[Paper Review] Time-dependent variational principle with controlled bond expansion for matrix product states

Jheng-Wei Li, Andreas Gleis+1 more|arXiv (Cornell University)|Aug 23, 2022

Numerical methods for differential equations92 references4 citations

TL;DR

This paper introduces Controlled Bond Expansion (CBE) for matrix product states (MPS) within the time-dependent variational principle (TDVP), enabling dynamic bond dimension adaptation during time evolution to reduce projection error while maintaining the efficiency and stability of one-site TDVP. CBE–TDVP achieves 2-site TDVP accuracy at nearly one-site computational cost, enabling long-time, accurate simulations of quantum dynamics with controlled error growth.

ABSTRACT

We present a controlled bond expansion (CBE) approach to simulate quantum dynamics based on the time-dependent variational principle (TDVP) for matrix product states. Our method alleviates the numerical difficulties of the standard, fixed-rank one-site TDVP integrator by increasing bond dimensions on the fly to reduce the projection error. This is achieved in an economical, local fashion, requiring only minor modifications of standard one-site TDVP implementations. We illustrate the performance of CBE--TDVP with several numerical examples on finite quantum lattices.

Motivation & Objective

• To address the fixed-rank limitation of standard one-site TDVP (1TDVP), which cannot track entanglement growth during time evolution.
• To develop a rank-adaptive integrator that maintains the numerical stability, unitarity, and energy conservation of 1TDVP while improving accuracy.
• To achieve 2-site TDVP-level accuracy with computational costs comparable to 1TDVP, avoiding the high overhead of traditional adaptive schemes.
• To control the TDVP projection error by dynamically expanding the MPS bond dimension using a localized, economical scheme.

Proposed method

• Introduce Controlled Bond Expansion (CBE) to identify and add subspaces missed by 1TDVP that contain significant weight from $H\Psi$, reducing projection error.
• Modify the standard 1TDVP algorithm by allowing bond dimensions to increase on the fly, using the CBE scheme from Ref. [54] to guide expansion.
• Use the tangent space projector $\mathcal{P}^{1\mathrm{s}}(t)$ to project $H\Psi(t)$ onto the MPS manifold, with dynamically adjusted bond dimensions.
• Apply the CBE scheme to compute the truncated complement $\widetilde{A}^{\mathrm{tr}}_{\ell}$, which captures components of $H\Psi$ orthogonal to the current MPS tangent space.
• Implement the method in a local, economical way requiring only minor modifications to standard 1TDVP code.
• Control bond expansion via a threshold $\widetilde{\epsilon}$ on singular values of the effective Hamiltonian, ensuring error remains bounded.

Experimental results

Research questions

• RQ1 Can bond dimension be adaptively increased during 1TDVP time evolution to reduce projection error without sacrificing computational efficiency?
• RQ2 Does CBE–TDVP achieve 2-site TDVP accuracy while preserving the unitary evolution and energy conservation of 1TDVP?
• RQ3 How does the error in CBE–TDVP evolve over long times, and can it be kept under control despite dynamic bond growth?
• RQ4 What is the computational cost of CBE–TDVP relative to 2TDVP, and does it scale favorably with system size and physical dimension $d$?
• RQ5 Can CBE–TDVP simulate symmetry-broken initial states (e.g., domain walls) that 1TDVP fails to capture due to fixed-rank constraints?

Key findings

• CBE–TDVP achieves a plateau in fidelity error $\delta F(\bar{t})$ of $6.7 \times 10^{-5}$ over long times, indicating controlled and non-accumulating error.
• Even with a relatively large expansion threshold $\widetilde{\epsilon} = 10^{-2}$, the error remains bounded and does not grow rapidly, demonstrating robustness.
• For $D_{\mathrm{max}} = 120$, the fidelity error plateaus at $6.7 \times 10^{-5}$, and increasing $D_{\mathrm{max}}$ to infinity only modestly reduces this value.
• CBE–TDVP outperforms 1TDVP on a domain wall initial state: while 1TDVP fails completely (fidelity drops to zero), CBE–TDVP maintains high fidelity.
• CBE–TDVP reduces CPU time by orders of magnitude compared to 2TDVP—achieving 1.5 time units in under a day, while 2TDVP requires two days for the same accuracy.
• The bond dimension growth in CBE–TDVP is slow and sub-exponential, unlike the rapid saturation seen in 2TDVP, leading to significant cost savings despite $D^3$ scaling.