Tensor-network variational diagonalization of quantum many-body spectra

Abstract

Complete many-body spectra encode thermodynamics, dynamical response, and quantum chaos, yet their exponential size places them beyond enumeration. We introduce tensor-network variational diagonalization (TNVD), which learns a binary eigenstate labeling, represents the resulting energy tensor as a spectrum matrix product state, and encodes the associated diagonalizing transformation in a finite-depth circuit. Together, these objects define a joint-compressibility problem, with polynomial contraction cost at fixed tensor resources. TNVD reproduces Ising spectra and samples the density of states from an MPS encoding of \(2^{100}\) levels; a random-label control reveals that the energy tensor's virtual entanglement depends critically on label organization. Across random-field Ising and XXZ chains, disorder drives both systems away from their clean integrable limits, yet similar level-statistics crossovers coexist with markedly different TNVD errors. In the harder XXZ case, low-energy discarded weights decay more slowly and the ground-state Schmidt tail is heavier, identifying Schmidt truncation -- rather than spectral chaos alone -- as the finite-resource bottleneck. TNVD thus provides a direct test of when complete spectra admit tractable tensor-network representations.