The scripts and data for the project are shown in three files. PEPS_contraction_lib.zip contains CTMRG, iTEBD methods for contracting the iPEPS, and also some exact diagonalization tools. PEPS_optim.zip contains code for optimizing the iPEPS, and Entanglement_spectrum_and_entropy.zip contains code for extraction of the entanglement spectrum and entropy. To use the code, one has to modify the file paths in the scripts.

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Diagonalizing large-scale quantum many-body Hamiltonians using variational quantum circuit and tensor network

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Physical review. B./Physical review. B

Zipping many-body quantum states: A scalable approach to diagonal entropy

Yu-Hsueh Chen, Tarun Grover

Published Oct 9, 2026• 0 citations

Publisher

DOI: 10.1103/85vz-n116

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Exact diagonalization (ED) provides complete access to many-body eigenenergies and eigenstates, yet its exponential cost confines it to small systems. We propose tensor network variational diagonalization (TNVD), which encodes the full eigenenergy spectrum in a matrix product state (MPS) while representing the corresponding eigenstates via a finite depth variational quantum circuit (VQC) acting on product states. TNVD thereby reduces diagonalization complexity from exponential to polynomial in system size $N$ . For the quantum Ising chain, TNVD accurately reproduces eigenenergies for $N\leq 16$ and, at sizes inaccessible to ED such as $N=100$ , directly samples them from a single MPS encoding all $2^N$ levels. A random label control shows that this compact $N$ -site representation depends on how the eigenenergies are organized in label space. TNVD further reveals that, at their integrable limits, the random field Ising and XXZ chains show comparable level spacing ratios and mean eigenstate entanglement entropies but markedly different Schmidt spectrum decay. This difference exposes distinct eigenstate entanglement structures that govern their classical simulability and the difficulty of finite depth quantum circuit preparation. Our work establishes TNVD as a scalable full spectrum diagonalization framework and its VQC as a quantum route to volume-law-entangled eigenstates that challenge efficient classical simulation.Exact diagonalization (ED) provides complete access to many-body eigenenergies and eigenstates, yet its exponential cost confines it to small systems. We propose tensor network variational diagonalization (TNVD), which encodes the full eigenenergy spectrum in a matrix product state (MPS) while representing the corresponding eigenstates via a finite depth variational quantum circuit (VQC) acting on product states. TNVD thereby reduces diagonalization complexity from exponential to polynomial in system size $N$. For the quantum Ising chain, TNVD accurately reproduces eigenenergies for $N\leq 16$ and, at sizes inaccessible to ED such as $N=100$, directly samples them from a single MPS encoding all $2^N$ levels. A random label control shows that this compact $N$-site representation depends on how the eigenenergies are organized in label space. TNVD further reveals that, at their integrable limits, the random field Ising and XXZ chains show comparable level spacing ratios and mean eigenstate entanglement entropies but markedly different Schmidt spectrum decay. This difference exposes distinct eigenstate entanglement structures that govern their classical simulability and the difficulty of finite depth quantum circuit preparation. Our work establishes TNVD as a scalable full spectrum diagonalization framework and its VQC as a quantum route to volume-law-entangled eigenstates that challenge efficient classical simulation.

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|t| \log|t|

, where

t

is the deviation from the critical point, while in a 2+1-D state with amplitudes proportional to the Boltzmann weight of the 2D Ising model, it follows a

t^2 \log|t|

scaling.

N

. For the quantum Ising chain, TNVD accurately reproduces eigenenergies for

N\leq 16

and, at sizes inaccessible to ED such as

N=100

, directly samples them from a single MPS encoding all

2^N

levels. A random label control shows that this compact

N

-site representation depends on how the eigenenergies are organized in label space. TNVD further reveals that, at their integrable limits, the random field Ising and XXZ chains show comparable level spacing ratios and mean eigenstate entanglement entropies but markedly different Schmidt spectrum decay. This difference exposes distinct eigenstate entanglement structures that govern their classical simulability and the difficulty of finite depth quantum circuit preparation. Our work establishes TNVD as a scalable full spectrum diagonalization framework and its VQC as a quantum route to volume-law-entangled eigenstates that challenge efficient classical simulation.