Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2001.04121 · Physical review. B./Physical review. B

Automatic differentiation of dominant eigensolver and its applications in quantum physics

Hao Xie, Jin-Guo Liu, Lei Wang

Published Jun 12, 2020• 33 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.101.245139

Related papers

Quantum computing fidelity susceptibility using automatic differentiation

Olivia Di Matteo, Richard M. Woloshyn

Automatic differentiation is an invaluable feature of machine learning and quantum machine learning software libraries. In this work, it is shown how quantum automatic differentiation can be used to solve the condensed-matter problem of computing fidelity susceptibility, a quantity whose value may be indicative of a phase transition in a system. Results are presented using simulations including hardware noise for small instances of the transverse-field Ising model, and a number of optimizations that can be applied are highlighted. Error mitigation (zero-noise extrapolation) is applied within the autodifferentiation framework to a number of gradient values required for the computation of fidelity susceptibility and a related quantity, the second derivative of the energy. Such computations are found to be highly sensitive to the additional statistical noise incurred by the error mitigation method.

arXiv:2207.06526•Physical Review A•Nov 28, 2022•5 citations

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

Chu Guo, Dario Poletti

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.

arXiv:2003.04295•Physical review. E•Jan 15, 2021•24 citations

Automatic differentiable numerical renormalization group

Jonas B. Rigo, Andrew K. Mitchell

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.

arXiv:2108.09575•Physical Review Research•Mar 25, 2022•14 citations

Implicit differentiation of variational quantum algorithms

Shahnawaz Ahmed, Nathan Killoran, Juan Carrasquilla

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.

arXiv:2211.13765•Nov 24, 2022•1 citation

Challenges with Implementing Differentiable Quantum Dynamics

Sri Hari Krishna Narayanan, Michael A. Perlin, Robert J. Lewis-Swan +4 more

Differentiable quantum dynamics require automatic differentiation of a complex-valued initial value problem, which numerically integrates a system of ordinary differential equations from a specified initial condition, as well as the eigendecomposition of a matrix. This work is a survey of existing differentiable programming frameworks for these tasks, finding that no framework natively supports our application requirements fully. We therefore demonstrate a need for broader support of complex-valued, differentiable numerical integration in scientific computing libraries. We further demonstrate that the derivatives of our quantum dynamics application can be computed through a combination of differentiable programming frameworks and handcoding.

arXiv:2406.06361•Society for Industrial and Applied Mathematics eBooks•Jan 1, 2026•0 citations

Differentiable Programming Tensor Networks

Haijun Liao, Jin-Guo Liu, Lei Wang +1 more

Differentiable programming is a fresh programming paradigm which composes parameterized algorithmic components and optimizes them using gradient search.The concept emerges from deep learning but is not limited to training neural networks.We present the theory and practice of programming tensor network algorithms in a fully differentiable way.By formulating the tensor network algorithm as a computation graph, one can compute higher-order derivatives of the program accurately and efficiently using automatic differentiation.We present essential techniques to differentiate through the tensor networks contraction algorithms, including numerical stable differentiation for tensor decompositions and efficient backpropagation through fixed-point iterations.As a demonstration, we compute the specific heat of the Ising model directly by taking the second-order derivative of the free energy obtained in the tensor renormalization group calculation.Next, we perform gradient-based variational optimization of infinite projected entangled pair states for the quantum antiferromagnetic Heisenberg model and obtain state-ofthe-art variational energy and magnetization with moderate efforts.Differentiable programming removes laborious human efforts in deriving and implementing analytical gradients for tensor network programs, which opens the door to more innovations in tensor network algorithms and applications.

arXiv:1903.09650•Physical Review X•Sep 5, 2019•251 citations
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.
Automatic differentiation is an invaluable feature of machine learning and quantum machine learning software libraries. In this work, it is shown how quantum automatic differentiation can be used to solve the condensed-matter problem of computing fidelity susceptibility, a quantity whose value may be indicative of a phase transition in a system. Results are presented using simulations including hardware noise for small instances of the transverse-field Ising model, and a number of optimizations that can be applied are highlighted. Error mitigation (zero-noise extrapolation) is applied within the autodifferentiation framework to a number of gradient values required for the computation of fidelity susceptibility and a related quantity, the second derivative of the energy. Such computations are found to be highly sensitive to the additional statistical noise incurred by the error mitigation method.
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.
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.
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.
Differentiable quantum dynamics require automatic differentiation of a complex-valued initial value problem, which numerically integrates a system of ordinary differential equations from a specified initial condition, as well as the eigendecomposition of a matrix. This work is a survey of existing differentiable programming frameworks for these tasks, finding that no framework natively supports our application requirements fully. We therefore demonstrate a need for broader support of complex-valued, differentiable numerical integration in scientific computing libraries. We further demonstrate that the derivatives of our quantum dynamics application can be computed through a combination of differentiable programming frameworks and handcoding.
Differentiable programming is a fresh programming paradigm which composes parameterized algorithmic components and optimizes them using gradient search.The concept emerges from deep learning but is not limited to training neural networks.We present the theory and practice of programming tensor network algorithms in a fully differentiable way.By formulating the tensor network algorithm as a computation graph, one can compute higher-order derivatives of the program accurately and efficiently using automatic differentiation.We present essential techniques to differentiate through the tensor networks contraction algorithms, including numerical stable differentiation for tensor decompositions and efficient backpropagation through fixed-point iterations.As a demonstration, we compute the specific heat of the Ising model directly by taking the second-order derivative of the free energy obtained in the tensor renormalization group calculation.Next, we perform gradient-based variational optimization of infinite projected entangled pair states for the quantum antiferromagnetic Heisenberg model and obtain state-ofthe-art variational energy and magnetization with moderate efforts.Differentiable programming removes laborious human efforts in deriving and implementing analytical gradients for tensor network programs, which opens the door to more innovations in tensor network algorithms and applications.