[Paper Review] Least Action Principle for the Real-Time Density Matrix Renormalization Group

Kouji Ueda, Chenglong Jin+3 more|arXiv (Cornell University)|Dec 19, 2006

Quantum many-body systems3 citations

TL;DR

This paper introduces a least action principle for the real-time density matrix renormalization group (DMRG) to restore time-reversal symmetry and reduce truncation error accumulation. By minimizing a discrete action functional based on the square of the time-evolution error, the method optimizes matrix product state (MPS) representations across all time slices simultaneously via iterative, parallel updates, leading to symmetric and more accurate time evolution.

ABSTRACT

A kind of least action principle is introduced for the discrete time evolution of one-dimensional quantum lattice models. Based on this principle, we obtain an optimal condition for the matrix product states on succeeding time slices generated by the real-time density matrix renormalization group method. This optimization can also be applied to classical simulations of quantum circuits. We discuss the time reversal symmetry in the fully optimized MPS.

Motivation & Objective

• Address the time-asymmetry in real-time DMRG caused by cumulative truncation errors during time evolution.
• Recover time-reversal symmetry in numerical time evolution by formulating a variational principle over the entire time interval.
• Minimize the accumulation of numerical errors from repeated renormalization steps in time-evolving MPS.
• Enable more accurate and symmetric time evolution by optimizing MPS at all time slices simultaneously through a global variational functional.
• Provide a framework for extending real-time DMRG to include adaptive time steps, symplectic integrators, and non-local or non-invertible operators via variational reformulation.

Proposed method

• Define a discrete action functional $ I = \int \left| \left( \frac{\partial}{\partial t} - \frac{H}{i\hbar} \right) |\Psi(t)\rangle \right|^2 dt $, which quantifies the deviation from the Schrödinger equation.
• Discretize time into $ N $ intervals with step size $ \Delta t $, and express the time evolution operator as $ \mathcal{T} = \exp(\Delta t H / i\hbar) $, approximating short-time evolution.
• Use matrix product states (MPS) to represent the evolving quantum state at each time slice, with local tensors optimized iteratively.
• Apply a variational optimization that minimizes the action functional across all time slices, treating the entire time evolution as a single variational problem.
• Implement parallel sweeps forward and backward in time, enabling symmetric optimization of MPS tensors at each time point.
• Utilize transfer matrix techniques and orthogonalization schemes (e.g., $ B[s] \Lambda^B B[s] $) to dynamically adjust bond dimension $ m $ and improve accuracy.

Experimental results

Research questions

• RQ1 How can time-reversal symmetry be restored in real-time DMRG simulations where truncation errors break this symmetry?
• RQ2 What variational principle can be used to minimize the cumulative error from repeated renormalization in time-evolving MPS?
• RQ3 Can a global optimization over all time slices improve the accuracy and symmetry of real-time DMRG time evolution?
• RQ4 What is the role of the action functional $ I = \int \mathcal{L} dt $, with $ \mathcal{L} = \left| \left( \frac{\partial}{\partial t} - \frac{H}{i\hbar} \right) |\Psi(t)\rangle \right|^2 $, in stabilizing time evolution?
• RQ5 How can the method be extended to handle non-local or non-invertible operators, or to incorporate adaptive time steps and symplectic integrators?

Key findings

• The least action principle restores time-reversal symmetry in real-time DMRG by minimizing a global action functional over all time slices, reducing asymmetry in backward and forward time evolution.
• The method prevents the exponential growth of numerical errors by avoiding the accumulation of truncation errors through global variational optimization.
• The action-based variational method allows for parallel computation of forward and backward time sweeps, improving computational efficiency and symmetry.
• The formulation enables dynamic adjustment of bond dimension $ m $ via the extended renormalized wave function $ B[s_\ell] \Lambda^B B[s_{\ell+1}] $, enhancing accuracy during evolution.
• The approach is extendable to systems with long-range interactions via multi-targeting schemes, and applicable to classical statistical models and quantum circuits with invertible transfer matrices.
• The method provides a variational framework that can be adapted to include symplectic integrators, adaptive time steps, and other advanced numerical techniques in real-time DMRG.