Non-Hermitian (NH) quantum many-body systems exhibit a rich array of physical phenomena, including NH skin effects and exceptional points, that remain largely inaccessible to existing numerical techniques. In this Letter, we investigate the application of variational Monte Carlo and neural network wave function representations to examine their ground-state properties. Due to the breakdown of the Rayleigh-Ritz variational principle in NH settings, we develop a self-consistent symmetric optimization framework based on variance minimization with a dynamically updated energy estimate. Our approach respects the biorthogonal structure of left and right eigenstates, and is further strengthened by exploiting system symmetries and pseudo-Hermiticity. Leveraging this tool, we probe and report accurate NH physical observables for a two-dimensional transverse-field Ising model with a complex longitudinal field, spanning both parity-time symmetric and broken phases. Lastly, we show, through extensive numerical evidence, that our method offers a scalable and flexible computational tool to investigate NH quantum many-body systems, beyond the reach of conventional numerical techniques such as the density-matrix renormalization group algorithm.
Non-Hermitian (NH) quantum many-body systems exhibit a rich array of physical phenomena, including NH skin effects and exceptional points, that remain largely inaccessible to existing numerical techniques. In this Letter, we investigate the application of variational Monte Carlo and neural network wave function representations to examine their ground-state properties. Due to the breakdown of the Rayleigh-Ritz variational principle in NH settings, we develop a self-consistent symmetric optimization framework based on variance minimization with a dynamically updated energy estimate. Our approach respects the biorthogonal structure of left and right eigenstates, and is further strengthened by exploiting system symmetries and pseudo-Hermiticity. Leveraging this tool, we probe and report accurate NH physical observables for a two-dimensional transverse-field Ising model with a complex longitudinal field, spanning both parity-time symmetric and broken phases. Lastly, we show, through extensive numerical evidence, that our method offers a scalable and flexible computational tool to investigate NH quantum many-body systems, beyond the reach of conventional numerical techniques such as the density-matrix renormalization group algorithm.
We introduce the Biorthogonal Variational Quantum Eigensolver (B-VQE), a quantum algorithm for simulating non-Hermitian many-body systems on noisy intermediate-scale quantum (NISQ) hardware. Non-Hermitian quantum matter exhibits exceptional points, parity-time symmetry breaking, and non-Hermitian skin effects, yet existing quantum algorithms often rely on costly post-selection procedures and are not designed to capture biorthogonal eigenstates.
B-VQE employs independent variational circuits to represent the left and right eigenstates of a non-Hermitian Hamiltonian and optimizes a biorthogonal objective function that directly tracks non-Hermitian phase transitions. The framework incorporates an Exceptional-Point Detector (EPD) that identifies exceptional points through a hardware-native coalescence metric and a Non-Hermitian Quantum Geometric Tensor (NH-QGT) readout that distinguishes state-topological and band-topological signatures in interacting many-body systems. To overcome the exponential overhead associated with conventional non-Hermitian simulation, we develop an importance-sampling mitigation strategy that removes the need for ancilla-based post-selection while retaining polynomial computational scaling.
We validate the approach on three representative models: a non-Hermitian Hubbard chain, a non-Hermitian XXZ spin chain, and a two-dimensional non-Hermitian (t)-(J) model. B-VQE achieves relative energy errors below (5\times10^{-3}) and locates exceptional points with high accuracy on noise-free simulations while resolving phase boundaries associated with localization, quantum scars, and skin-effect physics. These results establish B-VQE as a scalable NISQ methodology for constructing non-Hermitian many-body phase diagrams and exploring topological and critical phenomena in open quantum systems.
Non-Hermitian (NH) quantum systems have emerged as a powerful framework for describing open quantum systems, nonequilibrium dynamics, and engineered quantum optical materials. However, solving the ground-state properties of NH systems is challenging due to the exponential scaling of the Hilbert space and exotic phenomena such as the emergence of exceptional points. Another challenge arises from the limitations of traditional methods like exact diagonalization (ED). For the past decade, neural networks (NNs) have shown promise in approximating many-body wavefunctions, yet their application to NH systems remains largely unexplored. In this paper, we explore different NN architectures to investigate the ground-state properties of a parity-time-symmetric, one-dimensional NH, transverse field Ising model with a complex spectrum by employing a recurrent neural network (RNN), a restricted Boltzmann machine (RBM), and a multilayer perceptron (MLP). We construct the NN-based many-body wavefunctions and validate our approach by recovering the ground-state properties of the model for small system sizes, finding excellent agreement with ED. Furthermore, for larger system sizes, we demonstrate that the RNN outperforms both the RBM and MLP. However, we show that the accuracy of the RBM and MLP can be significantly improved through transfer learning, allowing them to perform comparably to the RNN for larger system sizes. These results highlight the potential of neural-network-based approaches—particularly for accurately capturing the low-energy physics of NH quantum systems in case of both weak and strong non-Hermiticity.
