Scheme for automatic differentiation of complex loss functions with applications in quantum physics

The past few years have seen a significant transfer of tools from machine learning to solve quantum physics problems. Automatic differentiation is one standard algorithm used to efficiently compute gradients of loss functions for generic neural networks. In this work we show how to extend automatic differentiation to the case of complex loss function in a way that can be readily implemented in existing frameworks and which is compatible with the common case of real loss functions. We then combine this tool with matrix product states and apply it to compute the ground state and the steady state of a close and an open quantum system. Compared to the traditional density matrix renormalization group algorithm, complex automatic differentiation allows both straightforward GPU accelerations as well as generalizations to different types of tensor and neural networks.

Figure 1

(a) Forward evaluation of a function $F(x_1,x_2)$ of two variables $x_1$ and $x_2$. (b) Backward evaluation of the gradient of the function $F(x_1,x_2)$.

Figure 2

Loss function values [see Eq. (22)] as a function of $n$, which denotes the ratio of iterations versus total number of iterations. We considered a long-range XXZ chain with $L=50, h=0.1, \alpha =1$, where the interaction strength $\mathrm{\Delta }=0$ for (a) and $\mathrm{\Delta }=1$ for (b). The blue and yellow dashed lines represent AD simulations results for $D=10$ and $D=20,$ respectively, and the black solid line represents the DMRG results with $D=30$. We have used 2000 iteration for AD and plotted the last 90 percent loss values.

Figure 3

Loss function values [see Eq. (30)] as a function of $n$, which denotes the ratio of iterations versus total number of iterations. We consider a boundary driven XXZ chain, where we have used $L=10$ and $\gamma =0$ in (a), $L=10$ and $\gamma =0.5$ in (b), $L=20$ and $\gamma =0$ in (c), and $L=20$ and $\gamma =0.5$ in (d). The other parameters used for boundary driven XXZ chain are $J=1, \mathrm{\Delta }=0.5, {\overline{n}}_1=0.6, {\overline{n}}_L=0.4, {\mathrm{\Lambda }}_1={\mathrm{\Lambda }}_L=1$. The blue and yellow dashed lines represent AD simulation results for $D=10$ and $D=20,$ respectively, while the blue and yellow solid lines represent DMRG results for $D=10$ and $D=20,$ respectively. We have used $20000$ iteration for AD and 2000 sweeps for DMRG algorithm in this case. The last 90 percent loss values for AD and DMRG are plotted.