}}} = 2 \times {10^{ - 10}} {\text{ s}} $, ${\varOmega _0} = $$ 10 {\text{ MHz}}$, $\delta = 600 {\text{ MHz}} $.
图 5 (a)多项式展开的驱动函数和傅里叶级数展开的驱动函数对比图, 黄色曲线为STAPINN方法找到的驱动曲线, 蓝色曲线为多项式拟合的驱动曲线, 绿色曲线为傅里叶级数拟合的驱动曲线; (b)利用Mathematica软件模拟此驱动下的布居转移情况, 蓝色和褐色实线为傅里叶级数展开的驱动函数对应的布居数反转曲线, 红色和绿色虚线为多项式展开的驱动函数对应的布居反转曲线. 参数选取为: $ {t_0} = 5 \times $$ {10^{ - 12}} {\text{ s}} $, ${t_{\text{f}}} = 2 \times {10^{ - 10}} {\text{ s}} $, ${\varOmega _0} = 10 {\text{ MHz}} $, $\delta = 600 {\text{ MHz}} $
Figure 5. (a) Comparison between the driver function of polynomial expansion and the driver function of Fourier series expansion. The yellow curve is the optimized driver curve found by STAPINN method, the blue curve is the driver curve of polynomial fitting expansion, and the green curve is the driver curve of Fourier series fitting expansion; (b) using Mathematica software to simulate the population transfer under this drive, the blue and brown solid lines are the population inversion curves corresponding to the driver function of Fourier series expansion, and the red and green dashed lines are the population inversion curves corresponding to the driver function of polynomial expansion. The parameters are selected as: $ {t_0} = 5 \times {10^{ - 12}} {\text{ s}} $, ${t_{\text{f}}} = 2 \times $$ {10^{ - 10}} {\text{ s}} $, ${\varOmega _0} = 10 {\text{ MHz}} $, $\delta = 600 {\text{ MHz}} $.
图 6 (a)不同训练次数下的驱动曲线图; (b)不同训练次数下的驱动实现的布居转移情况对比图. 参数选取为: $ {t_0} = 5 \times $$ {10^{ - 12}} {\text{ s}}$, ${t_{\text{f}}} = 2 \times {10^{ - 10}} {\text{ s}} $, ${\varOmega _0} = 10 {\text{ MHz}} $, $\delta = 600 {\text{ MHz}} $
Figure 6. (a) Drive curves under different training times; (b) comparison diagram of population transfer of the drive under different training times. Set the parameter to: $ {t_0} = 5 \times $$ {10^{ - 12}} {\text{ s}}$, ${t_{\text{f}}} = 2 \times {10^{ - 10}} {\text{ s}} $, ${\varOmega _0} = 10 {\text{ MHz}} $, $\delta = 600{\text{ }} {\text{ MHz}} $
图 7 保真度随参数$ {\varOmega _0} $和$ \delta $的变化
Figure 7. Variation of fidelity with parameter $ {\varOmega _0} $and $ \delta $.
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