Matrix product state algorithms must enlarge their bond spaces as entanglement grows and compress them to control cost. We formulate basis-update and Galerkin (BUG) time integration as a sequence of canonical MPS sweeps for Hamiltonians represented as matrix product operators. We show when two natural basis updates produce the same trial space and when transporting coefficients between successive bases preserves the represented state. Under these conditions, the existing first-order error bound for uncompressed tree-tensor-network BUG also applies to the alternating-endpoint MPS schedule. We verify the uncompressed implementation against an independent six-site calculation. We then compare BUG with two-site TDVP for 16-site transverse-field Ising and Haldane-Shastry dynamics. At matched timestep and truncation settings, BUG performs fewer local exponential actions and has lower runtime. These settings do not produce equal accuracy. The runtime versus accuracy curves cross for the Ising model and are close for the Haldane-Shastry model. The comparison therefore identifies model-dependent trade-offs rather than a general advantage for either method.
Matrix product state algorithms must enlarge their bond spaces as entanglement grows and compress them to control cost. We formulate basis-update and Galerkin (BUG) time integration as a sequence of canonical MPS sweeps for Hamiltonians represented as matrix product operators. We show when two natural basis updates produce the same trial space and when transporting coefficients between successive bases preserves the represented state. Under these conditions, the existing first-order error bound for uncompressed tree-tensor-network BUG also applies to the alternating-endpoint MPS schedule. We verify the uncompressed implementation against an independent six-site calculation. We then compare BUG with two-site TDVP for 16-site transverse-field Ising and Haldane-Shastry dynamics. At matched timestep and truncation settings, BUG performs fewer local exponential actions and has lower runtime. These settings do not produce equal accuracy. The runtime versus accuracy curves cross for the Ising model and are close for the Haldane-Shastry model. The comparison therefore identifies model-dependent trade-offs rather than a general advantage for either method.
Time evolution of tensor-network states is a central tool for studying nonequilibrium dynamics in strongly correlated quantum systems. In the past decade, using the time-dependent variational principle (TDVP) discretized by projector-splitting has become the gold standard for evolving matrix product states (MPS) with long-range Hamiltonians. More recently, a conceptually different class of discretizations of the TDVP, the class of Basis Update and Galerkin (BUG) integrators, has been developed within the framework of dynamical low-rank approximation, offering an alternative, more flexible approach. In this work, we formulate two BUG integrators explicitly for MPS. The BUG integrators result in "single-site" MPS integrators that are robust with respect to small singular values, avoid backward-in-time substeps, and naturally allow for the adaptive choice of bond dimensions. We consider the augmented BUG integrator of first order and the midpoint BUG integrator of second order and clarify their relation to the projector-splitting discretization of the TDVP. Using representative models of strongly correlated quantum systems, like the XY spin model with long-range interaction and two-dimensional lattice models of fermions and spins of various lengths, we perform a systematic numerical comparison between BUG and projector-splitting time discretizations of the continuous-time TDVP, focusing on scalability, numerical complexity, and accuracy. We find that BUG integrators, due to their reduced numerical cost and the flexibility of the concept, provide a competitive and, in certain regimes, advantageous alternative to the well-established time-stepping by projector-splitting in TDVP, especially in models with large local dimensions, and in cases where observables are extracted with high temporal resolution.
We study low-rank tensor methods for the numerical solution of Schrödinger's equation with time-independent and explicitly time-dependent Hamiltonians, motivated by large-scale simulations of many-body quantum systems and quantum computing devices subject to time-dependent control pulses. We outline the recent application of the "basis update and Galerkin" (BUG) method for tensor trains, and describe the established TDVP and TDVP-2 algorithms based on the time-dependent variational principle. For comparison, we also consider the BUG method in the Tucker format. All these approaches enable memory efficient representations of partially entangled quantum states and thereby mitigate the exponential cost of conventional state-vector formulations. The rank-adaptivity relies on the truncated singular value decomposition, in which the rank of a matrix is reduced by setting its smallest singular values to zero, based on a threshold parameter that controls the truncation error. Numerical experiments on representative time-independent and time-dependent Hamiltonian models quantify the tradeoff between accuracy and compression across methods, with particular attention to the interplay between the time-step and the truncation threshold, and how the computational effort scales with the number of sub-systems in the quantum system.
