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arXiv:2310.07600 · Physical review. B./Physical review. B

Hybrid quantum-classical algorithm for the transverse-field Ising model in the thermodynamic limit

Sumeet, Max Hörmann, Kai Phillip Schmidt

Published Oct 15, 2024• 9 citations
arXivHTMLPDF

Abstract

We describe a hybrid quantum-classical approach to treat quantum many-body systems in the thermodynamic limit. This is done by combining numerical linked-cluster expansions (NLCE) with the variational quantum eigensolver (VQE). Here, the VQE algorithm is used as a cluster solver within the NLCE. We test our hybrid quantum-classical algorithm ($\mathrm{NLCE}+\mathrm{VQE}$) for the ferromagnetic transverse-field Ising model on the one-dimensional chain and the two-dimensional square lattice. The calculation of ground-state energies on each open cluster demands a modified Hamiltonian variational ansatz for the VQE. One major finding is convergence of $\mathrm{NLCE}+\mathrm{VQE}$ to the conventional NLCE result in the thermodynamic limit when at least $N/2$ circuit repetitions (known as layers) are used in the VQE ansatz for each cluster with $N$ sites. Our approach demonstrates the fruitful connection of techniques known from correlated quantum many-body systems with hybrid algorithms explored on existing quantum-computing devices.

