Sparse autoregressive neural networks for classical spin systems

I. Biazzo; D. Wu; G. Carleo 

Efficient sampling and approximation of Boltzmann distributions involving large sets of binary variables, or spins, are pivotal in diverse scientific fields even beyond physics. Recent advances in generative neural networks have significantly impacted this domain. However, these neural networks are often treated as black boxes, with architectures primarily influenced by data-driven problems in computational science. Addressing this gap, we introduce a novel autoregressive neural network architecture named TwoBo, specifically designed for sparse two-body interacting spin systems. We directly incorporate the Boltzmann distribution into its architecture and parameters, resulting in enhanced convergence speed, superior free energy accuracy, and reduced trainable parameters. We perform numerical experiments on disordered, frustrated systems with more than 1000 spins on grids and random graphs, and demonstrate its advantages compared to previous autoregressive and recurrent architectures. Our findings validate a physically informed approach and suggest potential extensions to multivalued variables and many-body interaction systems, paving the way for broader applications in scientific research.

Machine Learning-Science And Technology


Vol. 5 , num. 2, p. 025074.

DOI : 10.1088/2632-2153/ad5783

Empirical Sample Complexity of Neural Network Mixed State Reconstruction

H. Zhao; G. Carleo; F. Vicentini 

Quantum state reconstruction using Neural Quantum States has been proposed as a viable tool to reduce quantum shot complexity in practical applications, and its advantage over competing techniques has been shown in numerical experiments focusing mainly on the noiseless case. In this work, we numerically investigate the performance of different quantum state reconstruction techniques for mixed states: the finite-temperature Ising model. We show how to systematically reduce the quantum resource requirement of the algorithms by applying variance reduction techniques. Then, we compare the two leading neural quantum state encodings of the state, namely, the Neural Density Operator and the positive operator-valued measurement representation, and illustrate their different performance as the mixedness of the target state varies. We find that certain encodings are more efficient in different regimes of mixedness and point out the need for designing more efficient encodings in terms of both classical and quantum resources.



Vol. 8 , p. 01840.

Neural-network quantum states for ultra-cold Fermi gases

J. Kim; G. M. Pescia; B. Fore; J. Nys; G. Carleo et al. 

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.|The theoretical description of ultra-cold Fermi gases is challenging due to the presence of strong, short-ranged interactions. This work introduces a Pfaffian-Jastrow neural-network quantum state that outperforms existing Slater-Jastrow frameworks and diffusion Monte Carlo methods.

Communications Physics


Vol. 7 , num. 1, p. 148.

DOI : 10.1038/s42005-024-01613-w

Adaptive projected variational quantum dynamics

D. Linteau; S. Barison; N. H. Lindner; G. Carleo 

We propose an adaptive quantum algorithm to prepare accurate variational time evolved wave functions. The method is based on the projected variational quantum dynamics (pVQD) algorithm, that performs a global optimization with linear scaling in the number of variational parameters. Instead of fixing a variational ansatz at the beginning of the simulation, the circuit is grown systematically during the time evolution. Moreover, the adaptive step does not require auxiliary qubits and the gate search can be performed in parallel on different quantum devices. We apply the algorithm, named adaptive pVQD, to the simulation of driven spin models and fermionic systems, where it shows an advantage when compared to both Trotterized circuits and nonadaptive variational methods. Finally, we use the shallower circuits prepared using the adaptive pVQD algorithm to obtain more accurate measurements of physical properties of quantum systems on hardware.

Physical Review Research


Vol. 6 , num. 2, p. 023130.

DOI : 10.1103/PhysRevResearch.6.023130

Physical and unphysical regimes of self-consistent many-body perturbation theory

K. Van Houcke; E. Kozik; R. Rossi; Y. Deng; F. Werner 

In the standard framework of self-consistent many-body perturbation theory, the skeleton series for the self-energy is truncated at a finite order N and plugged into the Dyson equation, which is then solved for the propagator G(N). We consider two examples of fermionic models, the Hubbard atom at half filling and its zero space-time dimensional simplified version. First, we show that G(N) converges when N -> infinity to a limit G(infinity), which coincides with the exact physical propagator G(exact) at small enough coupling, while G(infinity )not equal G(exact) at strong coupling. This follows from the findings of [Phys. Rev. Lett. 114, 156402 (2015)] and an additional subtle mathematical mechanism elucidated here. Second, we demonstrate that it is possible to discriminate between the G(infinity )= G(exac) and G(infinity )not equal G(exact) regimes thanks to a criterion which does not require the knowledge of G(exact), as proposed in [2].

Scipost Physics


Vol. 16 , num. 5, p. 133.

DOI : 10.21468/SciPostPhys.16.5.133

Overhead-constrained circuit knitting for variational quantum dynamics

G. Gentinetta; F. Metz; G. Carleo 

Simulating the dynamics of large quantum systems is a formidable yet vital pursuit for obtaining a deeper understanding of quantum mechanical phenomena. While quantum computers hold great promise for speeding up such simulations, their practical application remains hindered by limited scale and pervasive noise. In this work, we propose an approach that addresses these challenges by employing circuit knitting to partition a large quantum system into smaller subsystems that can each be simulated on a separate device. The evolution of the system is governed by the projected variational quantum dynamics (PVQD) algorithm, supplemented with constraints on the parameters of the variational quantum circuit, ensuring that the sampling overhead imposed by the circuit knitting scheme remains controllable. We test our method on quantum spin systems with multiple weakly entangled blocks each consisting of strongly correlated spins, where we are able to accurately simulate the dynamics while keeping the sampling overhead manageable. Further, we show that the same method can be used to reduce the circuit depth by cutting long-ranged gates.



Vol. 8 , p. 07857.

Variational quantum time evolution without the quantum geometric tensor

J. Gacon; J. Nys; R. Rossi; S. Woerner; G. Carleo 

Realand imaginary -time quantum state evolutions are crucial in physics and chemistry for exploring quantum dynamics, preparing ground states, and computing thermodynamic observables. On near -term devices, variational quantum time evolution is a promising candidate for these tasks, as the required circuit model can be tailored to the available devices’ capabilities. Due to the evaluation of the quantum geometric tensor (QGT), however, this approach quickly becomes infeasible for relevant system sizes. Here, we propose a dual formulation for variational time evolution, which replaces the calculation of the QGT by solving a fidelity -based optimization to compute updates to the dynamics in each time step. We demonstrate our algorithm for the time evolution of the Heisenberg Hamiltonian and show that it accurately reproduces the system dynamics at a fraction of the cost of standard variational quantum time evolution algorithms.

Physical Review Research


Vol. 6 , num. 1, p. 013143.

DOI : 10.1103/PhysRevResearch.6.013143

Variational quantum algorithm for ergotropy estimation in quantum many-body batteries

D. T. Hoang; F. Metz; A. Thomasen; T. D. Anh-Tai; T. Busch et al. 

Quantum batteries are predicted to have the potential to outperform their classical counterparts and are therefore an important element in the development of quantum technologies. Of particular interest is the role of correlations in many-body quantum batteries and how these can affect the maximal work extraction, quantified by the ergotropy. In this paper we simulate the charging process and work extraction of many-body quantum batteries on noisy intermediate-scale quantum devices and devise the variational quantum ergotropy (VQErgo) algorithm, which finds the optimal unitary operation that maximizes work extraction from the battery. We test VQErgo by calculating the ergotropy of a many-body quantum battery undergoing transverse field Ising dynamics following a sudden quench. We investigate the battery for different system sizes and charging times and analyze the minimum number of ansatz circuit repetitions needed for the variational optimization using both ideal and noisy simulators. We also discuss how the growth of long-range correlations can hamper the accuracy of VQErgo in larger systems, requiring increased repetitions of the ansatz circuit to reduce error. Finally, we optimize part of the VQErgo algorithm and calculate the ergotropy on one of IBM’s quantum devices.

Physical Review Research


Vol. 6 , num. 1, p. 013038.

