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arXiv:2307.02633 · Physical Review Research

Hybrid ground-state quantum algorithms based on neural Schrödinger forging

Paulin de Schoulepnikoff, Oriel Kiss, Sofia Vallecorsa, Giuseppe Carleo, Michele Grossi

Published Apr 4, 2024• 9 citations
arXivHTMLPDF

Abstract

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. Published by the American Physical Society 2024

DOI
10.1103/physrevresearch.6.023021

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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. Published by the American Physical Society 2024
The optimal use of quantum and classical computational techniques together is important to address problems that cannot be easily solved by quantum computations alone. This is the case of the ground state problem for quantum many-body systems. We show here that probabilistic generative models can work in conjunction with quantum algorithms to design hybrid quantum-classical variational ansätze that forge entanglement to lower quantum resource overhead. The variational ansätze comprise parametrized quantum circuits on two separate quantum registers, and a classical generative neural network that can entangle them by learning a Schmidt decomposition of the whole system. The method presented is efficient in terms of the number of measurements required to achieve fixed precision on expected values of observables. To demonstrate its effectiveness, we perform numerical experiments on the transverse field Ising model in one and two dimensions, and fermionic systems such as the t-V Hamiltonian of spinless fermions on a lattice.
Owing to their great expressivity and versatility, neural networks have gained attention for simulating large two-dimensional quantum many-body systems. However, their expressivity comes with the cost of a challenging optimization due to the in general rugged and complicated loss landscape. Here, we present a hybrid optimization scheme for neural quantum states (NQS), involving a data-driven pretraining with numerical or experimental data and a second, Hamiltonian-driven optimization stage. By using both projective measurements from the computational basis as well as expectation values from other measurement configurations such as spin-spin correlations, our pretraining gives access to the sign structure of the state, yielding improved and faster convergence that is robust w.r.t. experimental imperfections and limited datasets. We apply the hybrid scheme to the ground state search for the 2D transverse field Ising model and dipolar XY model on 6 × 6 and 10 × 10 square lattices with a patched transformer wave function, using numerical data as well as experimental data from a programmable Rydberg quantum simulator [Chen et al., Nature 616 (2023)], and show that the information from a second measurement basis highly improves the performance. Our work paves the way for a reliable and efficient optimization of neural quantum states.
In this thesis, I explore the domain of hybrid quantum-classical computation, the foremost approach for utilizing Noisy Intermediate-Scale Quantum (NISQ) devices. The opening chapter presents an overview of Variational Quantum Algorithms (VQAs), highlighting the primary algorithmic challenges. It offers an in-depth review of the Quantum Approximate Optimization Algorithm (QAOA), discussing its variants for ground state preparation. In the second chapter, we apply QAOA for the supervised learning of a simple Binary Neural Network. This model represents an idealized yet prototypical example of classical combinatorial optimization problems involving multi-spin interactions. In the following chapters, the discussion shifts toward quantum many-body ground state preparation, focusing on the one-dimensional Heisenberg XYZ model and the longitudinal-transverse-field Ising model (LTFIM). We have developed a novel technique that, at any point in the phase diagram, leverages the transferability of a specific class of optimal schedules from systems with small to those with larger numbers of qubits. This approach mitigates trainability issues, specifically vanishing gradients (Barren Plateaus). Next, we tailor a QAOA scheme to characterize a topological quantum phase transition within a lattice gauge theory model. This investigation is particularly significant due to its implications for high-energy physics and relevance to quantum error correction and surface codes. In the concluding chapter, I propose new stimulating research directions and help to identify core challenges and unresolved questions in variational quantum computing that transcend any particular application domain.
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Abstract Simulating physical systems with variational quantum algorithms is a well-studied approach, but it is challenging to implement in current devices due to demands in qubit number and circuit depth. However, limited knowledge of a system can reduce the cost of these algorithms. We demonstrate that the computational overhead in simulating many-body systems can be systematically decreased by leveraging the system’s symmetries, which guide the choice of iterative entanglement forging partitions aligned with the underlying entropy structure. To do so, we simulate a Fermi-Hubbard one-dimensional chain with a parametrized hopping term, as well as atomic nuclei 28{}^{28}28 28 Ne and 60{}^{60}60 60 Ti with the nuclear shell model. Within an adaptive variational quantum eigensolver setting, we propose an algorithm that employs partitions and subpartitions of the quantum circuit to reduce significantly the quantum resources compared to a fully-fledged simulation. Entropy-driven entanglement forging (EDEF) does not require access to the full statevector or exact entropies; in hardware it is implemented via standard EF expectation-reconstruction rules with crossed statistics, while statevector access is used here only as a classical simulation shortcut. We find that the maximum number of qubits can be reduced by up to a factor of four and we observe an order of magnitude decrease in the number of two-qubit gates compared to regular simulations. Our findings indicate that our method, entropy-driven entanglement forging, can be used to adjust quantum simulations to the limitations of noisy intermediate-scale quantum devices.
The use of artificial neural networks to represent quantum wave functions has recently attracted interest as a way to solve complex many-body problems. The potential of these variational parametrizations has been supported by analytical and numerical evidence in controlled benchmarks. While approaching the end of the early research phase in this field, it becomes increasingly important to show how neural-network states perform for models and physical problems that constitute a clear open challenge for other many-body computational methods. In this paper, we start addressing this aspect, concentrating on a presently unsolved model describing two-dimensional frustrated magnets. Using a fully convolutional neural network model as a variational ans\"atz, we study the frustrated spin-1/$2{J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model on the square lattice. We demonstrate that the resulting predictions for both ground-state energies and properties are competitive with, and often improve upon, existing state-of-the-art methods. In a relatively small region in the parameter space, corresponding to the maximally frustrated regime, our ans\"atz exhibits comparatively good but not the best performance. The gap between the complexity of the models adopted here and those routinely adopted in deep-learning applications is, however, still substantial, such that further improvements in future generations of neural-network quantum states are likely to be expected.