DOI
10.1103/physrevb.110.155128

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We describe a hybrid quantum-classical approach to treat quantum many-body systems in the thermodynamic limit. This is done by combining numerical linked-cluster expansions (NLCE) with the variational quantum eigensolver (VQE). Here, the VQE algorithm is used as a cluster solver within the NLCE. We test our hybrid quantum-classical algorithm (NLCE+VQE\mathrm{NLCE}+\mathrm{VQE}NLCE+VQE) for the ferromagnetic transverse-field Ising model on the one-dimensional chain and the two-dimensional square lattice. The calculation of ground-state energies on each open cluster demands a modified Hamiltonian variational ansatz for the VQE. One major finding is convergence of NLCE+VQE\mathrm{NLCE}+\mathrm{VQE}NLCE+VQE to the conventional NLCE result in the thermodynamic limit when at least N/2N/2N/2 circuit repetitions (known as layers) are used in the VQE ansatz for each cluster with NNN sites. Our approach demonstrates the fruitful connection of techniques known from correlated quantum many-body systems with hybrid algorithms explored on existing quantum-computing devices.
We propose a quantum algorithm for computing one-quasiparticle excitation energies in the thermodynamic limit by combining numerical linked-cluster expansions (NLCEs) and the variational quantum eigensolver (VQE). Our approach uses VQE to block-diagonalize the cluster Hamiltonian through a single-unitary transformation. This unitary is then postprocessed using the projective cluster-additive transformation (PCAT) to ensure cluster additivity, a key requirement for NLCE convergence. We benchmark our method on the transverse-field Ising model (TFIM) in one and two dimensions, and with longitudinal field, computing one-quasiparticle dispersions in the high-field polarized phase. We compare two cost function classes, trace minimization and variance based, demonstrating their effectiveness with the Hamiltonian variational ansatz (HVA). For pure TFIM, ⌈N/2⌉\lceil N/2 \rceil⌈N/2⌉ layers of HVA suffice: NLCE+VQE matches exact diagonalization. For TFIM with longitudinal field, where parity symmetry breaks and PCAT becomes essential, both ⌈N/2⌉\lceil N/2 \rceil⌈N/2⌉ and NNN layers of HVA converge with increasing cluster size, with NNN layers providing improved accuracy. Our results establish PCAT as a cluster-additive framework that extends variational quantum algorithms to excited-state calculations in the thermodynamic limit via NLCE. While demonstrated with VQE, the PCAT postprocessing approach, which requires only low-energy eigenspace information, applies to any quantum eigenstate preparation method.
We run a numerical linked-cluster expansion with a quantum algorithm (NLCE+QA), computing ground-state energies and one quasi-particle dispersions in the thermodynamic limit using a 20-qubit trapped-ion quantum processing unit (QPU). The NLCE+QA framework extracts thermodynamic-limit properties from small-cluster calculations, making it naturally suited for near-term quantum devices. Projector-based block-diagonalization schemes such as projective cluster-additive transformation (PCAT) are essential to NLCE+QA, and they involve matrix inversion and square root operations that amplify measurement noise. A central question is therefore whether current hardware can provide expectation values that are accurate enough to withstand non-linear classical post-processing. We explore this challenge for the transverse-field Ising model (TFIM) in one dimension, on a ladder geometry, as well as in a longitudinal field in one dimension. For the quantum algorithm, we consider adiabatic state preparation (ASP), as well as a variational quantum eigensolver (VQE) trained on a classical device. The final expectation values are obtained from the QPU, using a novel alternative to the Hadamard test that we name the CX-test. We explore the regimes currently attainable on quantum devices and comment on the improvements needed for quantum computers to achieve results beyond classical reach.
We study quantum quenches in the transverse-field Ising model defined on different lattice geometries such as chains, two- and three-leg ladders, and two-dimensional square lattices. Starting from fully polarized initial states, we consider the dynamics of the transverse and the longitudinal magnetization for quenches to weak, strong, and critical values of the transverse field. To this end, we rely on an efficient combination of numerical linked cluster expansions (NLCEs) and a forward propagation of pure states in real time. As a main result, we demonstrate that NLCEs comprising solely rectangular clusters provide a promising approach to study the real-time dynamics of two-dimensional quantum many-body systems directly in the thermodynamic limit. By comparing to existing data from the literature, we unveil that NLCEs yield converged results on time scales which are competitive to other state-of-the-art numerical methods.
The Variational Quantum Eigensolver (VQE) is a leading hybrid quantum-classical algorithm for simulating many-body systems in the NISQ era. Its effectiveness, however, depends on the faithful preparation of eigenstates, which becomes challenging in degenerate and strongly entangled regimes. We study this problem using the transverse-field Ising model (TFIM) with periodic boundary conditions in one, two, and three dimensions, considering systems of up to 27 qubits. We employ different ansatzes: the hardware-efficient EfficientSU2 from Qiskit, the physics-inspired Hamiltonian Variational Ansatz (HVA) and HVA with symmetry breaking, and benchmark their performance using energy variance, entanglement entropy, spin correlations, and magnetization.
We demonstrate that numerical linked cluster expansions (NLCE) yield a powerful approach to calculate time-dependent correlation functions for quantum many-body systems in one dimension. As a paradigmatic example, we study the dynamics of the spin current in the spin-1/2XXZ1/2 XXZ1/2XXZ chain for different values of anisotropy, as well as in the presence of an integrability-breaking next-nearest neighbor interaction. For short to intermediate time scales, we unveil that NLCE yields a convergence towards the thermodynamic limit already for small cluster sizes, which is much faster than in direct calculations of the autocorrelation function for systems with open or periodic boundary conditions. Most importantly, we show that the range of accessible cluster sizes in NLCE can be extended by evaluating the contributions of larger clusters by means of a pure-state approach based on the concept of dynamical quantum typicality (DQT). Even for moderate computational effort, this combination of DQT and NLCE provides a competitive alternative to existing state-of-the-art techniques, which may be applied in higher dimensions as well.
The Variational Quantum Eigensolver (VQE) is a leading hybrid quantum-classical algorithm for simulating many-body systems in the NISQ era. Its effectiveness, however, depends on the faithful preparation of eigenstates, which becomes challenging in degenerate and strongly entangled regimes. We study this problem using the transverse-field Ising model (TFIM) with periodic boundary conditions in one, two, and three dimensions, considering systems of up to 27 qubits. We employ different ansatzes: the hardware-efficient EfficientSU2 from Qiskit, the physics-inspired Hamiltonian Variational Ansatz (HVA) and HVA with symmetry breaking, and benchmark their performance using energy variance, entanglement entropy, spin correlations, and magnetization.