DOI : 10.1103/PhysRevResearch.6.013038

The complexity of quantum support vector machines

G. Gentinetta; A. Thomsen; D. Sutter; S. Woerner 

Quantum support vector machines employ quantum circuits to define the kernel function. It has been shown that this approach offers a provable exponential speedup compared to any known classical algorithm for certain data sets. The training of such models corresponds to solving a convex optimization problem either via its primal or dual formulation. Due to the probabilistic nature of quantum mechanics, the training algorithms are affected by statistical uncertainty, which has a major impact on their complexity. We show that the dual problem can be solved in O(M4.67/epsilon 2) quantum circuit evaluations, where M denotes the size of the data set and epsilon the solution accuracy compared to the ideal result from exact expectation values, which is only obtainable in theory. We prove under an empirically motivated assumption that the kernelized primal problem can alternatively be solved in O(min{M2/epsilon 6, 1/epsilon 10}) evaluations by employing a generalization of a known classical algorithm called PEGASOS. Accompanying empirical results demonstrate these analytical complexities to be essentially tight. In addition, we investigate a variational approximation to quantum support vector machines and show that their heuristic training achieves considerably better scaling in our experiments.



Vol. 8 , p. 1225.

DOI : 10.22331/q-2024-01-11-1225

Hybrid ground-state quantum algorithms based on neural Schrödinger forging

P. de Schoulepnikoff; O. Kiss; S. Vallecorsa; G. Carleo; M. Grossi 

Entanglement forging based variational algorithms leverage the bipartition of quantum systems for addressing ground-state problems. The primary limitation of these approaches lies in the exponential summation required over the numerous potential basis states, or bitstrings, when performing the Schmidt decomposition of the whole system. To overcome this challenge, we propose a method for entanglement forging employing generative neural networks to identify the most pertinent bitstrings, eliminating the need for the exponential sum. Through empirical demonstrations on systems of increasing complexity, we show that the proposed algorithm achieves comparable or superior performance compared to the existing standard implementation of entanglement forging. Moreover, by controlling the amount of required resources, this scheme can be applied to larger as well as non-permutation-invariant systems, where the latter constraint is associated with the Heisenberg forging procedure. We substantiate our findings through numerical simulations conducted on spin models exhibiting one-dimensional rings, two-dimensional triangular lattice topologies, and nuclear shell model configurations.

Physical Review Research


Vol. 6 , num. 2, p. 023021.

DOI : 10.1103/PhysRevResearch.6.023021

Scalable Quantum Algorithms for Noisy Quantum Computers

J. S. Gacon / Director(s) : G. Carleo; S. Woerner 

Quantum computing not only holds the potential to solve long-standing problems in quantum physics, but also to offer speed-ups across a broad spectrum of other fields. Access to a computational space that incorporates quantum effects, such as superposition and entanglement, enables the derivation of promising quantum algorithms for important tasks, including preparing the ground state of a quantum system or predicting its evolution over time. Successfully tackling these tasks promises insights into significant theoretical and technological questions, such as superconductivity and the design of new materials. The aim of quantum algorithms is to use a series of quantum operations in a quantum circuit to solve a problem beyond the reach of classical computers. However, the noise and limited scale of current quantum computers restricts these circuits to moderate sizes and depths. As a result, many prominent algorithms are currently infeasible to run for problem sizes of practical interest. In response, recent research focused on variational quantum algorithms, which allow the selection of circuits that act within a quantum device’s capabilities. Yet, these algorithms can require the execution of a large number of circuits, leading to prohibitively long computation times. This doctoral thesis develops two main techniques to reduce these quantum computational resource requirements, with the goal of scaling up application sizes on current quantum processors. The first approach is based on stochastic approximations of computationally costly quantities, such as quantum circuit gradients or the quantum geometric tensor (QGT). The second method takes a different perspective on the QGT, leading to a potentially more efficient description of time evolution on current quantum computers. Both techniques rely on maintaining available information and only computing necessary corrections, instead of re-computing possibly redundant data. The main focus of application for our algorithms is the simulation of quantum systems, broadly defined as including the preparation of ground and thermal states, and the real- and imaginary-time propagation of a system. The developed subroutines, however, can further be utilized in the fields of optimization or machine learning. Our algorithms are benchmarked on a range of representative models, such as Ising or Heisenberg spin models, both in numerical simulations and experiments on the hardware. In combination with error mitigation techniques, the latter is scaled up to 27 qubits; into a regime that variational quantum algorithms are challenging to scale to on noisy quantum computers without our algorithms.

Lausanne: EPFL


p. 233.

DOI : 10.5075/epfl-thesis-11132


Dissipative boundary state preparation

F. Yang; P. Molignini; E. J. Bergholtz 

We devise a generic and experimentally accessible recipe to prepare boundary states of topological or non topological quantum systems through an interplay between coherent Hamiltonian dynamics and local dissipation. Intuitively, our recipe harnesses the spatial structure of boundary states which vanish on sublattices where losses are suitably engineered. This yields unique nontrivial steady states that populate the targeted boundary states with infinite lifetimes while all other states are exponentially damped in time. Remarkably, applying loss only at one boundary can yield a unique steady state localized at the very same boundary. We detail our construction and rigorously derive full Liouvillian spectra and dissipative gaps in the presence of a spectral mirror symmetry for a one-dimensional Su-Schrieffer-Heeger model and a two-dimensional Chern insulator. We outline how our recipe extends to generic noninteracting systems.

Physical Review Research


Vol. 5 , num. 4, p. 043229.

DOI : 10.1103/PhysRevResearch.5.043229

A rapidly mixing Markov chain from any gapped quantum many-body system

S. Bravyi; G. Carleo; D. Gosset; Y. Liu 

We consider the computational task of sampling a bit string x from a distribution pi(x) = |(x|0)|2, where 0 is the unique ground state of a local Hamiltonian H. Our main result describes a direct link between the inverse spectral gap of H and the mixing time of an associated continuous-time Markov Chain with steady state pi. The Markov Chain can be implemented efficiently whenever ratios of ground state amplitudes (y|0)/(x|0) are efficiently computable, the spectral gap of H is at least inverse polynomial in the system size, and the starting state of the chain satisfies a mild technical condition that can be efficiently checked. This extends a previously known relationship between sign-problem free Hamiltonians and Markov chains. The tool which enables this generalization is the so-called fixed-node Hamiltonian construction, previously used in Quantum Monte Carlo simulations to address the fermionic sign problem. We implement the proposed sampling algorithm numerically and use it to sample from the ground state of Haldane-Shastry Hamiltonian with up to 56 qubits. We observe empirically that our Markov chain based on the fixed node Hamiltonian mixes more rapidly than the standard Metropolis-Hastings Markov chain.



Vol. 7 , p. 1173.

DOI : 10.22331/q-2023-11-07-1173

Bayes-optimal inference for spreading processes on random networks

D. Ghio; A. L. M. Aragon; I. Biazzo; L. Zdeborova 

We consider a class of spreading processes on networks, which generalize commonly used epidemic models such as the SIR model or the SIS model with a bounded number of reinfections. We analyze the related problem of inference of the dynamics based on its partial observations. We analyze these inference problems on random networks via a message-passing inference algorithm derived from the belief propagation (BP) equations. We investigate whether said algorithm solves the problems in a Bayes-optimal way, i.e., no other algorithm can reach a better performance. For this, we leverage the so-called Nishimori conditions that must be satisfied by a Bayes-optimal algorithm. We also probe for phase transitions by considering the convergence time and by initializing the algorithm in both a random and an informed way and comparing the resulting fixed points. We present the corresponding phase diagrams. We find large regions of parameters where even for moderate system sizes the BP algorithm converges and satisfies closely the Nishimori conditions, and the problem is thus conjectured to be solved optimally in those regions. In other limited areas of the space of parameters, the Nishimori conditions are no longer satisfied and the BP algorithm struggles to converge. No sign of a phase transition is detected, however, and we attribute this failure of optimality to finite-size effects. The article is accompanied by a Python implementation of the algorithm that is easy to use or adapt.

Physical Review E


Vol. 108 , num. 4, p. 044308.

