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arXiv:2305.15868 · Physical review. B./Physical review. B

Accurate determination of low-energy eigenspectra with multitarget matrix product states

Xuan Li, Zongsheng Zhou, Guanglei Xu, Runze Chi, Yibin Guo, Tong Liu, Haijun Liao, Tao Xiang

Published Jan 9, 2024• 2 citations
arXivHTMLPDF

Abstract

Determining the low-energy eigenspectra of quantum many-body systems is a long-standing challenge in physics. In this paper, we solve this problem by introducing two algorithms to determine low-energy eigenstates based on a compact matrix product state (MPS) representation of multiple targeted eigenstates. The first algorithm utilizes a canonicalization approach that takes advantage of the imaginary-time evolution of multitarget MPSs, offering faster convergence and ease of implementation. The second algorithm employs a variational approach that optimizes local tensors on the Grassmann manifold, capable of achieving higher accuracy. These algorithms can be used independently or combined to enhance convergence speed and accuracy. We apply them to the transverse-field Ising model and demonstrate that the calculated low-energy eigenspectra agree remarkably well with the exact solution. Moreover, the eigenenergies exhibit uniform convergence in gapped phases, suggesting that the low-energy excited eigenstates have nearly the same level of accuracy as the ground state. Our results highlight the accuracy and versatility of multitarget MPS-based algorithms for determining low-energy eigenspectra and their potential applications in quantum many-body physics.