We develop a novel time-evolution algorithm for matrix product states based on the recently introduced hybrid tensor network (hTN) framework. We retain the tensors close to the boundary on the classical computer and offload the highly entangled inner ones to the quantum computer. In our variant, we employ the Basis Update and Galerkin (BUG) integrator to time-evolve the classical tensors, and we develop a coupling scheme between the classical and quantum parts. Our framework admits modular combination with any quantum time-evolution method, such as (classically pre-optimized) Trotterization. The ratio of classical and quantum tensor degrees of freedom can be dynamically adjusted during the time evolution, which can be advantageous when the classical memory requirements become prohibitive. The quantum and classical components can run in parallel during a single time step and are not constrained by synchronization barriers or mid-circuit measurements. We describe the detailed steps and pseudocode for our algorithm specialized for tensor networks originating from the matrix product state Ansatz.
Abstract. Computing the numerical solution to high-dimensional tensor differential equations can lead to prohibitive computational costs and memory requirements. To reduce the memory and computational footprint, dynamical low-rank approximation (DLRA) has proven to be a promising approach. DLRA represents the solution as a low-rank tensor factorization and evolves the resulting low-rank factors in time. A central challenge in DLRA is to find time integration schemes that are robust to the arising small singular values. A robust parallel basis update and Galerkin integrator, which simultaneously evolves all low-rank factors, has recently been derived for matrix differential equations. This work extends the parallel low-rank matrix integrator to Tucker tensors and general tree tensor networks, yielding an algorithm in which all bases and connecting tensors are evolved in parallel over a time step. We formulate the algorithm, provide a robust error bound, and demonstrate the efficiency of the new integrators for problems in quantum many-body physics, uncertainty quantification, and radiative transfer. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://github.com/JonasKu/Publication-Parallel-BUG-for-TTNs and in the supplementary materials ( Publication-Parallel-BUG-for-TTNs-main.zip [112 KB]). [Formula: see text]
Out-of-equilibrium dynamics of non-integrable Hamiltonian many-body quantum systems are characterized by highly entangled wave functions. Near-maximal entanglement arises in systems exhibiting thermalization or pre-thermalization, where the system converges to a steady state with a fixed energy density. Classical simulation of the time dependence of such wave functions requires exponential resources. However, typical computations aim to estimate expectation values of local operators and correlation functions to some expected precision. For thermalizing systems at sufficiently high energy densities, such computations can be done without storing the full wave function by instead simulating the evolution of the local operator, which requires significantly fewer resources. Nonetheless, constructing such resource-efficient classical algorithms remains a challenge for intermediate energy densities, where simulating both the wave function and operator evolution is costly. In this paper, we propose a heuristic approach to accelerate the convergence of Matrix Product State (MPS) simulations of expectation values, applicable across a broad range of energy densities. We estimate the desired observables by rescaling the MPS results at low bond dimensions with a factor that depends on the fidelity of the MPS wave function. Using this technique, we simulated the dynamics of the two-dimensional Transverse-Field Ising Model (TFIM) on a 7×8 grid with periodic boundary conditions, using a maximum bond dimension of χ=4096 on a single A100 GPU, as well as the dynamics of the two-dimensional XY model on grids of size up to 9×9. We compare our TFIM results to similar simulations on a digital quantum processor [R. Haghshenas et al., Nature 653, 56 (2026)], demonstrating excellent agreement and confirming the predictive power of our method.
Simulating quantum many-body systems (QMBS) is one of the long-standing, highly non-trivial challenges in condensed matter physics and quantum information due to the exponentially growing size of the system's Hilbert space. To date, tensor networks have been an essential tool for studying such quantum systems, owing to their ability to efficiently capture the entanglement properties of the systems they represent. One of the well-known tensor network architectures, namely matrix product states (MPS), is the standard method for simulating one-dimensional QMBS. Here, we propose a simple, yet efficient, method to augment the already available MPS algorithms to simulate the dynamics of time-dependent Hamiltonians with better accuracy and a faster convergence rate, giving a second-order convergence compared to the first-order convergence of the standard method. We apply our proposed method to simulate the dynamics of a chain of single spins associated with nitrogen-vacancy color centers in diamonds, which has potential applications for practical and scalable quantum technologies, and find that our method improves the average error for a system of few NV centers by a factor of about 1000 for moderate step sizes. Our work paves the way for efficient simulation of QMBS under the influence of time-dependent Hamiltonians.