Neural-network Quantum States Research Articles

A simple linear algebra identity to optimize large-scale neural network quantum states

Neural-network architectures have been increasingly used to represent quantum many-body wave functions. These networks require a large number of variational parameters and are challenging to optimize using traditional methods, as gradient descent. Stochastic reconfiguration (SR) has been effective with a limited number of parameters, but becomes impractical beyond a few thousand parameters. Here, we leverage a simple linear algebra identity to show that SR can be employed even in the deep learning scenario. We demonstrate the effectiveness of our method by optimizing a Deep Transformer architecture with 3 × 105 parameters, achieving state-of-the-art ground-state energy in the J1–J2 Heisenberg model at J2/J1 = 0.5 on the 10 × 10 square lattice, a challenging benchmark in highly-frustrated magnetism. This work marks a significant step forward in the scalability and efficiency of SR for neural-network quantum states, making them a promising method to investigate unknown quantum phases of matter, where other methods struggle.

Neural-network quantum states for many-body physics

Neural-network quantum states for many-body physics

Second-order optimization strategies for neural network quantum states.

The Variational Monte Carlo (VMC) method has recently seen important advances through the use of neural network quantum states. While more and more sophisticated ansatze have been designed to tackle a wide variety of quantum many-body problems, modest progress has been made on the associated optimization algorithms. In this work, we revisit the Kronecker-Factored Approximate Curvature (KFAC), an optimizer that has been used extensively in a variety of simulations. We suggest improvements in the scaling and the direction of this optimizer and find that they substantially increase its performance at a negligible additional cost. We also reformulate the VMC approach in a game theory framework, to propose a novel optimizer based on decision geometry. We find that on a practical test case for continuous systems, this new optimizer consistently outperforms any of the KFAC improvements in terms of stability, accuracy and speed of convergence. Beyond VMC, the versatility of this approach suggests that decision geometry could provide a solid foundation for accelerating a broad class of machine learning algorithms. This article is part of the theme issue 'The liminal position of Nuclear Physics: from hadrons to neutron stars'.

Improved optimization for the neural-network quantum states and tests on the chromium dimer.

The advent of Neural-network Quantum States (NQS) has significantly advanced wave function ansatz research, sparking a resurgence in orbital space variational Monte Carlo (VMC) exploration. This work introduces three algorithmic enhancements to reduce computational demands of VMC optimization using NQS: an adaptive learning rate algorithm, constrained optimization, and block optimization. We evaluate the refined algorithm on complex multireference bond stretches of H2O and N2 within the cc-pVDZ basis set and calculate the ground-state energy of the strongly correlated chromium dimer (Cr2) in the Ahlrichs SV basis set. Our results achieve superior accuracy compared to coupled cluster theory at a relatively modest CPU cost. This work demonstrates how to enhance optimization efficiency and robustness using these strategies, opening a new path to optimize large-scale restricted Boltzmann machine-based NQS more effectively and marking a substantial advancement in NQS's practical quantum chemistry applications.

Liouville-space neural network representation of density matrices

Neural network quantum states such as ansatz wave functions have shown a great deal of promise for finding the ground state of spin models. Recently, work has focused on extending this idea to mixed states for simulating the dynamics of open systems. Most approaches so far have used a purification ansatz where a copy of the system Hilbert space is added, which when traced out gives the correct density matrix. Here we instead present an extension of the restricted Boltzmann machine which directly represents the density matrix in Liouville space. This allows the compact representation of states which appear in mean-field theory. We benchmark our approach on two different versions of the dissipative transverse-field Ising model, which show our ansatz is able to compete with other state-of-the-art approaches. Published by the American Physical Society 2024

Efficiency of neural-network state representations of one-dimensional quantum spin systems

Neural-network state representations of quantum many-body systems are attracting great attention and more rigorous quantitative analysis of their expressibility and complexity is warranted. Our analysis of the restricted Boltzmann machine (RBM) state representation of one-dimensional (1D) quantum spin systems provides new insight into their computational complexity. We define a class of long-range-fast-decay (LRFD) RBM states with quantifiable upper bounds on truncation errors and provide numerical evidence for a large class of 1D quantum systems that may be approximated by LRFD RBMs of at most polynomial complexities. These results lead us to conjecture that the ground states of a wide range of quantum systems may be exactly represented by LRFD RBMs or a variant of them, even in cases where other state representations become less efficient. At last, we provide the relations between multiple typical state manifolds. Our work proposes a paradigm for doing complexity analysis for generic long-range RBMs which naturally yields a further classification of this manifold. This paradigm and our characterization of their nonlocal structures may pave the way for understanding the natural measure of complexity for quantum many-body states described by RBMs and are generalizable for higher-dimensional systems and deep neural-network quantum states. Published by the American Physical Society 2024

