Obtaining accurate representations of the eigenstates of an array of coupled superconducting qubits is a crucial step in the design of circuit quantum electrodynamics- (circuit-QED) based quantum processors. However, exact diagonalization of the device Hamiltonian is challenging for system sizes beyond tens of qubits. Here, we employ a tensor network method based on a variant of the density matrix renormalization group (DMRG) algorithm, DMRG-X, to efficiently obtain localized eigenstates of a two-dimensional transmon array without the need to first compute lower-energy states. We also introduce MTDMRG-X, a new algorithm that combines DMRG-X with multitarget DMRG to efficiently compute excited states even in regimes with strong eigenstate hybridization. We showcase the use of these methods for the analysis of long-range couplings in a multitransmon Hamiltonian including qubits and couplers, and we discuss eigenstate localization. These developments facilitate the design and parameter optimization of large-scale superconducting quantum processors.
Obtaining accurate representations of the eigenstates of an array of coupled superconducting qubits is a crucial step in the design of circuit quantum electrodynamics- (circuit-QED) based quantum processors. However, exact diagonalization of the device Hamiltonian is challenging for system sizes beyond tens of qubits. Here, we employ a tensor network method based on a variant of the density matrix renormalization group (DMRG) algorithm, DMRG-X, to efficiently obtain localized eigenstates of a two-dimensional transmon array without the need to first compute lower-energy states. We also introduce MTDMRG-X, a new algorithm that combines DMRG-X with multitarget DMRG to efficiently compute excited states even in regimes with strong eigenstate hybridization. We showcase the use of these methods for the analysis of long-range couplings in a multitransmon Hamiltonian including qubits and couplers, and we discuss eigenstate localization. These developments facilitate the design and parameter optimization of large-scale superconducting quantum processors.
We benchmark a variant of the recently introduced density matrix renormalization group (DMRG)-X algorithm against exact results for the localized random field XX chain. We find that the eigenstates obtained via DMRG-X exhibit a highly accurate l-bit description for system sizes much bigger than the direct, many-body, exact diagonalization in the spin variables is able to access. We take advantage of the underlying free fermion description of the XX model to accurately test the strengths and limitations of this algorithm for large system sizes. We discuss the theoretical constraints on the performance of the algorithm from the entanglement properties of the eigenstates, and its actual performance at different values of disorder. A small but significant improvement to the algorithm is also presented, which helps significantly with convergence. We find that, at high entanglement, DMRG-X shows a bias towards eigenstates with low entanglement, but can be improved with increased bond dimension. This result suggests that one must be careful when applying the algorithm for interacting many-body localized spin models near a transition.This article is part of the themed issue 'Breakdown of ergodicity in quantum systems: from solids to synthetic matter'.
The phenomenon of many-body localization generalizes Anderson localization to interacting systems. Understanding this conceptually novel phenomenon requires a study of many-body eigenstates at finite energy densities. This is a very challenging task since the most efficient numerical methods such as, e.g., the density matrix renormalization group method, can only access the ground state and low lying excitations. In this work, the authors introduce a unitary tensor network based variational method that approximately finds all many-body eigenstates of fully localized Hamiltonians and scales polynomially with system size. The usefulness of their approach is demonstrated by considering the Heisenberg chain in a strongly disordered magnetic field.