DOI : 10.1103/PhysRevE.108.044308

Unbiasing time-dependent Variational Monte Carlo by projected quantum evolution

A. Sinibaldi; C. Giuliani; G. Carleo; F. Vicentini 

We analyze the accuracy and sample complexity of variational Monte Carlo approaches to simulate the dynamics of many-body quantum systems classically. By systematically studying the relevant stochastic estimators, we are able to: (i) prove that the most used scheme, the time-dependent Variational Monte Carlo (tVMC), is affected by a systematic statistical bias or exponential sample complexity when the wave function contains some (possibly approximate) zeros, an important case for fermionic systems and quantum information protocols; (ii) show that a different scheme based on the solution of an optimization problem at each time step is free from such problems; (iii) improve the sample complexity of this latter approach by several orders of magnitude with respect to previous proofs of concept. Finally, we apply our advancements to study the high-entanglement phase in a protocol of non-Clifford unitary dynamics with local random measurements in 2D, first benchmarking on small spin lattices and then extending to large systems.



Vol. 7 , p. 1131.

DOI : 10.22331/q-2023-10-10-1131

Learning ground states of gapped quantum Hamiltonians with Kernel Methods

C. Giuliani; F. Vicentini; R. Rossi; G. Carleo 

Neural network approaches to approximate the ground state of quantum hamiltonians require the numerical solution of a highly nonlinear optimization problem. We introduce a statistical learning approach that makes the optimization trivial by using kernel methods. Our scheme is an approximate realization of the power method, where supervised learning is used to learn the next step of the power iteration. We show that the ground state properties of arbitrary gapped quantum hamiltonians can be reached with polynomial resources under the assumption that the supervised learning is efficient. Using kernel ridge regression, we provide numerical evidence that the learning assumption is verified by applying our scheme to find the ground states of several prototypical interacting many-body quantum systems, both in one and two dimensions, showing the flexibility of our approach.



Vol. 7 , p. 1096.

DOI : 10.22331/q-2023-08-29-1096

From tensor-network quantum states to tensorial recurrent neural networks

D. Wu; R. Rossi; F. Vicentini; G. Carleo 

We show that any matrix product state (MPS) can be exactly represented by a recurrent neural network (RNN) with a linear memory update. We generalize this RNN architecture to two-dimensional lattices using a multilinear memory update. It supports perfect sampling and wave-function evaluation in polynomial time, and can represent an area law of entanglement entropy. Numerical evidence shows that it can encode the wave function using a bond dimension lower by orders of magnitude when compared to MPS, with an accuracy that can be systematically improved by increasing the bond dimension.

Physical Review Research


Vol. 5 , num. 3, p. L032001.

DOI : 10.1103/PhysRevResearch.5.L032001

Self-correcting quantum many-body control using reinforcement learning with tensor networks

F. Metz; M. Bukov 

Optimal control of quantum many-body systems is needed to make use of quantum technologies, but is challenging due to the exponentially large dimension of the Hilbert space as a function of the number of qubits. Metz and Bukov propose a framework combining matrix product states and reinforcement learning that allows control of a larger number of interacting quantum particles than achievable with standard neural-network-based methods.

Nature Machine Intelligence


Vol. 5 , num. 7, p. 780-791.

DOI : 10.1038/s42256-023-00687-5

Quantum circuits for solving local fermion-to-qubit map- pings

J. Nys; G. Carleo 

Local Hamiltonians of fermionic systems on a lattice can be mapped onto local qubit Hamiltonians. Maintaining the lo-cality of the operators comes at the ex-pense of increasing the Hilbert space with auxiliary degrees of freedom. In order to retrieve the lower-dimensional physical Hilbert space that represents fermionic de-grees of freedom, one must satisfy a set of constraints. In this work, we intro-duce quantum circuits that exactly satisfy these stringent constraints. We demon-strate how maintaining locality allows one to carry out a Trotterized time-evolution with constant circuit depth per time step. Our construction is particularly advanta-geous to simulate the time evolution op-erator of fermionic systems in d>1 di-mensions. We also discuss how these families of circuits can be used as vari-ational quantum states, focusing on two approaches: a first one based on gen-eral constant-fermion-number gates, and a second one based on the Hamiltonian variational ansatz where the eigenstates are represented by parametrized time -evolution operators. We apply our meth-ods to the problem of finding the ground state and time-evolved states of the t -V model.



Vol. 7 , p. 930.

DOI : 10.22331/q-2023-02-21-930

The autoregressive neural network architecture of the Boltzmann distribution of pairwise interacting spins systems

I. Biazzo 

Autoregressive Neural Networks (ARNNs) have shown exceptional results in generation tasks across image, language, and scientific domains. Despite their success, ARNN architectures often operate as black boxes without a clear connection to underlying physics or statistical models. This research derives an exact mapping of the Boltzmann distribution of binary pairwise interacting systems in autoregressive form. The parameters of the ARNN are directly related to the Hamiltonian’s couplings and external fields, and commonly used structures like residual connections and recurrent architecture emerge from the derivation. This explicit formulation leverages statistical physics techniques to derive ARNNs for specific systems. Using the Curie-Weiss and Sherrington-Kirkpatrick models as examples, the proposed architectures show superior performance in replicating the associated Boltzmann distributions compared to commonly used designs. The findings foster a deeper connection between physical systems and neural network design, paving the way for tailored architectures and providing a physical lens to interpret existing ones.

Communications Physics


Vol. 6 , num. 1, p. 296.

DOI : 10.1038/s42005-023-01416-5

Phenomenological theory of variational quantum ground-state preparation

N. Astrakhantsev; G. Mazzola; I. Tavernelli; G. Carleo 

The variational approach is a cornerstone of computational physics, considering both conventional and quantum computing computational platforms. The variational quantum eigensolver algorithm aims to prepare the ground state of a Hamiltonian exploiting parametrized quantum circuits that may offer an advantage compared to classical trial states used, for instance, in quantum Monte Carlo or tensor network calculations. While, traditionally, the main focus has been on developing better trial circuits, we show that the algorithm’s success, if optimized within stochastic gradient descent (SGD) or quantum natural gradient descent (QNGD), crucially depends on other parameters such as the learning rate, the number Ns of measurements to estimate the gradient components, and the Hamiltonian gap Delta. Within the standard SGD or QNGD, we first observe the existence of a finite Ns value below which the optimization is impossible, and the energy variance resembles the behavior of the specific heat in second-order phase transitions. Second, when Ns is above such threshold level, and learning is possible, we develop a phenomenological model that relates the fidelity of the state preparation with the optimization hyperparameters and Delta. More specifically, we observe that the computational resources scale as 1/Delta 2, and we propose a symmetry enhancement of the variational ansatz as a way to increase the closing gap. We test our understanding on several instances of two-dimensional frustrated quantum magnets, which are believed to be the most promising candidates for near-term quantum advantage through variational quantum simulations.

Physical Review Research


Vol. 5 , num. 3, p. 033225.

DOI : 10.1103/PhysRevResearch.5.033225

Quantum process tomography with unsupervised learning and tensor networks

G. Torlai; C. J. Wood; A. Acharya; G. Carleo; J. Carrasquilla et al. 

The impressive pace of advance of quantum technology calls for robust and scalable techniques for the characterization and validation of quantum hardware. Quantum process tomography, the reconstruction of an unknown quantum channel from measurement data, remains the quintessential primitive to completely characterize quantum devices. However, due to the exponential scaling of the required data and classical post-processing, its range of applicability is typically restricted to one- and two-qubit gates. Here, we present a technique for performing quantum process tomography that addresses these issues by combining a tensor network representation of the channel with a data-driven optimization inspired by unsupervised machine learning. We demonstrate our technique through synthetically generated data for ideal one- and two-dimensional random quantum circuits of up to 10 qubits, and a noisy 5-qubit circuit, reaching process fidelities above 0.99 using several orders of magnitude fewer (single-qubit) measurement shots than traditional tomographic techniques. Our results go far beyond state-of-the-art, providing a practical and timely tool for benchmarking quantum circuits in current and near-term quantum computers.

Nature Communications


Vol. 14 , num. 1, p. 2858.

