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Study of nanostructures coupled with a superconducting lead via the neural-network quantum states methods

Jana KODRLOVÁ, Martin Žonda, Martin Friák

Published Jan 1, 2025• 0 citations
Publisher

Abstract

The Neural-Network Quantum States (NNQS) method is rapidly emerging as a powerful tool for investigating quantum many-body physics.By combining variational Monte Carlo techniques with neural network-based variational functions, this approach leverages the remarkable advancements in deep learning achieved in recent years.While substantial progress has been made in simulating magnetic systems on lattices, simple molecules, and even continuous systems, the ab initio simulation of complex strongly correlated electron systems continues to pose significant challenges.With the help of the generalized atomic limit (GAL) -a recently developed model describing a system of quantum dots coupled to a superconducting lead -we attempt to show the efficiency and accuracy of NNQS, mainly the Restricted Boltzmann Machine, by comparing the acquired results to other available methods such as exact diagonalization.The simultaneous study of the system's properties such as the energy spectrum and quantum phase transitions could bring advancements in electronics, sensors or the design of high-quality qubits used in quantum computers.

DOI
10.37904/metal.2025.5156

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The Neural-Network Quantum States (NNQS) method is rapidly emerging as a powerful tool for investigating quantum many-body physics.By combining variational Monte Carlo techniques with neural network-based variational functions, this approach leverages the remarkable advancements in deep learning achieved in recent years.While substantial progress has been made in simulating magnetic systems on lattices, simple molecules, and even continuous systems, the ab initio simulation of complex strongly correlated electron systems continues to pose significant challenges.With the help of the generalized atomic limit (GAL) -a recently developed model describing a system of quantum dots coupled to a superconducting lead -we attempt to show the efficiency and accuracy of NNQS, mainly the Restricted Boltzmann Machine, by comparing the acquired results to other available methods such as exact diagonalization.The simultaneous study of the system's properties such as the energy spectrum and quantum phase transitions could bring advancements in electronics, sensors or the design of high-quality qubits used in quantum computers.
The Neural-Network Quantum States (NNQS) method is rapidly emerging as a powerful tool for investigating \nquantum many-body physics. By combining variational Monte Carlo techniques with neural network-based \nvariational functions, this approach leverages the remarkable advancements in deep learning achieved in \nrecent years. While substantial progress has been made in simulating magnetic systems on lattices, simple \nmolecules, and even continuous systems, the ab initio simulation of complex strongly correlated electron \nsystems continues to pose significant challenges. With the help of the generalized atomic limit (GAL) – a \nrecently developed model describing a system of quantum dots coupled to a superconducting lead – we \nattempt to show the efficiency and accuracy of NNQS, mainly the Restricted Boltzmann Machine, by comparing \nthe acquired results to other available methods such as exact diagonalization. The simultaneous study of the \nsystem’s properties such as the energy spectrum and quantum phase transitions could bring advancements \nin electronics, sensors or the design of high-quality qubits used in quantum computers.
Systems of quantum dots coupled to superconducting (SC) leads designed for practical applications are becoming increasingly complicated, ren- dering traditional analytical and numerical methods insufficient. Therefore, we tested new Neural Network Quantum States (NNQS) methods based on various network architectures and on variational Monte Carlo. To simplify their appli- cation, a new Chain Expansion method developed in our research group, which constructs effective models, was used. We applied these methods to systems of one to three quantum dots coupled to one to three SC leads, benchmarking them against conventional techniques and analyzing their phase diagrams. While at least one NNQS variant typically yielded correct predictions, no single method performed consistently well across all studied configurations. Thus, we conclude that current NNQS methods still face significant limitations when systematically studying complex hybrid su- perconducting nanostructures. This thesis also introduces a focused analytical study of effective models for fundamental quantum dot configurations. 1
We introduce an efficient approach to implement neural network quantum states (NNQS) as trial wave functions in auxiliary-field quantum Monte Carlo (AFQMC). NNQS are a recently developed class of variational ansätze capable of flexibly representing many-body wave functions, though they often incur a high computational cost during optimization. AFQMC, on the other hand, is a powerful stochastic projector approach for ground-state calculations, but it normally requires an approximate constraint via a trial wave function or trial density matrix, whose quality affects the accuracy. Recently, it has been shown (Xiao et al., arXiv2505.18519) that a broad class of highly correlated wave functions can be integrated into AFQMC through stochastic sampling techniques. In this work, we apply this approach and present a direct integration of NNQS with AFQMC, allowing NNQS to serve as high-quality trial wave functions for AFQMC with manageable computational cost. We test the NNQS-AFQMC method on the challenging nitrogen molecule (N 2 ) at stretched geometries. Our results demonstrate that AFQMC with an NNQS trial wave function can attain near-exact total energies, highlighting the potential of AFQMC with NNQS to overcome longstanding challenges in strongly correlated electronic structure calculations. We also outline future research directions for improving this promising methodology.
Various ensembles of quantum dots embedded onto a superconducting surface have been addressed within the recently derived generalized atomic limit model. We study how the model is able to capture the emerging phenomena such as Andreev bound states and quantum phase transitions. This is done through the methods of exact diagonaliza- tion with the Lanczos algorithm and variational Monte Carlo, where the wavefunction is approximated by the machine learning based approach of restricted Boltzmann machines, known as neural network quantum states. We demonstrate that these systems exhibit complex phase diagrams and a rich spectrum of bound states within the superconduct- ing gap. The advantages and limitations of both methods are analyzed providing an important foundation for future research.
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Neural network quantum state (NNQS) has emerged as a promising candidate for quantum many-body problems, but its practical applications are often hindered by the high cost of sampling and local energy calculation. We develop a high-performance NNQS method for ab initio electronic structure calculations. The major innovations include: (1) A transformer based architecture as the quantum wave function ansatz; (2) A data-centric parallelization scheme for the variational Monte Carlo (VMC) algorithm which preserves data locality and well adapts for different computing architectures; (3) A parallel batch sampling strategy which reduces the sampling cost and achieves good load balance; (4) A parallel local energy evaluation scheme which is both memory and computationally efficient; (5) Study of real chemical systems demonstrates both the superior accuracy of our method compared to state-of-the-art and the strong and weak scalability for large molecular systems with up to 120 spin orbitals.