Reweighted Time-Evolving Block Decimation for Improved Quantum Dynamics Simulations

TL;DR

TEBD-based simulations of 1D quantum dynamics with MPDOs struggle to conserve global quantities and maintain important few-body correlations due to uniform truncation. The authors introduce Reweighted TEBD (rTEBD), which reweights the MPDO in a Pauli basis by a factor $γ^{-n}$ to prioritize low-weight observables, with separate schemes for bosonic/spin and fermionic systems that preserve unitarity aside from truncation. Benchmarking on free-fermion and interacting-spin models shows that rTEBD achieves superior preservation of trace, energy density, and long-range correlations compared to MPDO-TEBD and is competitive with or better than MPS-TEBD, with optimal $γ$ typically near 1.6. The method is simple to implement and can be extended to open systems, imaginary-time evolution, and other tensor-network time-evolution approaches, offering a practical improvement for quantum dynamics simulations.

Abstract

We introduce a simple yet significant improvement to the time-evolving block decimation (TEBD) tensor network algorithm for simulating the time dynamics of strongly correlated one-dimensional (1D) mixed quantum states. The efficiency of 1D tensor network methods stems from using a product of matrices to express either: the coefficients of a wavefunction, yielding a matrix product state (MPS); or the expectation values of a density matrix, yielding a matrix product density operator (MPDO). To avoid exponential computational costs, TEBD truncates the matrix dimension while simulating the time evolution. However, when truncating a MPDO, TEBD does not favor the likely more important low-weight expectation values, such as $\langle c_i^\dagger c_j \rangle$, over the exponentially many high-weight expectation values, such as $\langle c_{i_1}^\dagger c^\dagger_{i_2} \cdots c_{i_n} \rangle$ of weight $n$, despite the critical importance of the low-weight expectation values. Motivated by this shortcoming, we propose a reweighted TEBD (rTEBD) algorithm that deprioritizes high-weight expectation values by a factor of $γ^{-n}$ during the truncation. This simple modification (which only requires reweighting certain matrices by a factor of $γ$ in the MPDO) makes rTEBD significantly more accurate than the TEBD time-dependent simulation of an MPDO, and competive with and sometimes better than TEBD using MPS. Furthermore, by prioritizing low-weight expectation values, rTEBD preserves conserved quantities to high precision.

Figures (15)

• Figure 1: Representation of a wavefunction of a 6-site qubit chain using a matrix product state.
• Figure 2: Time evolution of a MPS using TEBD. $U$ are the two-qubit unitaries that act in a Trotter decomposed Childs_2021 brickwork-like circuit.
• Figure 3: Representation of a density matrix $\rho$ in terms of a matrix product density operator (MPDO).
• Figure 4: Representation of a MPDO in the Pauli basis. The pair of dimension-2 legs of each $A_j$ matrix is combined into a single dimension-4 leg to index the 4 Pauli matrices ($\openone,\sigma^x,\sigma^y,\sigma^z$). Each pair of two-qubit unitaries, $U$ and $U^\dagger$, are also combined via tensor product, $U \otimes U^\dagger$, to obtain a super-operator that acts on a pair of $A_j$.
• Figure 5: Time evolution of a reweighted MPDO defined in Eqn. \ref{['eq:mpdo_rewt']} using rTEBD. $\Tilde{\mathcal{U}}$ are the two-qubit super-operators defined in the reweighted pauli basis (Eqn. \ref{['eq:boson_rewt']}) that act in a Trotter decomposed Childs_2021 brickwork-like circuit.