DOI : 10.1038/s41467-023-38332-9

Dilute neutron star matter from neural-network quantum states

B. Fore; J. M. Kim; G. Carleo; M. Hjorth-Jensen; A. Lovato et al. 

Low-density neutron matter is characterized by fascinating emergent quantum phenomena, such as the formation of Cooper pairs and the onset of superfluidity. We model this density regime by capitalizing on the expressivity of the hidden-nucleon neural-network quantum states combined with variational Monte Carlo and stochastic reconfiguration techniques. Our approach is competitive with the auxiliary-field diffusion Monte Carlo method at a fraction of the computational cost. Using a leading-order pionless effective field theory Hamiltonian, we compute the energy per particle of infinite neutron matter and compare it with those obtained from highly realistic interactions. In addition, a comparison between the spin-singlet and triplet two-body distribution functions indicates the emergence of pairing in the 1S0 channel.

Physical Review Research


Vol. 5 , num. 3, p. 033062.

DOI : 10.1103/PhysRevResearch.5.033062

Ab initio quantum chemistry with neural-network wavefunctions

J. Hermann; J. Spencer; K. Choo; A. Mezzacapo; W. M. C. Foulkes et al. 

Deep learning methods outperform human capabilities in pattern recognition and data processing problems and now have an increasingly important role in scientific discovery. A key application of machine learning in molecular science is to learn potential energy surfaces or force fields from ab initio solutions of the electronic Schrodinger equation using data sets obtained with density functional theory, coupled cluster or other quantum chemistry (QC) methods. In this Review, we discuss a complementary approach using machine learning to aid the direct solution of QC problems from first principles. Specifically, we focus on quantum Monte Carlo methods that use neural-network ansatzes to solve the electronic Schrodinger equation, in first and second quantization, computing ground and excited states and generalizing over multiple nuclear configurations. Although still at their infancy, these methods can already generate virtually exact solutions of the electronic Schrodinger equation for small systems and rival advanced conventional QC methods for systems with up to a few dozen electrons.

Nature Reviews Chemistry


DOI : 10.1038/s41570-023-00516-8

Variational quantum algorithm for unconstrained black box binary optimization: Application to feature selection

C. Zoufal; R. V. Mishmash; N. Sharma; N. Kumar; A. Sheshadri et al. 

We introduce a variational quantum algorithm to solve unconstrained black box binary optimization problems, i.e., problems in which the objective function is given as black box. This is in contrast to the typical setting of quantum algorithms for optimization where a classical objective function is provided as a given Quadratic Unconstrained Binary Optimization problem and mapped to a sum of Pauli operators. Furthermore, we provide theoretical justification for our method based on convergence guarantees of quantum imaginary time evolution.To investigate the performance of our algorithm and its potential advantages, we tackle a challenging real-world optimization problem: feature selection. This refers to the problem of selecting a subset of relevant features to use for constructing a predictive model such as fraud detection. Optimal feature selection-when formulated in terms of a generic loss function-offers little structure on which to build classical heuristics, thus resulting primarily in ‘greedy methods’. This leaves room for (near-term) quantum algorithms to be competitive to classical state-of-the-art approaches. We apply our quantum-optimization-based feature selection algorithm, termed VarQFS, to build a predictive model for a credit risk data set with 20 and 59 input features (qubits) and train the model using quantum hardware and tensor-network-based numerical simu-lations, respectively. We show that the quantum method produces competitive and in certain aspects even better performance compared to traditional feature selection techniques used in today’s industry.



Vol. 7 , p. 1-23.

DOI : 10.22331/q-2023-01-26-909

Continuous-variable neural network quantum states and the quantum rotor model

J. Stokes; S. De; S. Veerapaneni; G. Carleo 

We initiate the study of neural network quantum state algorithms for analyzing continuous-variable quantum systems in which the quantum degrees of freedom correspond to coordinates on a smooth manifold. A simple family of continuous-variable trial wavefunctions is introduced which naturally generalizes the restricted Boltzmann machine (RBM) wavefunction introduced for analyzing quantum spin systems. By virtue of its simplicity, the same variational Monte Carlo training algorithms that have been developed for ground state determination and time evolution of spin systems have natural analogues in the continuum. We offer a proof of principle demonstration in the context of ground state determination of a stoquastic quantum rotor Hamiltonian. Results are compared against those obtained from partial differential equation (PDE) based scalable eigensolvers. This study serves as a benchmark against which future investigation of continuous-variable neural quantum states can be compared, and points to the need to consider deep network architectures and more sophisticated training algorithms.

Quantum Machine Intelligence


Vol. 5 , num. 1, p. 12.

DOI : 10.1007/s42484-023-00100-9


Two-dimensional Hubbard model at finite temperature: Weak, strong, and long correlation regimes

F. Simkovic; R. Rossi; M. Ferrero 

We investigate the momentum-resolved spin and charge susceptibilities, as well as the chemical potential and double occupancy in the two-dimensional Hubbard model as functions of doping, temperature, and interaction strength. Through these quantities, we identify a weak-coupling regime, a strong-coupling regime with short-range correlations and an intermediate-coupling regime with long magnetic correlation lengths. In the spin channel, we observe an additional crossover from commensurate to incommensurate correlations. In contrast, we find charge correlations to be only short ranged for all studied temperatures, which suggests that the spin and charge responses are decoupled. These findings were obtained by a connected determinant diagrammatic Monte Carlo algorithm for the computation of double expansions, which we introduce in this paper. This permits us to obtain numerically exact results at temperatures as low as T = 0.067 and interactions up to U = 7, while working on arbitrarily large lattices. Our method also allows us to gain physical insights from investigating the analytic structure of perturbative series. We connect to previous work by studying smaller lattice geometries and report substantial finite-size effects.

Physical Review Research


Vol. 4 , num. 4, p. 043201.

DOI : 10.1103/PhysRevResearch.4.043201

Variational dynamics as a ground-state problem on a quantum computer

S. Barison; F. Vicentini; I. Cirac; G. Carleo 

We propose a variational quantum algorithm to study the real-time dynamics of quantum systems as a ground -state problem. The method is based on the original proposal of Feynman and Kitaev to encode time into a register of auxiliary qubits. We prepare the Feynman-Kitaev Hamiltonian acting on the composed system as a qubit operator and find an approximate ground state using the variational quantum eigensolver. We apply the algorithm to the study of the dynamics of a transverse-field Ising chain with an increasing number of spins and time steps, proving a favorable scaling in terms of the number of two-qubit gates. Through numerical experiments, we investigate its robustness against noise, showing that the method can be used to evaluate dynamical properties of quantum systems and detect the presence of dynamical quantum phase transitions by measuring Loschmidt echoes.

Physical Review Research


Vol. 4 , num. 4, p. 043161.

DOI : 10.1103/PhysRevResearch.4.043161

Variational solutions to fermion-to-qubit mappings in two spatial dimensions

J. Nys; G. Carleo 

Through the introduction of auxiliary fermions, or an enlarged spin space, one can map local fermion Hamiltonians onto local spin Hamiltonians, at the expense of introducing a set of additional constraints. We present a variational MonteCarlo framework to study fermionic systems through higher-dimensional (>1D) Jordan-Wigner transformations. We provide exact solutions to the parity and Gauss-law constraints that are encountered in bosonization procedures. We study the t-V model in 2D and demonstrate how both the ground state and the low-energy excitation spectra can be retrieved in combination with neural network quantum state ansatze.



Vol. 6 , p. 833.

DOI : 10.22331/q-2022-10-13-833

RG-Flow: a hierarchical and explainable flow model based on renormalization group and sparse prior

H-Y. Hu; D. Wu; Y-Z. You; B. Olshausen; Y. Chen 

Flow-based generative models have become an important class of unsupervised learning approaches. In this work, we incorporate the key ideas of renormalization group (RG) and sparse prior distribution to design a hierarchical flow-based generative model, RG-Flow, which can separate information at different scales of images and extract disentangled representations at each scale. We demonstrate our method on synthetic multi-scale image datasets and the CelebA dataset, showing that the disentangled representations enable semantic manipulation and style mixing of the images at different scales. To visualize the latent representations, we introduce receptive fields for flow-based models and show that the receptive fields of RG-Flow are similar to those of convolutional neural networks. In addition, we replace the widely adopted isotropic Gaussian prior distribution by the sparse Laplacian distribution to further enhance the disentanglement of representations. From a theoretical perspective, our proposed method has O(log L) complexity for inpainting of an image with edge length L, compared to previous generative models with O(L-2) complexity.

