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arXiv:2309.02519 · Physical review. B./Physical review. B

Efficient simulation of low-temperature physics in one-dimensional gapless systems

Yuya Kusuki, Kotaro Tamaoka, Zixia Wei, Yasushi Yoneta

Published Jul 29, 2024• 0 citations
arXivHTMLPDF

Abstract

We discuss the computational efficiency of the finite-temperature simulation with minimally entangled typical thermal states (METTS). To argue that METTS can be efficiently represented as matrix product states, we present an analytic upper bound for the average entanglement R\'enyi entropy of METTS for a R\'enyi index $0<q\ensuremath{\le}1$. In particular, for one-dimensional (1D) gapless systems described by conformal field theories, the upper bound scales as $O(c{N}^{0}log\ensuremath{\beta})$ where $c$ is the central charge and $N$ is the system size. Furthermore, we numerically find that the average R\'enyi entropy exhibits a universal behavior characterized by the central charge and is roughly given by half of the analytic upper bound. Based on these results, we show that METTS can provide a speedup compared to employing the purification method to analyze thermal equilibrium states at low temperatures in 1D gapless systems.

DOI
10.1103/physrevb.110.l041122

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We discuss the computational efficiency of the finite-temperature simulation with minimally entangled typical thermal states (METTS). To argue that METTS can be efficiently represented as matrix product states, we present an analytic upper bound for the average entanglement R'enyi entropy of METTS for a R'enyi index 0<q\ensuremath≤10<q\ensuremath{\le}10<q\ensuremath≤1. In particular, for one-dimensional (1D) gapless systems described by conformal field theories, the upper bound scales as O(cN0log\ensuremathβ)O(c{N}^{0}log\ensuremath{\beta})O(cN0log\ensuremathβ) where ccc is the central charge and NNN is the system size. Furthermore, we numerically find that the average R'enyi entropy exhibits a universal behavior characterized by the central charge and is roughly given by half of the analytic upper bound. Based on these results, we show that METTS can provide a speedup compared to employing the purification method to analyze thermal equilibrium states at low temperatures in 1D gapless systems.
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We discuss a method based on sampling minimally entangled typical thermal states (METTS) that can simulate finite temperature quantum systems with a computational cost comparable to ground state DMRG. Detailed implementations of each step of the method are presented, along with efficient algorithms for working with matrix product states and matrix product operators. We furthermore explore how properties of METTS can reveal characteristic order and excitations of systems and discuss why METTS form an efficient basis for sampling. Finally, we explore the extent to which the average entanglement of a METTS ensemble is minimal.
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Minimally Entangled Typical Thermal States for Classical and Quantum Simulation of 1+1-Dimensional Z2\mathbb Z_2Z2​ Lattice Gauge Theory at Finite Temperature and Density
Simulating strongly coupled gauge theories at finite temperature and density is a longstanding challenge in nuclear and high-energy physics that also has fundamental implications for condensed matter physics. In this work, we use minimally entangled typical thermal state (METTS) approaches to facilitate both classical and quantum computational studies of such systems. METTS techniques combine classical random sampling with imaginary time evolution, which can be performed on either a classical or a quantum computer, to estimate thermal averages of observables. We study 1+1-dimensional Z2\mathbb{Z}_2Z2​ gauge theory coupled to spinless fermionic matter, which maps onto a local quantum spin chain. We benchmark both a classical matrix-product-state implementation of METTS and a recently proposed adaptive variational approach that is a promising candidate for implementation on near-term quantum devices, focusing on the equation of state as well as on various measures of fermion confinement. Of particular importance is the choice of basis for obtaining new METTS samples, which impacts both the classical sampling complexity (a key factor in both classical and quantum simulation applications) and complexity of circuits used in the quantum computing approach. Our work sets the stage for future studies of strongly coupled gauge theories with both classical and quantum hardware.