[Paper Review] Efficient simulation of many-body localized systems

Yichen Huang|arXiv (Cornell University)|Aug 19, 2015

Quantum many-body systems48 references3 citations

TL;DR

This paper presents a rigorous, polynomial-time classical algorithm to efficiently simulate finite-energy-density properties in one-dimensional many-body localized (MBL) systems using matrix product operators (MPOs). By leveraging a logarithmic light cone condition and energy filtering, it constructs a diagonal density operator supported on a microcanonical ensemble with inverse polynomial bandwidth, enabling efficient simulation of MBL dynamics and phase transitions without relying on strong assumptions like local integrals of motion.

ABSTRACT

An efficient numerical method is developed using the matrix product formalism for computing the properties at finite energy densities in one-dimensional (1D) many-body localized (MBL) systems. Arguing that any efficient (possibly quantum) algorithm can only have a polynomially small energy resolution, we propose a (rigorous) polynomial-time (classical) algorithm that outputs a diagonal density operator supported on a microcanonical ensemble of an inverse polynomial bandwidth. The proof uses no other conditions for MBL but assumes that the effect of any local perturbation (e.g., injecting conserved charges) is restricted to a region whose radius grows logarithmically with time. A non-optimal version of this algorithm efficiently simulates the quantum phase estimation algorithm in 1D MBL systems; a heuristic version of the algorithm can be easily coded and used to, e.g., detect energy-tuned dynamical quantum phase transitions between MBL phases. We extend the algorithm to two and higher spatial dimensions using the projected entangled pair formalism.

Motivation & Objective

• To develop an efficient classical algorithm for computing properties of 1D MBL systems at finite energy densities.
• To overcome the exponential Hilbert space scaling that limits exact diagonalization.
• To provide a rigorous simulation method based on minimal assumptions, specifically the logarithmic light cone condition.
• To extend the method to two and higher dimensions using projected entangled pair operators (PEPOs).
• To enable detection of energy-tuned dynamical quantum phase transitions in MBL phases.

Proposed method

• The algorithm simulates real-time evolution using the matrix product state (MPS) formalism, which efficiently captures local dynamics under the logarithmic light cone condition.
• It applies an energy filtering technique that constructs an interference pattern to select eigenstates within an inverse polynomial energy bandwidth.
• The method outputs a diagonal density operator in MPO form, representing a microcanonical ensemble of eigenstates with controlled energy resolution.
• A heuristic variant uses the TEBD algorithm for time evolution, which is efficient but non-rigorous due to truncation errors.
• For higher dimensions, the algorithm constructs a PEPO representation of the diagonal density operator with quasi-polynomial bond dimension.
• The approach avoids reliance on strong MBL characterizations such as local integrals of motion or area-law entanglement, instead assuming only the logarithmic spread of local perturbations.

Experimental results

Research questions

• RQ1 Can a classical algorithm efficiently simulate MBL systems at finite energy densities without requiring exact eigenstates?
• RQ2 What minimal physical condition on information propagation enables efficient classical simulation of MBL systems?
• RQ3 How can one construct a diagonal density operator with inverse polynomial energy resolution in MBL systems using tensor network methods?
• RQ4 Can the algorithm detect dynamical quantum phase transitions in MBL phases through energy-tuned observables?
• RQ5 Is it possible to extend the MPO-based simulation to two and higher dimensions using PEPOs with controlled computational cost?

Key findings

• The algorithm runs in polynomial time and produces a diagonal density operator in MPO form that approximates a microcanonical ensemble with inverse polynomial energy resolution.
• The method is rigorously justified under the assumption that local perturbations spread logarithmically in time, a minimal condition for MBL.
• The heuristic version of the algorithm successfully simulates the quantum phase estimation algorithm in 1D MBL systems, enabling detection of energy-tuned dynamical quantum phase transitions.
• In two and higher dimensions, the algorithm constructs a PEPO representation of the diagonal density operator with quasi-polynomial bond dimension, though physical observables cannot yet be computed efficiently from this PEPO.
• The approach does not require assumptions such as local integrals of motion or area-law entanglement, making it applicable under a broader, more fundamental condition.
• The algorithm demonstrates that even exact eigenstates cannot be efficiently prepared due to exponentially small level spacing, justifying the use of inverse polynomial resolution as a practical and rigorous target.