Machine Learning-Science And Technology


Vol. 3 , num. 3, p. 035009.

DOI : 10.1088/2632-2153/ac8393

Neural-network quantum states for periodic systems in continuous space

G. Pescia; J. Han; A. Lovato; J. Lu; G. Carleo 

We introduce a family of neural quantum states for the simulation of strongly interacting systems in the presence of spatial periodicity. Our variational state is parametrized in terms of a permutationally invariant part described by the Deep Sets neural-network architecture. The input coordinates to the Deep Sets are periodically transformed such that they are suitable to directly describe periodic bosonic systems. We show example applications to both one- and two-dimensional interacting quantum gases with Gaussian interactions, as well as to He-4 confined in a one-dimensional geometry. For the one-dimensional systems we find very precise estimations of the ground-state energies and the radial distribution functions of the particles. In two dimensions we obtain good estimations of the ground-state energies, comparable to results obtained from more conventional methods.

Physical Review Research


Vol. 4 , num. 2, p. 023138.

DOI : 10.1103/PhysRevResearch.4.023138

Deep learning exotic hadrons

L. Ng; L. Bibrzycki; J. Nys; C. Fernandez-Ramirez; A. Pilloni et al. 

We perform the first amplitude analysis of experimental data using deep neural networks to determine the nature of an exotic hadron. Specifically, we study the line shape of the P-c(4312) signal reported by the LHCb collaboration, and we find that its most likely interpretation is that of a virtual state. This method can be applied to other near-threshold resonance candidates.

Physical Review D


Vol. 105 , num. 9, p. L091501.

DOI : 10.1103/PhysRevD.105.L091501

Hamiltonian reconstruction as metric for variational studies

K. Zhang; S. Lederer; K. Choo; T. Neupert; G. Carleo et al. 

Variational approaches are among the most powerful techniques to approximately solve quantum many-body problems. These encompass both variational states based on tensor or neural networks, and parameterized quantum circuits in variational quantum eigensolvers. However, self-consistent evaluation of the quality of variational wavefunctions is a notoriously hard task. Using a recently developed Hamiltonian reconstruction method, we propose a multi-faceted approach to evaluating the quality of neural-network based wavefunctions. Specifically, we consider convolutional neural network (CNN) and restricted Boltzmann machine (RBM) states trained on a square lattice spin-1/2 J(1) – J(2) Heisenberg model. We find that the reconstructed Hamiltonians are typically less frustrated, and have easy-axis anisotropy near the high frustration point. In addition, the reconstructed Hamiltonians suppress quantum fluctuations in the large J(2) limit. Our results highlight the critical importance of the wavefunction’s symmetry. Moreover, the multi-faceted insight from the Hamiltonian reconstruction reveals that a variational wave function can fail to capture the true ground state through suppression of quantum fluctuations.

Scipost Physics


Vol. 13 , num. 3, p. 063.

DOI : 10.21468/SciPostPhys.13.3.063

Neural tensor contractions and the expressive power of deep neural quantum states

O. Sharir; A. Shashua; G. Carleo 

We establish a direct connection between general tensor networks and deep feed-forward artificial neural networks. The core of our results is the construction of neural-network layers that efficiently perform tensor contractions and that use commonly adopted nonlinear activation functions. The resulting deep networks feature a number of edges that closely match the contraction complexity of the tensor networks to be approximated. In the context of many-body quantum states, this result establishes that neural-network states have strictly the same or higher expressive power than practically usable variational tensor networks. As an example, we show that all matrix product states can be efficiently written as neural-network states with a number of edges polynomial in the bond dimension and depth that is logarithmic in the system size. The opposite instead does not hold true, and our results imply that there exist quantum states that are not efficiently expressible in terms of matrix product states or projected entangled pair states but that are instead efficiently expressible with neural network states.

Physical Review B


Vol. 106 , num. 20, p. 205136.

DOI : 10.1103/PhysRevB.106.205136

Hidden-nucleons neural-network quantum states for the nuclear many-body problem

A. Lovato; C. Adams; G. Carleo; N. Rocco 

We generalize the hidden-fermion family of neural network quantum states to encompass both continuous and discrete degrees of freedom and solve the nuclear many-body Schrodinger equation in a systematically improvable fashion. We demonstrate that adding hidden nucleons to the original Hilbert space considerably augments the expressivity of the neural-network architecture compared to the Slater-Jastrow ansatz. The benefits of explicitly encoding in the wave function point symmetries such as parity and timereversal are also discussed. Leveraging on improved optimization methods and sampling techniques, the hidden-nucleon ansatz achieves an accuracy comparable to the numericallyexact hyperspherical harmonic method in light nuclei and to the auxiliary field diffusion Monte Carlo in 16O. Thanks to its polynomial scaling with the number of nucleons, this method opens the way to highly-accurate quantum Monte Carlo studies of medium-mass nuclei.

Physical Review Research


Vol. 4 , num. 4, p. 043178.

DOI : 10.1103/PhysRevResearch.4.043178

Fermionic wave functions from neural-network constrained hidden states

J. Robledo Moreno; G. Carleo; A. Georges; J. Stokes 

We introduce a systematically improvable family of variational wave functions for the simulation of strongly correlated fermionic systems. This family consists of Slater determinants in an augmented Hilbert space involving “hidden” additional fermionic degrees of freedom. These determinants are projected onto the physical Hilbert space through a constraint that is optimized, together with the single-particle orbitals, using a neural network parameterization. This construction draws inspiration from the success of hidden-particle representations but overcomes the limitations associated with the mean-field treatment of the constraint often used in this context. Our construction provides an extremely expressive family of wave functions, which is proved to be universal. We apply this construction to the ground-state properties of the Hubbard model on the square lattice, achieving levels of accuracy that are competitive with those of state-of-the-art variational methods.

Proceedings of the National Academy of Sciences


Vol. 119 , num. 32, p. e2122059119.

DOI : 10.1073/pnas.2122059119

Matrix product states with backflow correlations

G. Lami; G. Carleo; M. Collura 

By taking inspiration from the backflow transformation for correlated systems, we introduce a tensor network Ansatz which extends the well-established matrix product state representation of a quantum many-body wave function. This structure provides enough resources to ensure that states in dimensions larger than or equal to one obey an area law for entanglement. It can be efficiently manipulated to address the ground-state search problem by means of an optimization scheme which mixes tensor-network and variational Monte Carlo algorithms. We benchmark the Ansatz against spin models both in one and two dimensions, demonstrating high accuracy and precision. We finally employ our approach to study the challenging S = 1/2 two-dimensional (2D) J(1) -J(2) model, that it is with the state-of-the-art methods in 2D.

Physical Review B


Vol. 106 , num. 8, p. L081111.

DOI : 10.1103/PhysRevB.106.L081111

Role of stochastic noise and generalization error in the time propagation of neural-network quantum states

D. Hofmann; G. Fabiani; J. H. Mentink; G. Carleo; M. A. Sentef 

Neural-network quantum states (NQS) have been shown to be a suitable variational ansatz to simulate out-of-equilibrium dynamics in two-dimensional systems using timedependent variational Monte Carlo (t-VMC). In particular, stable and accurate time propagation over long time scales has been observed in the square-lattice Heisenberg model using the Restricted Boltzmann machine architecture. However, achieving similar performance in other systems has proven to be more challenging. In this article, we focus on the two-leg Heisenberg ladder driven out of equilibrium by a pulsed excitation as a benchmark system. We demonstrate that unmitigated noise is strongly amplified by the nonlinear equations of motion for the network parameters, which causes numerical instabilities in the time evolution. As a consequence, the achievable accuracy of the simulated dynamics is a result of the interplay between network expressiveness and measures required to remedy these instabilities. We show that stability can be greatly improved by appropriate choice of regularization. This is particularly useful as tuning of the regularization typically imposes no additional computational cost. Inspired by machine learning practice, we propose a validation-set based diagnostic tool to help determining optimal regularization hyperparameters for t-VMC based propagation schemes. For our benchmark, we show that stable and accurate time propagation can be achieved in regimes of sufficiently regularized variational dynamics.