Neural-network quantum states for ultra-cold Fermi gases

Ultra-cold Fermi gases exhibit a rich array of quantum mechanical properties, including the transition from a fermionic superfluid Bardeen-Cooper-Schrieffer (BCS) state to a bosonic superfluid Bose-Einstein condensate (BEC). While these properties can be precisely probed experimentally, accurately describing them poses significant theoretical challenges due to strong pairing correlations and the non-perturbative nature of particle interactions. In this work, we introduce a Pfaffian-Jastrow neural-network quantum state featuring a message-passing architecture to efficiently capture pairing and backflow correlations. We benchmark our approach on existing Slater-Jastrow frameworks and state-of-the-art diffusion Monte Carlo methods, demonstrating a performance advantage and the scalability of our scheme. We show that transfer learning stabilizes the training process in the presence of strong, short-ranged interactions, and allows for an effective exploration of the BCS-BEC crossover region. Our findings highlight the potential of neural-network quantum states as a promising strategy for investigating ultra-cold Fermi gases.

Benchmarking discrete truncated Wigner approximation and neural network quantum states with the exact dynamics in a Rydberg atomic chain

We benchmark the discrete truncated Wigner approximation (DTWA) and Neural quantum states (NQS) based on restricted Boltzmann-like machines with the exact excitation and correlation dynamics in a chain of ten Rydberg atoms. The initial state is where all atoms are in their electronic ground state. We characterize the excitation dynamics using the maximum and time-averaged number of Rydberg excitations. DTWA results are different from the exact dynamics for large Rydberg-Rydberg interactions. In contrast, by increasing the number of hidden spins, the NQS can be improved but still limited to short-time dynamics. Interestingly, irrespective of interaction strengths, the time-averaged number of excitations obtained using NQS is in excellent agreement with the exact results. Concerning the calculation of quantum correlations, for instance, second-order bipartite and average two-site Rényi entropies, NQS looks more promising. Finally, we discuss the existence of a power law scaling for the initial growth of average two-site Rényi entropy.

Supervised training of neural-network quantum states for the next-nearest neighbor Ising model

Different neural network architectures can be unsupervisedly or supervisedly trained to represent quantum states. We explore and compare different strategies for the supervised training of feed forward neural network quantum states. We empirically and comparatively evaluate the performance of feed forward neural network quantum states in different phases of matter for variants of the architecture, for different hyper-parameters, and for two different loss functions, to which we refer as mean-squared error and overlap, respectively. We consider the next-nearest neighbor Ising model for the diversity of its phases and focus on its paramagnetic, ferromagnetic, and pair-antiferromagnetic phases. We observe that the overlap loss function allows better training of the model across all phases, provided a rescaling of the neural network.

Machine learning one-dimensional spinless trapped fermionic systems with neural-network quantum states

Machine learning one-dimensional spinless trapped fermionic systems with neural-network quantum states

Neural network quantum states analysis of the Shastry-Sutherland model

We utilize neural network quantum states (NQS) to investigate the ground state properties of the Heisenberg model on a Shastry-Sutherland lattice using the variational Monte Carlo method. We show that already relatively simple NQSs can be used to approximate the ground state of this model in its different phases and regimes. We first compare several types of NQSs with each other on small lattices and benchmark their variational energies against the exact diagonalization results. We argue that when precision, generality, and computational costs are taken into account, a good choice for addressing larger systems is a shallow restricted Boltzmann machine NQS. We then show that such NQS can describe the main phases of the model in zero magnetic field. Moreover, NQS based on a restricted Boltzmann machine correctly describes the intriguing plateaus forming in magnetization of the model as a function of increasing magnetic field.

Scalable imaginary time evolution with neural network quantum states

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_2J1−J2, 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.

A Nonstochastic Optimization Algorithm for Neural-Network Quantum States.