Abstract The physical properties of a quantum many-body system can, in principle, be determined by diagonalizing the respective Hamiltonian, but the dimensions of its matrix representation scale exponentially with the number of degrees of freedom. Hence, only small systems that are described through simple models can be tackled via exact diagonalization. To overcome this limitation, numerical methods based on the renormalization group paradigm that restrict the quantum many-body problem to a manageable subspace of the exponentially large full Hilbert space have been put forth. A striking example is the density-matrix renormalization group (DMRG), which has become the reference numerical method to obtain the low-energy properties of one-dimensional quantum systems with short-range interactions. Here, we provide a pedagogical introduction to DMRG, presenting both its original formulation and its modern tensor-network-based version. This colloquium sets itself apart from previous contributions in two ways. First, didactic code implementations are provided to bridge the gap between conceptual and practical understanding. Second, a concise and self-contained introduction to the tensor-network methods employed in the modern version of DMRG is given, thus allowing the reader to effortlessly cross the deep chasm between the two formulations of DMRG without having to explore the broad literature on tensor networks. We expect this pedagogical review to find wide readership among students and researchers who are taking their first steps in numerical simulations via DMRG. Graphic abstract
The success of polynomial-time tensor network methods for computing ground states of certain quantum local Hamiltonians has recently been given a sound theoretical basis by Arad et al. [Math. Phys. 356, 65 (2017)]. The convergence proof, however, relies on ``rigorous renormalization group'' (RRG) techniques which differ fundamentally from existing algorithms. We introduce a practical adaptation of the RRG procedure which, while no longer theoretically guaranteed to converge, finds matrix product state ansatz approximations to the ground spaces and low-lying excited spectra of local Hamiltonians in realistic situations. In contrast to other schemes, RRG does not utilize variational methods on tensor networks. Rather, it operates on subsets of the system Hilbert space by constructing approximations to the global ground space in a treelike manner. We evaluate the algorithm numerically, finding similar performance to density matrix renormalization group (DMRG) in the case of a gapped nondegenerate Hamiltonian. Even in challenging situations of criticality, large ground-state degeneracy, or long-range entanglement, RRG remains able to identify candidate states having large overlap with ground and low-energy eigenstates, outperforming DMRG in some cases.
One of the most fascinating issues in modern condensed matter physics is to understand highly-correlated electronic structures and propose their novel device designs toward the reduced carbon-dioxide future. Among various developed numerical approaches for highly-correlated electrons, the density matrix renormalization group (DMRG) has been widely accepted as the most promising numerical scheme compared to Monte Carlo and exact diagonalization in terms of accuracy and accessible system size. In fact, DMRG almost perfectly resolves one-dimensional chain like long quantum systems. In this paper, we suggest its extended approach toward higher-dimensional systems by high-performanc computing techniques. The computing target in DMRG is a huge non-uniform sparse matrix diagonalization. In order to efficiently parallelize the part, we implement communication step doubling together with reuse of the mid-point data between the doubled two steps to avoid severe bottleneck of all-to-all communications essential for the diagonalization. The technique is successful even for clusters composed of more than 1000 cores and offers a trustworthy exploration way for two-dimensional highly-correlated systems.
Quantum computing offers the potential for computational abilities that can go beyond classical machines. However, they are still limited by several challenges such as noise, decoherence, and gate errors. As a result, efficient classical simulation of quantum circuits is vital not only for validating and benchmarking quantum hardware but also for gaining deeper insights into the behavior of quantum algorithms. A promising framework for classical simulation is provided by tensor networks. Recently, the density-matrix renormalization group (DMRG) algorithm was developed for simulating quantum circuits using matrix product states (MPS). Although MPS are efficient for representing quantum states with one-dimensional correlation structures, the fixed linear geometry restricts the expressive power of the MPS. In this work, we extend the DMRG algorithm for simulating quantum circuits to tree tensor networks (TTNs). The framework employs a variational compression scheme that optimizes the TTN to approximate the evolved quantum state. To benchmark the method, we simulate random circuits and the quantum approximate optimization algorithm (QAOA) with various two-qubit gate connectivities. For the random circuits, we devise treelike gate layouts that are suitable for TTN and show that TTN requires less memory than MPS for the simulations. For the QAOA circuits, a naive TTN construction that exploits graph structure significantly improves the simulation fidelities. Our findings show that the DMRG algorithm with TTNs provides a promising framework for simulating quantum circuits, particularly when gate connectivities exhibit clustering or a hierarchical structure.