Scipost Physics


Vol. 12 , num. 5, p. 165.

DOI : 10.21468/SciPostPhys.12.5.165

Exploring quantum perceptron and quantum neural network structures with a teacher-student scheme

A. Gratsea; P. Huembeli 

Near-term quantum devices can be used to build quantum machine learning models, such as quantum kernel methods and quantum neural networks (QNN), to perform classification tasks. There have been many proposals on how to use variational quantum circuits as quantum perceptrons or as QNNs. The aim of this work is to introduce a teacher-student scheme that could systematically compare any QNN architectures and evaluate their relative expressive power. Specifically, the teacher model generates the datasets mapping random inputs to outputs which then have to be learned by the student models. This way, we avoid training on arbitrary data sets and allow to compare the learning capacity of different models directly via the loss, the prediction map, the accuracy and the relative entropy between the prediction maps. Here, we focus particularly on a quantum perceptron model inspired by the recent work of Tacchino et al. (2019) and compare it to the data re-uploading scheme that was originally introduced by Perez-Salinas et al. (2020). We discuss alterations of the perceptron model and the formation of deep QNN to better understand the role of hidden units and the non-linearities in these architectures.

Quantum Machine Intelligence


Vol. 4 , num. 1, p. 2.

DOI : 10.1007/s42484-021-00058-6

The physics of energy-based models

P. Huembeli; J. M. Arrazola; N. Killoran; M. Mohseni; P. Wittek 

Energy-based models (EBMs) are experiencing a resurgence of interest in both the physics community and the machine learning community. This article provides an intuitive introduction to EBMs, without requiring any background in machine learning, connecting elementary concepts from physics with basic concepts and tools in generative models, and finally giving a perspective where current research in the field is heading. This article, in its original form, was written as an online lecture note in HTML and Javascript and contains interactive graphics. We recommend the reader to also visit the interactive version.

Quantum Machine Intelligence


Vol. 4 , num. 1, p. 1(2022).

DOI : 10.1007/s42484-021-00057-7

Nuclei with Up to A=6 Nucleons with Artificial Neural Network Wave Functions

A. Gnech; C. Adams; N. Brawand; G. Carleo; A. Lovato et al. 

The ground-breaking works of Weinberg have opened the way to calculations of atomic nuclei that are based on systematically improvable Hamiltonians. Solving the associated many-body Schrodinger equation involves non-trivial difficulties, due to the non-perturbative nature and strong spin-isospin dependence of nuclear interactions. Artificial neural networks have proven to be able to compactly represent the wave functions of nuclei with up to A = 4 nucleons. In this work, we extend this approach to Li-6 and He-6 nuclei, using as input a leading-order pionless effective field theory Hamiltonian. We successfully benchmark their binding energies, point-nucleon densities, and radii with the highly-accurate hyperspherical harmonics method.

Few-Body Systems


Vol. 63 , num. 1, p. 7.

DOI : 10.1007/s00601-021-01706-0

Hessian-based toolbox for reliable and interpretable machine learning in physics

A. Dawid; P. Huembeli; M. Tomza; M. Lewenstein; A. Dauphin 

Machine learning (ML) techniques applied to quantum many-body physics have emerged as a new research field. While the numerical power of this approach is undeniable, the most expressive ML algorithms, such as neural networks, are black boxes: The user does neither know the logic behind the model predictions nor the uncertainty of the model predictions. In this work, we present a toolbox for interpretability and reliability, agnostic of the model architecture. In particular, it provides a notion of the influence of the input data on the prediction at a given test point, an estimation of the uncertainty of the model predictions, and an extrapolation score for the model predictions. Such a toolbox only requires a single computation of the Hessian of the training loss function. Our work opens the road to the systematic use of interpretability and reliability methods in ML applied to physics and, more generally, science.

Machine Learning-Science And Technology


Vol. 3 , num. 1, p. 015002.

DOI : 10.1088/2632-2153/ac338d


Unbiased Monte Carlo cluster updates with autoregressive neural networks

D. Wu; R. Rossi; G. Carleo 

Efficient sampling of complex high-dimensional probability distributions is a central task in computational science. Machine learning methods like autoregressive neural networks, used with Markov chain Monte Carlo sampling, provide good approximations to such distributions but suffer from either intrinsic bias or high variance. In this Letter, we propose a way to make this approximation unbiased and with low variance. Our method uses physical symmetries and variable-size cluster updates which utilize the structure of autoregressive factorization. We test our method for first- and second-order phase transitions of classical spin systems, showing its viability for critical systems and in the presence of metastable states.

Physical Review Research


Vol. 3 , num. 4, p. L042024.

DOI : 10.1103/PhysRevResearch.3.L042024

Mott Insulating States with Competing Orders in the Triangular Lattice Hubbard Model

A. Wietek; R. Rossi; F. Simkovic; M. Klett; P. Hansmann et al. 

The physics of the triangular lattice Hubbard model exhibits a rich phenomenology, ranging from a metal-insulator transition, intriguing thermodynamic behavior, and a putative spin liquid phase at intermediate coupling, ultimately becoming a magnetic insulator at strong coupling. In this multimethod study, we combine a finite-temperature tensor network method, minimally entangled thermal typical states (METTS), with two Green-function-based methods, connected-determinant diagrammatic Monte Carlo and cellular dynamical mean-field theory, to establish several aspects of this model. We elucidate the evolution from the metallic to the insulating regime from the complementary perspectives brought by these different methods. We compute the full thermodynamics of the model on a width-four cylinder using METTS in the intermediate to strong coupling regime. We find that the insulating state hosts a large entropy at intermediate temperatures, which increases with the strength of the coupling. Correspondingly, and consistently with a thermodynamic Maxwell relation, the double occupancy has a minimum as a function of temperature which is the manifestation of the Pomeranchuk effect of increased localization upon heating. The intermediate coupling regime is found to exhibit both pronounced chiral as well as stripy antiferromagnetic spin correlations. We propose a scenario in which time-reversal symmetry-broken states compete with stripy-spin states at lowest temperatures.

Physical Review X


Vol. 11 , num. 4, p. 041013.

DOI : 10.1103/PhysRevX.11.041013

An efficient quantum algorithm for the time evolution of parameterized circuits

S. Barison; F. Vicentini; G. Carleo 

We introduce a novel hybrid algorithm to simulate the real-time evolution of quantum systems using parameterized quantum circuits. The method, named “projected – Variational Quantum Dynamics” (p-VQD) realizes an iterative, global projection of the exact time evolution onto the parameterized manifold. In the small time step limit, this is equivalent to the McLachlan’s variational principle. Our approach is efficient in the sense that it exhibits an optimal linear scaling with the total number of variational parameters. Furthermore, it is global in the sense that it uses the variational principle to optimize all parameters at once. The global nature of our approach then significantly extends the scope of existing efficient variational methods, that instead typically rely on the iterative optimisation of a restricted subset of variational parameters. Through numerical experiments, we also show that our approach is particularly advantageous over existing global optimisation algorithms based on the time-dependent variational principle that, due to a demanding quadratic scaling with parameter numbers, are unsuitable for large parameterized quantum circuits.



Vol. 5 , p. 512.

