Showing papers by "Giuseppe Carleo published in 2021"

Quantum Simulators: Architectures and Opportunities

Abstract: Quantum simulators are a promising technology on the spectrum of quantum devices from specialized quantum experiments to universal quantum computers. These quantum devices utilize entanglement and many-particle behaviors to explore and solve hard scientific, engineering, and computational problems. Rapid development over the last two decades has produced more than 300 quantum simulators in operation worldwide using a wide variety of experimental platforms. Recent advances in several physical architectures promise a golden age of quantum simulators ranging from highly optimized special purpose simulators to flexible programmable devices. These developments have enabled a convergence of ideas drawn from fundamental physics, computer science, and device engineering. They have strong potential to address problems of societal importance, ranging from understanding vital chemical processes, to enabling the design of new materials with enhanced performance, to solving complex computational problems. It is the position of the community, as represented by participants of the NSF workshop on "Programmable Quantum Simulators," that investment in a national quantum simulator program is a high priority in order to accelerate the progress in this field and to result in the first practical applications of quantum machines. Such a program should address two areas of emphasis: (1) support for creating quantum simulator prototypes usable by the broader scientific community, complementary to the present universal quantum computer effort in industry; and (2) support for fundamental research carried out by a blend of multi-investigator, multi-disciplinary collaborations with resources for quantum simulator software, hardware, and education.

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

Abstract: 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 Schr\"odinger 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 $A\ensuremath{\le}4$ 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.

An efficient quantum algorithm for the time evolution of parameterized circuits

Abstract: 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 optimization of a restricted subset of variational parameters. Through numerical experiments, we also show that our approach is particularly advantageous over existing global optimization 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.

Classical variational simulation of the Quantum Approximate Optimization Algorithm

Abstract: 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.

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

Abstract: 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 RVB-like many-variable Monte Carlo ansatz and convolutional neural network quantum states for (variational) calculations with up to $4\times 4^3$ and $4 \times 3^3$ spins, respectively. We demonstrate that these techniques yield consistent results, allowing for reliable extrapolations to the thermodynamic limit. 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 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.

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

Abstract: Novel numerical simulations suggest that the frustrated Heisenberg regime of a pyrochlore lattice cannot host a quantum spin liquid, contrary to many expectations.

An efficient quantum algorithm for the time evolution of parameterized circuits

Abstract: 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 optimization of a restricted subset of variational parameters. Through numerical experiments, we also show that our approach is particularly advantageous over existing global optimization 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.

Classical variational simulation of the Quantum Approximate Optimization Algorithm

Abstract: 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.

Simultaneous Perturbation Stochastic Approximation of the Quantum Fisher Information

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

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

Abstract: 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 non-linear activation functions. The resulting deep networks feature a number of edges that closely matches 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 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 practically usable PEPS, but that are instead efficiently expressible with neural network states.

Natural evolution strategies and variational Monte Carlo

Abstract: A notion of quantum natural evolution strategies is introduced, which provides a geometric synthesis of a number of known quantum/classical algorithms for performing classical black-box optimization. Recent work of Gomes et al. [2019] on heuristic combinatorial optimization using neural quantum states is pedagogically reviewed in this context, emphasizing the connection with natural evolution strategies. The algorithmic framework is illustrated for approximate combinatorial optimization problems, and a systematic strategy is found for improving the approximation ratios. In particular it is found that natural evolution strategies can achieve approximation ratios competitive with widely used heuristic algorithms for Max-Cut, at the expense of increased computation time.

Unbiased Monte Carlo cluster updates with autoregressive neural networks

Abstract: The paper shows that variationally-trained neural networks can generate efficient cluster update strategies for Markov-chain Monte Carlo in classical statistical physics.

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

Abstract: 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 time-dependent 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 by itself is sufficient to cause 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 regularization required to remedy these instabilities. Inspired by machine learning practice, we propose a validation-set based diagnostic tool to help determining the 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.

Simultaneous Perturbation Stochastic Approximation of the Quantum Fisher Information

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

Unbiased Monte Carlo Cluster Updates with Autoregressive Neural Networks

Abstract: Efficient sampling of complex high-dimensional probability densities is a central task in computational science. Machine Learning techniques based on autoregressive neural networks have been recently shown to provide good approximations of probability distributions of interest in physics. In this work, we propose a systematic way to remove the intrinsic bias associated with these variational approximations, combining it with Markov-chain Monte Carlo in an automatic scheme to efficiently generate cluster updates, which is particularly useful for models for which no efficient cluster update scheme is known. Our approach is based on symmetry-enforced cluster updates building on the neural-network representation of conditional probabilities. We demonstrate that such finite-cluster updates are crucial to circumvent ergodicity problems associated with global neural updates. We test our method for first- and second-order phase transitions in classical spin systems, proving in particular its viability for critical systems, or in the presence of metastable states.

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

Abstract: We initiate the study of neural-network quantum state algorithms for analyzing continuous-variable lattice quantum systems in first quantization. A simple family of continuous-variable trial wavefunctons 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.

Nuclei with up to $\boldsymbol{A=6}$ nucleons with artificial neural network wave functions

Abstract: 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 $^6$Li and $^6$He 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.

Hamiltonian reconstruction as metric for variational studies

Abstract: Variational approaches are among the most powerful modern 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. Furthermore, 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.