DOI
10.1103/physrevb.109.045115

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Determining the low-energy eigenspectra of quantum many-body systems is a long-standing challenge in physics. In this paper, we solve this problem by introducing two algorithms to determine low-energy eigenstates based on a compact matrix product state (MPS) representation of multiple targeted eigenstates. The first algorithm utilizes a canonicalization approach that takes advantage of the imaginary-time evolution of multitarget MPSs, offering faster convergence and ease of implementation. The second algorithm employs a variational approach that optimizes local tensors on the Grassmann manifold, capable of achieving higher accuracy. These algorithms can be used independently or combined to enhance convergence speed and accuracy. We apply them to the transverse-field Ising model and demonstrate that the calculated low-energy eigenspectra agree remarkably well with the exact solution. Moreover, the eigenenergies exhibit uniform convergence in gapped phases, suggesting that the low-energy excited eigenstates have nearly the same level of accuracy as the ground state. Our results highlight the accuracy and versatility of multitarget MPS-based algorithms for determining low-energy eigenspectra and their potential applications in quantum many-body physics.
Numerical methods based on matrix product states (MPSs) are currently the standard for calculating the ground-state properties of (quasi-)one-dimensional quantum many-body systems. While the properties of the low-lying excitations in such systems are often studied in this MPS framework through by means of time-evolution simulations, we can also look at their by directly calculating eigenstates corresponding to these excitations. The so-called MPS excitation ansatz is a powerful method for finding such eigenstates with a single-particle character in the thermodynamic limit. Although this excitation ansatz has been used quite extensively, a general method for calculating expectation values for these states is lacking in the literature: We aim to fill this gap by presenting a recursive algorithm to calculate arbitrary observables expressed as matrix product operators. This method concisely encapsulates existing methods for—as well as extensions to—the excitation ansatz, such as excitations with a larger spatial support and multiparticle excitations, and is robust enough to handle further innovations. We demonstrate the versatility of our method by studying the low-lying excitations in the spin-1 Heisenberg chain and the one-dimensional Hubbard model, looking at how the excitations converge in the former, while in the latter, we present a refined method of targeting single-particle excitations inside a continuum by minimizing the energy rather than the energy itself. We hope that this technique will foster further advancements with the excitation ansatz.
A key property of many-body localized Hamiltonians is the area law entanglement of even highly excited eigenstates. Matrix product states (MPS) can be used to efficiently represent low entanglement (area law) wave functions in one dimension. An important application of MPS is the widely used density matrix renormalization group (DMRG) algorithm for finding ground states of one-dimensional Hamiltonians. Here, we develop two algorithms, the shift-and-invert MPS (SIMPS) and excited state DMRG which find highly excited eigenstates of many-body localized Hamiltonians. Excited state DMRG uses a modified sweeping procedure to identify eigenstates, whereas SIMPS applies the inverse of the shifted Hamiltonian to a MPS multiple times to project out the targeted eigenstate. To demonstrate the power of these methods, we verify the breakdown of the eigenstate thermalization hypothesis in the many-body localized phase of the random field Heisenberg model, show the saturation of entanglement in the many-body localized phase, and generate local excitations.
Understanding the excitation spectrum in two-dimensional quantum many-body systems has long been a formidable challenge. In this study, we propose an innovative approach by introducing an excitation Ansatz based on an infinite matrix product state (MPS) with a helix structure. The use of infinite uniform MPS states allows us to accurately extract key properties such as energy, degeneracy, spectrum weight, and scaling behavior of low-energy excited states simultaneously. To validate the effectiveness of our method, we apply it to the critical point of the transverse-field Ising model. The scaling exponent of the energy gap extracted aligns closely with conformal bootstrap results. Subsequently, we extend our method to the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model on a square lattice. Our findings reveal that the degeneracy of lowest-energy excitations serves as a reliable metric for distinguishing different phases. The phase boundaries identified by our method are consistent with previous findings. The present method provides a promising avenue for studying the excitation spectrum of two-dimensional quantum many-body systems.
We propose a framework to design concurrently a frustration-free quantum many-body Hamiltonian and its numerically exact ground states on a sufficiently large finite-size cluster in one and two dimensions using an elementary matrix product state (MPS) representation. Our approach strategically chooses a local cluster Hamiltonian, which is arranged to overlap with neighboring clusters on a designed lattice. The frustration-free Hamiltonian is given as the sum of the cluster Hamiltonians by ensuring that there exists a state that has its local submanifolds as the lowest-energy eigenstate of every cluster. The key to find such a solution is a systematic protocol, which projects out excited states on every cluster using MPS and effectively entangles the cluster states. This algorithm not only offers an exact solution to general frustration-free Hamiltonian, but even provide the excited eigenstates by setting a proper target. It has several advantages, including the ability to achieve the calculation of states at nearly equal cost in one and two dimensions including those belonging to gapless or long-range entangled classes, flexibility in designing Hamiltonians unbiasedly across various forms of models, and numerically feasible validation of exactness through energy calculations. It enables the exploration of exact phase boundaries and the analysis of even a spatially nonuniform random system, providing platforms for quantum simulations and benchmarks.
The determination of the ground and low-lying excited states is critical in many studies of quantum chemistry and condensed-matter physics. Recent theoretical work proposes a variational quantum eigensolver using ancillary qubits to generate entanglement in the variational circuits, which avoids complex ansatz circuits and successive measurements in the previous algorithms. In this work, we employ the ancilla-entangled variational quantum eigensolver to simultaneously compute multiple low-lying eigenenergies and eigenstates of the H2 molecule and three- and five-spin transverse field Ising models (TFIMs) on a superconducting quantum processor. We obtain the potential energy curves of H2 and show an indication of antiferromagnetic to paramagnetic phase transition in the TFIMs from the average absolute magnetization. Our experiments demonstrate that the algorithm is capable of simultaneously determining multiple eigenenergies and eigenstates of many-body systems with high efficiency and accuracy and with less computational resources.
Finding the eigenstates of a many-body Hamiltonian is a fundamental challenge in physics and computational science. Since the search space grows exponentially with system size, numerous classical and quantum algorithms have been developed to address this problem. A practical strategy is to identify a physics-informed low-dimensional subspace that effectively accommodates the low-lying eigenstates, thereby reducing the computational complexity. In this paper, we propose a recursive module-coupling algorithm, which iteratively treats a system as a composition of locally-coupled smaller modules, with low-energy subspace estimated successively according to the same recursive structure. Unlike the density matrix renormalization group (DMRG) approach that optimizes a global matrix product state through repeated local sweeps and obtains excited states sequentially, our algorithm constructs a physically tailored variational basis from module eigenstates and obtains several low-energy states on an equal footing, leading to substantial speedups if targeting moderate accuracy. Our proposed method further leads naturally to a recursive quantum variational algorithm, providing a systematic and modular circuit-construction framework compatible with contemporary gate-based quantum architectures. At each recursive level, block encoders are trained to map logical basis states onto the retained physical subspace, within which a variational circuit is subsequently optimized. Such a quantum-circuit implementation provides not only a quantum multistate eigensolver, but also a systematic prescription for hierarchically constructing quantum state-preparation circuits. Classical simulations demonstrate the accuracy and efficiency of the proposed method, whereas experiments on IBM quantum processors show that eigenstate preparation with reasonable fidelities is achievable even in the current NISQ era.