DOI : 10.22331/q-2021-07-28-512

Gauge Equivariant Neural Networks for Quantum Lattice Gauge Theories

D. Luo; G. Carleo; B. K. Clark; J. Stokes 

Gauge symmetries play a key role in physics appearing in areas such as quantum field theories of the fundamental particles and emergent degrees of freedom in quantum materials. Motivated by the desire to efficiently simulate many-body quantum systems with exact local gauge invariance, gauge equivariant neural-network quantum states are introduced, which exactly satisfy the local Hilbert space constraints necessary for the description of quantum lattice gauge theory with Z(d) gauge group and non-Abelian Kitaev DoG thorn models on different geometries. Focusing on the special case of Z(2) gauge group on a periodically identified square lattice, the equivariant architecture is analytically shown to contain the loop-gas solution as a special case. Gauge equivariant neural-network quantum states are used in combination with variational quantum Monte Carlo to obtain compact descriptions of the ground state wave function for the Z(2) theory away from the exactly solvable limit, and to demonstrate the confining or deconfining phase transition of the Wilson loop order parameter.

Physical Review Letters


Vol. 127 , num. 27, p. 276402.

DOI : 10.1103/PhysRevLett.127.276402

Broken-Symmetry Ground States of the Heisenberg Model on the Pyrochlore Lattice

N. Astrakhantsev; T. Westerhout; A. Tiwari; K. Choo; A. Chen et al. 

The spin-1/2 Heisenberg model on the pyrochlore lattice is an iconic frustrated three-dimensional spin system with a rich phase diagram. Besides hosting several ordered phases, the model is debated to possess a spin-liquid ground state when only nearest-neighbor antiferromagnetic interactions are present. Here, we contest this hypothesis with an extensive numerical investigation using both exact diagonalization and complementary variational techniques. Specifically, we employ a resonating-valence-bond-like, manyvariable, Monte Carlo ansatz and convolutional neural network quantum states for (variational) calculations with up to 4 x 43 and 4 x 33 spins, respectively. We demonstrate that these techniques yield consistent results, allowing for reliable extrapolations to the thermodynamic limit. We consider the (2; j2/j1) parameter space, with j2, j1 being nearest and next-to-nearest neighbor interactions and 2 the XXZ interaction anisotropy. Our main results are (1) the determination of the phase transition between the putative spin-liquid phase and the neighboring magnetically ordered phase and (2) a careful characterization of the ground state in terms of symmetry-breaking tendencies. We find clear indications of a dimer order with spontaneously broken inversion and rotational symmetry, calling the scenario of a featureless quantum spin liquid into question. Our work showcases how many-variable variational techniques can be used to make progress in answering challenging questions about three-dimensional frustrated quantum magnets.

Physical Review X


Vol. 11 , num. 4, p. 041021.

DOI : 10.1103/PhysRevX.11.041021

Simultaneous Perturbation Stochastic Approximation of the Quantum Fisher Information

J. Gacon; C. Zoufal; G. Carleo; S. Woerner 

The Quantum Fisher Information matrix (QFIM) is a central metric in promising algorithms, such as Quantum Natural Gradient Descent and Variational Quantum Imaginary Time Evolution. Computing the full QFIM for a model with d parameters, however, is computation-ally expensive and generally requires O(d(2)) function evaluations. To remedy these increasing costs in high-dimensional parameter spaces, we propose using simultaneous perturbation stochastic approximation techniques to approximate the QFIM at a constant cost. We present the resulting algorithm and successfully apply it to prepare Hamiltonian ground states and train Variational Quantum Boltzmann Machines.



Vol. 5 , p. 267.

DOI : 10.22331/q-2021-10-20-567

Variational Monte Carlo Calculations of A <= 4 Nuclei with an Artificial Neural-Network Correlator Ansatz

C. Adams; G. Carleo; A. Lovato; N. Rocco 

The complexity of many-body quantum wave functions is a central aspect of several fields of physics and chemistry where nonperturbative interactions are prominent. Artificial neural networks (ANNs) have proven to be a flexible tool to approximate quantum many-body states in condensed matter and chemistry problems. In this work we introduce a neural-network quantum state ansatz to model the ground-state wave function of light nuclei, and approximately solve the nuclear many-body Schrodinger equation. Using efficient stochastic sampling and optimization schemes, our approach extends pioneering applications of ANNs in the field, which present exponentially scaling algorithmic complexity. We compute the binding energies and point-nucleon densities of A <= 4 nuclei as emerging from a leading-order pionless effective field theory Hamiltonian. We successfully benchmark the ANN wave function against more conventional parametrizations based on two- and three-body Jastrow functions, and virtually exact Green's function Monte Carlo results.

Physical Review Letters


Vol. 127 , num. 2, p. 022502.

DOI : 10.1103/PhysRevLett.127.022502

Classical variational simulation of the Quantum Approximate Optimization Algorithm

M. Medvidovic; G. Carleo 

A key open question in quantum computing is whether quantum algorithms can potentially offer a significant advantage over classical algorithms for tasks of practical interest. Understanding the limits of classical computing in simulating quantum systems is an important component of addressing this question. We introduce a method to simulate layered quantum circuits consisting of parametrized gates, an architecture behind many variational quantum algorithms suitable for near-term quantum computers. A neural-network parametrization of the many-qubit wavefunction is used, focusing on states relevant for the Quantum Approximate Optimization Algorithm (QAOA). For the largest circuits simulated, we reach 54 qubits at 4 QAOA layers, approximately implementing 324 RZZ gates and 216 RX gates without requiring large-scale computational resources. For larger systems, our approach can be used to provide accurate QAOA simulations at previously unexplored parameter values and to benchmark the next generation of experiments in the Noisy Intermediate-Scale Quantum (NISQ) era.

Npj Quantum Information


Vol. 7 , num. 1, p. 101.

DOI : 10.1038/s41534-021-00440-z

Machine learning toolbox for quantum many body physics

F. Vicentini 

Nature Reviews Physics


Vol. 3 , num. 3, p. 156-156.

DOI : 10.1038/s42254-021-00285-7

Quantum Simulators: Architectures and Opportunities

E. Altman; K. R. Brown; G. Carleo; L. D. Carr; E. Demler et al. 

PRX Quantum


Vol. 2 , num. 1, p. 017003.

DOI : 10.1103/PRXQuantum.2.017003


Deep Learning the Hohenberg-Kohn Maps of Density Functional Theory

J. R. Moreno; G. Carleo; A. Georges 

Physical Review Letters


Vol. 125 , num. 7, p. 076402.

DOI : 10.1103/PhysRevLett.125.076402

Quantum Natural Gradient

J. Stokes; J. Izaac; N. Killoran; G. Carleo 



Vol. 4 , p. 269.

DOI : 10.22331/q-2020-05-25-269

Deep Autoregressive Models for the Efficient Variational Simulation of Many-Body Quantum Systems

O. Sharir; Y. Levine; N. Wies; G. Carleo; A. Shashua 

Physical Review Letters


Vol. 124 , num. 2, p. 020503.

DOI : 10.1103/PhysRevLett.124.020503

Precise measurement of quantum observables with neural-network estimators

G. Torlai; G. Mazzola; G. Carleo; A. Mezzacapo 

Physical Review Research


Vol. 2 , num. 2, p. 022060(R).

DOI : 10.1103/PhysRevResearch.2.022060

Fermionic neural-network states for ab-initio electronic structure

K. Choo; A. Mezzacapo; G. Carleo 

Nature Communications


Vol. 11 , num. 1, p. 2368.

DOI : 10.1038/s41467-020-15724-9

Natural evolution strategies and variational Monte Carlo

T. Zhao; G. Carleo; J. Stokes; S. Veerapaneni 

Machine Learning: Science and Technology


Vol. 2 , num. 2, p. 02LT01.

DOI : 10.1088/2632-2153/abcb50

Phases of two-dimensional spinless lattice fermions with first-quantized deep neural-network quantum states

J. Stokes; J. R. Moreno; E. A. Pnevmatikakis; G. Carleo 

First-quantized deep neural network techniques are developed for analyzing strongly coupled fermionic systems on the lattice. Using a Slater-Jastrow-inspired ansatz which exploits deep residual networks with convolutional residual blocks, we approximately determine the ground state of spinless fermions on a square lattice with nearest-neighbor interactions. The flexibility of the neural-network ansatz results in a high level of accuracy when compared with exact diagonalization results on small systems, both for energy and correlation functions. On large systems, we obtain accurate estimates of the boundaries between metallic and charge-ordered phases as a function of the interaction strength and the particle density.

