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arXiv:2005.06829 · Physical Review Research

Thermal pure quantum matrix product states recovering a volume law entanglement

Atsushi Iwaki, Akira Shimizu, Chisa Hotta

Published May 25, 2021• 26 citations
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Abstract

We propose a way to construct a thermal pure quantum matrix product state (TPQ-MPS) that can simulate finite-temperature quantum many-body systems with a minimal numerical cost comparable to the matrix product algorithm for the ground state.The MPS was originally designed for the wave function with area law entanglement.However, by attaching the auxiliary sites to the edges of the random matrix product state, we find that the degree of entanglement is automatically tuned so as to recover the volume law of the entanglement entropy that characterizes the TPQ state.The finite-temperature physical quantities of the transverse Ising and the spin-1/2 Heisenberg chains evaluated by a TPQ-MPS show excellent agreement even for bond dimension ∼10-20 with those of the exact results.

DOI
10.1103/physrevresearch.3.l022015

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We propose a way to construct a thermal pure quantum matrix product state (TPQ-MPS) that can simulate finite-temperature quantum many-body systems with a minimal numerical cost comparable to the matrix product algorithm for the ground state.The MPS was originally designed for the wave function with area law entanglement.However, by attaching the auxiliary sites to the edges of the random matrix product state, we find that the degree of entanglement is automatically tuned so as to recover the volume law of the entanglement entropy that characterizes the TPQ state.The finite-temperature physical quantities of the transverse Ising and the spin-1/2 Heisenberg chains evaluated by a TPQ-MPS show excellent agreement even for bond dimension ∼10-20 with those of the exact results.
Abstract We present several features of a thermal pure quantum matrix product state (TPQ-MPS). TPQ-MPS can simulate finite temperature quantum many-body systems with a numerical cost comparable to the matrix product algorithm for the ground state in one-dimensional systems. Starting from a random matrix product state with auxiliary sites attached to the edges of the system, we successively operate the Hamiltonian with a shift-constant and obtain a series of TPQ-MPS that represents the microcanonical pure state at the highest temperature to the one at the lowest available temperature. How the parameters of the TPQ-MPS protocol influence the effective bond dimension χ of the matrix and the temperature distribution of these microcanonical states is examined.
Out-of-equilibrium dynamics of non-integrable Hamiltonian many-body quantum systems are characterized by highly entangled wave functions. Near-maximal entanglement arises in systems exhibiting thermalization or pre-thermalization, where the system converges to a steady state with a fixed energy density. Classical simulation of the time dependence of such wave functions requires exponential resources. However, typical computations aim to estimate expectation values of local operators and correlation functions to some expected precision. For thermalizing systems at sufficiently high energy densities, such computations can be done without storing the full wave function by instead simulating the evolution of the local operator, which requires significantly fewer resources. Nonetheless, constructing such resource-efficient classical algorithms remains a challenge for intermediate energy densities, where simulating both the wave function and operator evolution is costly. In this paper, we propose a heuristic approach to accelerate the convergence of Matrix Product State (MPS) simulations of expectation values, applicable across a broad range of energy densities. We estimate the desired observables by rescaling the MPS results at low bond dimensions with a factor that depends on the fidelity of the MPS wave function. Using this technique, we simulated the dynamics of the two-dimensional Transverse-Field Ising Model (TFIM) on a 7×87\times87×8 grid with periodic boundary conditions, using a maximum bond dimension of χ=4096χ= 4096χ=4096 on a single A100 GPU, as well as the dynamics of the two-dimensional XY model on grids of size up to 9×99\times99×9. We compare our TFIM results to similar simulations on a digital quantum processor [R. Haghshenas et al., Nature 653, 56 (2026)], demonstrating excellent agreement and confirming the predictive power of our method.
We present the first successful application of the matrix product state (MPS) representing a thermal quantum pure state (TPQ) in equilibrium in two spatial dimensions over almost the entire temperature range. We use the Kitaev honeycomb model as a prominent example hosting a quantum spin liquid (QSL) ground state to target the two specific-heat peaks previously solved nearly exactly using the free Majorana fermionic description. Starting from the high-temperature random state, our TPQ-MPS framework on a cylinder precisely reproduces these peaks, showing that the quantum many-body description based on spins can still capture the emergent itinerant Majorana fermions in a {\mathbb Z}_2 ℤ 2 gauge field. The truncation process efficiently discards the high-energy states, eventually reaching the long-range entangled topological state approaching the exact ground state for a given finite size cluster. An advantage of TPQ-MPS over exact diagonalization or purification-based methods is its lowered numerical cost coming from a reduced effective Hilbert space even at finite temperature.
A standard approach to generate random pure quantum states relies on sampling from the Haar measure. However, the entanglement properties of such states present a fundamental challenge for their general applicability. Here, we introduce the σσσ-ensembles $\unicode{x2013}$ a family of random quantum states with only a single control parameter. Crucially, these states are designed such that they can be tuned between volume-law and area-law behavior, which has been a major obstacle thus far. We construct representatives of this ensemble by imposing a probability distribution on the eigenvalues of the successive subsystems, and subsequently reconstructing a compatible global state using the matrix product state (MPS) formalism. Due to their area-law entanglement, our approach circumvents the intractability of Haar-random pure states in classical simulations of quantum systems and is more representative of typical Hamiltonian ground states.
Entanglement is the hallmark of quantum physics, yet its characterization in interacting many-body systems at thermal equilibrium remains one of the most important challenges in quantum statistical physics. We prove that the Gibbs state of any quantum spin chain can be exactly decomposed into a mixture of matrix product states with a bond dimension that is independent of the system size, at any finite temperature. As a consequence, the Schmidt number, arguably the most stringent measure of bipartite entanglement, is strictly finite for thermal states, even in the thermodynamic limit. Our decomposition is explicit and is accompanied by an efficient classical algorithm to sample the resulting matrix product states.
Strong entanglement limits the reach of tensor-network descriptions of quantum many-body systems. We study Gaussian-augmented matrix product states (GAMPS), formed by applying a free-fermionic (Gaussian) unitary to a matrix product state. To explore the capabilities of this family, we consider random GAMPS}, a minimally structured ensemble obtained by independently sampling random matrix product states and Gaussian unitaries. Exploiting the algebraic structure of Gaussian unitaries, we develop a replica tensor-network method that computes averages over the random GAMPS ensemble for hundreds of qubits. Random GAMPS exhibit nearly maximal volume-law entanglement already at modest bond dimension, while other quantum resource quantifiers rapidly approach Haar-random state values as the bond dimension increases. The entanglement scaling suggests an exponential reduction in the bond dimension needed to approach the volume-law entropy compared with unaugmented matrix product states. We find a state two-design behavior of random GAMPS at fixed accuracy with a bond dimension independent of system size. Our results reveal how Gaussian augmentation expands the expressive power of tensor networks.