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.
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.
Tensor renormalization group (TRG) constitutes an important methodology for accurate simulations of strongly correlated lattice models. Facilitated by the automatic differentiation technique widely used in deep learning, we propose a uniform framework of differentiable TRG ($\ensuremath{\partial}\mathrm{TRG}$) that can be applied to improve various TRG methods, in an automatic fashion. $\ensuremath{\partial}\mathrm{TRG}$ systematically extends the essential concept of second renormalization [Phys. Rev. Lett. 103, 160601 (2009)] where the tensor environment is computed recursively in the backward iteration. Given the forward TRG process, $\ensuremath{\partial}\mathrm{TRG}$ automatically finds the gradient of local tensors through backpropagation, with which one can deeply ``train'' the tensor networks. We benchmark $\ensuremath{\partial}\mathrm{TRG}$ in solving the square-lattice Ising model, and we demonstrate its power by simulating one- and two-dimensional quantum systems at finite temperature. The global optimization as well as GPU acceleration renders $\ensuremath{\partial}\mathrm{TRG}$ a highly efficient and accurate many-body computation approach.
Several quantities important in condensed matter physics, quantum information, and quantum chemistry, as well as quantities required in meta-optimization of machine learning algorithms, can be expressed as gradients of implicitly defined functions of the parameters characterizing the system. Here, we show how to leverage implicit differentiation for gradient computation through variational quantum algorithms and explore applications in condensed matter physics, quantum machine learning, and quantum information. A function defined implicitly as the solution of a quantum algorithm, e.g., a variationally obtained ground- or steady-state, can be automatically differentiated using implicit differentiation while being agnostic to how the solution is computed. We apply this notion to the evaluation of physical quantities in condensed matter physics such as generalized susceptibilities studied through a variational quantum algorithm. Moreover, we develop two additional applications of implicit differentiation -- hyperparameter optimization in a quantum machine learning algorithm, and the variational construction of entangled quantum states based on a gradient-based maximization of a geometric measure of entanglement. Our work ties together several types of gradient calculations that can be computed using variational quantum circuits in a general way without relying on tedious analytic derivations, or approximate finite-difference methods.
Machine learning techniques have recently gained prominence in physics, yielding a host of new results and insights. One key concept is that of backpropagation, which computes the exact gradient of any output of a program with respect to any input. This is achieved efficiently within the differentiable programming paradigm, which utilizes automatic differentiation (AD) of each step of a computer program and the chain rule. A classic application is in training neural networks. Here, we apply this methodology instead to the numerical renormalization group (NRG), a powerful technique in computational quantum many-body physics. We demonstrate how derivatives of NRG outputs with respect to Hamiltonian parameters can be accurately and efficiently obtained. Physical properties can be calculated using this differentiable NRG scheme---for example, thermodynamic observables from derivatives of the free energy. Susceptibilities can be computed by adding source terms to the Hamiltonian, but still evaluated with AD at precisely zero field. As an outlook, we briefly discuss the derivatives of dynamical quantities and a possible route to the vertex.
We investigate the automatic differentiation of dominant eigensolver where only a small proportion of eigenvalues and corresponding eigenvectors are obtained. Back-propagation through the dominant eigensolver involves solving certain low-rank linear systems without direct access to the full spectrum of the problem. Furthermore, the backward pass can be conveniently differentiated again, which implies that in principle one can obtain arbitrarily higher-order derivatives of the dominant eigendecomposition process. These results allow for the construction of an efficient dominant eigensolver primitive, which has wide applications in quantum physics. As a demonstration, we compute second-order derivative of the ground-state energy and fidelity susceptibility of one-dimensional transverse-field Ising model through the exact diagonalization approach. We also calculate the ground-state energy of the same model in the thermodynamic limit by performing gradient-based optimization of uniform matrix product states. By programming these computational tasks in a fully differentiable way, one can efficiently handle the dominant eigendecomposition of very large matrices while still sharing various advantages of differentiable programming paradigm, notably, the generic nature of the implementation and free of tedious human efforts of deriving gradients analytically.
The representation of a quantum wave function as a neural network quantum state (NQS) provides a powerful variational ansatz for finding the ground states of many-body quantum systems. Nevertheless, due to the complex variational landscape, traditional methods often employ the computation of quantum geometric tensor, consequently complicating optimization techniques. Contributing to efforts aiming to formulate alternative methods, we introduce an approach that bypasses the computation of the metric tensor and instead relies exclusively on first-order gradient descent with Euclidean metric. This allows for the application of larger neural networks and the use of more standard optimization methods from other machine learning domains. Our approach leverages the principle of imaginary time evolution by constructing a target wave function derived from the Schrödinger equation, and then training the neural network to approximate this target. We make this method adaptive and stable by determining the optimal time step and keeping the target fixed until the energy of the NQS decreases. We demonstrate the benefits of our scheme via numerical experiments with 2D J_1-J_2 J 1 − J 2 , Heisenberg model, which showcase enhanced stability and energy accuracy in comparison to direct energy loss minimization. Importantly, our approach displays competitiveness with the well-established density matrix renormalization group method and NQS optimization with stochastic reconfiguration.
This paper introduces a novel automatic differentiation algorithm based on quantum neural networks (QNNs). Traditional automatic differentiation methods often rely on manual adjustment of parameters, lacking inherent adaptability. This work proposes a new approach utilizing the unique properties of QNNs to automate gradient calculation, ultimately aiming to enhance the training efficiency of neural networks. The core mechanism involves incorporating quantum circuits within the network architecture to enable efficient and adaptive gradient computation. We detail the algorithm's design, explain the underlying quantum principles employed, and present preliminary experimental results demonstrating its improved performance compared to existing methods.