Physical Review B


Vol. 102 , num. 20, p. 205122.

DOI : 10.1103/PhysRevB.102.205122


Ground state phase diagram of the one-dimensional Bose-Hubbard model from restricted Boltzmann machines

K. McBrian; G. Carleo; E. Khatami 

Journal of Physics: Conference Series


p. 012005

DOI : 10.1088/1742-6596/1290/1/012005

Restricted Boltzmann machines in quantum physics

R. G. Melko; G. Carleo; J. Carrasquilla; J. I. Cirac 

Nature Physics


Vol. 15 , num. 9, p. 887-892.

DOI : 10.1038/s41567-019-0545-1

Machine learning and the physical sciences

G. Carleo; I. Cirac; K. Cranmer; L. Daudet; M. Schuld et al. 

Reviews of Modern Physics


Vol. 91 , num. 4, p. 045002.

DOI : 10.1103/RevModPhys.91.045002

Neural-Network Approach to Dissipative Quantum Many-Body Dynamics

M. J. Hartmann; G. Carleo 

Physical Review Letters


Vol. 122 , num. 25, p. 250502.

DOI : 10.1103/PhysRevLett.122.250502

NetKet: A machine learning toolkit for many-body quantum systems

G. Carleo; K. Choo; D. Hofmann; J. E. Smith; T. Westerhout et al. 



Vol. 10 , p. 100311.

DOI : 10.1016/j.softx.2019.100311

Two-dimensional frustrated J1−J2 model studied with neural network quantum states

K. Choo; T. Neupert; G. Carleo 

Physical Review B


Vol. 100 , num. 12, p. 125124.

DOI : 10.1103/PhysRevB.100.125124


Neural-network states for the classical simulation of quantum computing

B. Jónsson; B. Bauer; G. Carleo 

Simulating quantum algorithms with classical resources generally requires exponential resources. However, heuristic classical approaches are often very efficient in approximately simulating special circuit structures, for example with limited entanglement, or based on one-dimensional geometries. Here we introduce a classical approach to the simulation of general quantum circuits based on neural-network quantum states (NQS) representations. Considering a set of universal quantum gates, we derive rules for exactly applying single-qubit and two-qubit Z rotations to NQS, whereas we provide a learning scheme to approximate the action of Hadamard gates. Results are shown for the Hadamard and Fourier transform of entangled initial states for systems sizes and total circuit depths exceeding what can be currently simulated with state-of-the-art brute-force techniques. The overall accuracy obtained by the neural-network states based on Restricted Boltzmann machines is satisfactory, and offers a classical route to simulating highly-entangled circuits. In the test cases considered, we find that our classical simulations are comparable to quantum simulations affected by an incoherent noise level in the hardware of about 10−3 per gate.


Symmetries and Many-Body Excitations with Neural-Network Quantum States

K. Choo; G. Carleo; N. Regnault; T. Neupert 

Physical Review Letters


Vol. 121 , num. 16, p. 167204.

DOI : 10.1103/PhysRevLett.121.167204

Universal scaling laws for correlation spreading in quantum systems with short- and long-range interactions

L. Cevolani; J. Despres; G. Carleo; L. Tagliacozzo; L. Sanchez-Palencia 

Physical Review B


Vol. 98 , num. 2, p. 024302.

DOI : 10.1103/PhysRevB.98.024302

Learning hard quantum distributions with variational autoencoders

A. Rocchetto; E. Grant; S. Strelchuk; G. Carleo; S. Severini 

npj Quantum Information


Vol. 4 , num. 1, p. 28.

DOI : 10.1038/s41534-018-0077-z

Single-atom-resolved probing of lattice gases in momentum space

H. Cayla; C. Carcy; Q. Bouton; R. Chang; G. Carleo et al. 

Physical Review A


Vol. 97 , num. 6, p. 061609(R).

DOI : 10.1103/PhysRevA.97.061609

Constructing exact representations of quantum many-body systems with deep neural networks

G. Carleo; Y. Nomura; M. Imada 

Nature Communications


Vol. 9 , num. 1, p. 5322.

DOI : 10.1038/s41467-018-07520-3

Neural-network quantum state tomography

G. Torlai; G. Mazzola; J. Carrasquilla; M. Troyer; R. Melko et al. 

Nature Physics


Vol. 14 , num. 5, p. 447-450.

DOI : 10.1038/s41567-018-0048-5


Unitary Dynamics of Strongly Interacting Bose Gases with the Time-Dependent Variational Monte Carlo Method in Continuous Space

G. Carleo; L. Cevolani; L. Sanchez-Palencia; M. Holzmann 

Physical Review X


DOI : 10.1103/PhysRevX.7.031026

Nonstoquastic Hamiltonians and quantum annealing of an Ising spin glass

L. Hormozi; E. W. Brown; G. Carleo; M. Troyer 

Physical Review B


Vol. 95 , num. 18, p. 184416.

DOI : 10.1103/PhysRevB.95.184416

Solving the quantum many-body problem with artificial neural networks

G. Carleo; M. Troyer 



Vol. 355 , num. 6325, p. 602-606.

DOI : 10.1126/science.aag2302


Spreading of correlations in exactly solvable quantum models with long-range interactions in arbitrary dimensions

L. Cevolani; G. Carleo; L. Sanchez-Palencia 

New Journal of Physics


Vol. 18 , num. 9, p. 093002.

DOI : 10.1088/1367-2630/18/9/093002

Mott transition for strongly interacting one-dimensional bosons in a shallow periodic potential

G. Boéris; L. Gori; M. D. Hoogerland; A. Kumar; E. Lucioni et al. 

Physical Review A


Vol. 93 , num. 1, p. 011601(R).

DOI : 10.1103/PhysRevA.93.011601


Protected quasilocality in quantum systems with long-range interactions

L. Cevolani; G. Carleo; L. Sanchez-Palencia 

Physical Review A


Vol. 92 , num. 4, p. 041603(R).

DOI : 10.1103/PhysRevA.92.041603


Quench-Induced Breathing Mode of One-Dimensional Bose Gases

B. Fang; G. Carleo; A. Johnson; I. Bouchoule 

Physical Review Letters


Vol. 113 , num. 3, p. 035301.

DOI : 10.1103/PhysRevLett.113.035301

Light-cone effect and supersonic correlations in one- and two-dimensional bosonic superfluids

G. Carleo; F. Becca; L. Sanchez-Palencia; S. Sorella; M. Fabrizio 

Physical Review A


DOI : 10.1103/PhysRevA.89.031602


Universal Superfluid Transition and Transport Properties of Two-Dimensional Dirty Bosons

G. Carleo; G. Boéris; M. Holzmann; L. Sanchez-Palencia 

Physical Review Letters


Vol. 111 , num. 5, p. 050406.

DOI : 10.1103/PhysRevLett.111.050406


Localization and Glassy Dynamics Of Many-Body Quantum Systems

G. Carleo; F. Becca; M. Schiró; M. Fabrizio 

Scientific Reports


Vol. 2 , num. 1, p. 243.

DOI : 10.1038/srep00243


Itinerant ferromagnetic phase of the Hubbard model

G. Carleo; S. Moroni; F. Becca; S. Baroni 

Physical Review B


Vol. 83 , num. 6, p. 060411(R).

DOI : 10.1103/PhysRevB.83.060411


Reptation quantum Monte Carlo algorithm for lattice Hamiltonians with a directed-update scheme

G. Carleo; F. Becca; S. Moroni; S. Baroni 

Physical Review E


DOI : 10.1103/PhysRevE.82.046710


Bose-Einstein Condensation in Quantum Glasses

G. Carleo; M. Tarzia; F. Zamponi 

Physical Review Letters


DOI : 10.1103/PhysRevLett.103.215302

Zero-temperature dynamics of solidH4efrom quantum Monte Carlo simulations

G. Carleo; S. Moroni; S. Baroni 

Physical Review B


p. 094301

DOI : 10.1103/PhysRevB.80.094301