Solving non-Hermitian quantum many-body systems on a quantum computer by minimizing the variational energy is challenging as the energy can be complex. Here, we propose a variational quantum algorithm for solving the non-Hermitian Hamiltonian by minimizing a type of energy variance, where zero variance can naturally determine the eigenvalues and the associated left and right eigenstates. Moreover, the energy is set as a parameter in the cost function and can be tuned to scan the whole spectrum efficiently by using a two-step optimization scheme. Through numerical simulations, we demonstrate the algorithm for preparing the left and right eigenstates, verifying the biorthogonal relations, as well as evaluating the observables. We also investigate the impact of quantum noise on our algorithm and show that its performance can be largely improved using error mitigation techniques. Therefore, our work suggests an avenue for solving non-Hermitian quantum many-body systems with variational quantum algorithms on near-term noisy quantum computers.
Variational Monte Carlo calculations have recently reached state-of-the-art accuracy in the approximation of ground state properties of quantum many-body systems. Making use of flexible neural quantum states and automatic differentiation has bypassed traditional computational obstacles such as reliance on basis sets. In this paper, we propose a neural quantum state architecture capable of representing symmetric bosonic wavefunctions in Fock space, enabling the study of systems with variable particle number. By supplementing our variational state with Monte Carlo sampling and geometric optimization, we demonstrate competitive variational energies across an array of one- and two-dimensional systems, converging to the physical boson number under a set chemical potential. Our approach enables accurate estimates of one-body reduced density matrices, opening access to observables such as condensate fractions and radial density profiles from first principles. Our method opens the door to numerical predictions of key measurable quantities in practical grand canonical systems.
Understanding strongly correlated quantum many-body systems requires dealing with a largeconfiguration space comprised of a set of all possible measurable configurations. The total number of configurations grows exponentially with the size of the system, which makes exact numerical calculations impossible for even moderately sized systems. Since the advent of high-temperature superconductivity and the subsequent interest in strongly correlated phases of matter, progress in the field has been marked by ingenuity of the field to overcome these computational limitations. The first techniques to emerge were inspired by quantum chemistry and based on exact diagonalization[1], which is limited to small clusters, and different variants of quantum Monte Carlo[2], that suffers from the sign problem in fermionic and frustrated systems. Circa 1992, the density matrix renormalization group (DMRG) [3, 4, 5, 6, 7] became one of the major breakthroughs of the past few decades. Despite the success of DMRG for one-dimensional(1D) and quasi-one-dimensional geometries, extensions to actual two-dimensional systems remain challenging and applications are constrained to long cylinders and strips. One promising alternative is to combine one the oldest methods, variational wave functions (VWFs), with neural networks, which have been used to recently with great success in machine learning. The focus of this dissertation is on developing novel variational Monte Carlo algorithms to make neural network variational wave functions applicable to many problems in quantum many-body physics. These algorithms draw from existing variational wave-function, tensor network, and machine learning methods. We calculate dynamical spectral functions using the vector correction and Chebyshev expansion method. We do finite temperature calculations with minimally entangled thermal states. Finally, we do ground state calculations for the frustrated J1−J2 Heisenberg model. We benchmark these results using restricted Boltzmann machines as the neural network for the the variational wave-functions.--Author's abstract
In this work, we introduce a complementary optimization method for progressive and adaptive state search (COMPASS) based on biorthogonal adaptive recurrent neural quantum states. Our approach combines an adaptive autoregressive architecture with a biorthogonal variational Monte Carlo scheme as well as a complementary optimization scheme that alternates between energy and variance minimization. This enables the stable convergence to ground-state eigenpairs, while avoiding Markov chain sampling through exact autoregressive generation. We demonstrate that for parity-time(PT)-symmetric Hamiltonians, unconstrained complex ansatze can spontaneously break PT symmetry during optimization, even in the unbroken phase, leading to spurious imaginary energies. Real-valued ansatze, on the other hand, naturally constrain the optimization to the correct physical manifold. Conversely, for generic non-Hermitian (NH) Hamiltonians without symmetry protection and complex spectra, complex ansatze are essential for capturing complex ground-state properties. Our results establish that physically-informed ansatz selection is crucial for reliable NH simulations. By combining adaptive architectures, biorthogonal optimization, and symmetry-aware modeling, this framework enables a direct study of 1D and 2D NH many-body systems without Hermitian embeddings or adiabatic continuation. Applying this framework to systems with frustrated magnetism, we show that gap frustration provides a quantitative shield against NH spectral instability, with the frustration gap setting a critical threshold for PT-symmetry breaking. Also, complexifying the frustration coupling itself generates a new topologically nontrivial network of diabolic level crossings, controlled by the phase of the complex coupling, that has no Hermitian analog. We term this novel spectral topology in NH frustrated systems the diabolic ring.