Neural-network quantum states (NQS) employ artificial neural networks to encode many-body wave functions in a second quantization through variational Monte Carlo (VMC). They have recently been applied to accurately describe electronic wave functions of molecules and have shown the challenges in efficiency compared with traditional quantum chemistry methods. Here, we introduce a general nonstochastic optimization algorithm for NQS in chemical systems, which deterministically generates a selected set of important configurations simultaneously with energy evaluation of NQS. This method bypasses the need for Markov-chain Monte Carlo within the VMC framework, thereby accelerating the entire optimization process. Furthermore, this newly developed nonstochastic optimization algorithm for NQS offers comparable or superior accuracy compared to its stochastic counterpart and ensures more stable convergence. The application of this model to test molecules exhibiting strong electron correlations provides further insight into the performance of NQS in chemical systems and opens avenues for future enhancements.

Ground state properties of quantum skyrmions described by neural network quantum states

Ground state properties of quantum skyrmions described by neural network quantum states

Lee-Yang theory of quantum phase transitions with neural network quantum states

Lee-Yang theory of quantum phase transitions with neural network quantum states

Dilute neutron star matter from neural-network quantum states

Dilute neutron star matter from neural-network quantum states

Infinite neural network quantum states: entanglement and training dynamics

We study infinite limits of neural network quantum states (-NNQS), which exhibit representation power through ensemble statistics, and also tractable gradient descent dynamics. Ensemble averages of entanglement entropies are expressed in terms of neural network correlators, and architectures that exhibit volume-law entanglement are presented. The analytic calculations of entanglement entropy bound are tractable because the ensemble statistics are simplified in the Gaussian process limit. A general framework is developed for studying the gradient descent dynamics of neural network quantum states (NNQS), using a quantum state neural tangent kernel (QS-NTK). For -NNQS the training dynamics is simplified, since the QS-NTK becomes deterministic and constant. An analytic solution is derived for quantum state supervised learning, which allows an -NNQS to recover any target wavefunction. Numerical experiments on finite and infinite NNQS in the transverse field Ising model and Fermi Hubbard model demonstrate excellent agreement with theory. -NNQS opens up new opportunities for studying entanglement and training dynamics in other physics applications, such as in finding ground states.

Ground state search by local and sequential updates of neural network quantum states

Neural network quantum states are a promising tool to analyze complex quantum systems given their representative power. It can however be difficult to optimize efficiently and effectively the parameters of this type of ansatz. Here we propose a local optimization procedure which, when integrated with stochastic reconfiguration, outperforms previously used global optimization approaches. Specifically, we analyze both the ground state energy and the correlations for the nonintegrable tilted Ising model with restricted Boltzmann machines. We find that sequential local updates can lead to faster convergence to states which have energy and correlations closer to those of the ground state, depending on the size of the portion of the neural network which is locally updated. To show the generality of the approach we apply it to both 1D and 2D nonintegrable spin systems.

Amplitude Ratios and Neural Network Quantum States

Neural Network Quantum States (NQS) represent quantum wavefunctions by artificial neural networks. Here we study the wavefunction access provided by NQS defined in [Science, 355, 6325, pp. 602-606 (2017)] and relate it to results from distribution testing. This leads to improved distribution testing algorithms for such NQS. It also motivates an independent definition of a wavefunction access model: the amplitude ratio access. We compare it to sample and sample and query access models, previously considered in the study of dequantization of quantum algorithms. First, we show that the amplitude ratio access is strictly stronger than sample access. Second, we argue that the amplitude ratio access is strictly weaker than sample and query access, but also show that it retains many of its simulation capabilities. Interestingly, we only show such separation under computational assumptions. Lastly, we use the connection to distribution testing algorithms to produce an NQS with just three nodes that does not encode a valid wavefunction and cannot be sampled from.

Solving the nuclear pairing model with neural network quantum states.

We present a variational Monte Carlo method that solves the nuclear many-body problem in the occupation number formalism exploiting an artificial neural network representation of the ground-state wave function. A memory-efficient version of the stochastic reconfiguration algorithm is developed to train the network by minimizing the expectation value of the Hamiltonian. We benchmark this approach against widely used nuclear many-body methods by solving a model used to describe pairing in nuclei for different types of interaction and different values of the interaction strength. Despite its polynomial computational cost, our method outperforms coupled-cluster and provides energies that are in excellent agreement with the numerically exact full configuration-interaction values.