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arXiv:2408.13616 · Physical Review Letters

Unusual Energy Spectra of Matrix Product States

J. Maxwell Silvester, Giuseppe Carleo, Steven R. White

Published Mar 26, 2025• 4 citations
arXivHTMLPDF

Abstract

In approximate ground states obtained from imaginary-time evolution, the spectrum of the state-its decomposition into exact energy eigenstates-falls off exponentially with the energy. Here we consider the energy spectra of approximate matrix product ground states, such as those obtained with the density matrix renormalization group. Despite the high accuracy of these states, contributions to the spectra are roughly constant out to surprisingly high energy, with an increase in the bond dimension reducing the amplitude but not the extent of these high-energy tails. The unusual spectra appear to be a general feature of compressed wavefunctions, independent of boundary or dimensionality, and are also observed in neural network wavefunctions. The unusual spectra can have a strong effect on sampling-based methods, yielding large fluctuations. The energy variance, which can be used to extrapolate observables to eliminate truncation error, is subject to these large fluctuations when sampled. Nevertheless, we devise a sampling-based variance approach which gives excellent and efficient extrapolations.

DOI
10.1103/physrevlett.134.126503

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In approximate ground states obtained from imaginary-time evolution, the spectrum of the state-its decomposition into exact energy eigenstates-falls off exponentially with the energy. Here we consider the energy spectra of approximate matrix product ground states, such as those obtained with the density matrix renormalization group. Despite the high accuracy of these states, contributions to the spectra are roughly constant out to surprisingly high energy, with an increase in the bond dimension reducing the amplitude but not the extent of these high-energy tails. The unusual spectra appear to be a general feature of compressed wavefunctions, independent of boundary or dimensionality, and are also observed in neural network wavefunctions. The unusual spectra can have a strong effect on sampling-based methods, yielding large fluctuations. The energy variance, which can be used to extrapolate observables to eliminate truncation error, is subject to these large fluctuations when sampled. Nevertheless, we devise a sampling-based variance approach which gives excellent and efficient extrapolations.
The density-matrix renormalization-group method is very effective at finding ground states of one-dimensional (1D) quantum systems in practice, but it is a heuristic method, and there is no known proof for when it works. In this article we describe an efficient classical algorithm which provably finds a good approximation of the ground state of 1D systems under well-defined conditions. More precisely, our algorithm finds a matrix product state of bond dimension DDD whose energy approximates the minimal energy such states can achieve. The running time is exponential in DDD, and so the algorithm can be considered tractable even for DDD, which is logarithmic in the size of the chain. The result also implies trivially that the ground state of any local commuting Hamiltonian in 1D can be approximated efficiently; we improve this to an exact algorithm.
We show how to efficiently simulate pure quantum states in one dimensional systems that have both finite energy density and vanishingly small energy fluctuations. We do so by studying the performance of a tensor network algorithm that produces matrix product states whose energy variance decreases as the bond dimension increases. Our results imply that variances as small as ∝1/log⁡N can be achieved with polynomial bond dimension. With this, we prove that there exist states with a very narrow support in the bulk of the spectrum that still have moderate entanglement entropy, in contrast with typical eigenstates that display a volume law. Our main technical tool is the Berry-Esseen theorem for spin systems, a strengthening of the central limit theorem for the energy distribution of product states. We also give a simpler proof of that theorem, together with slight improvements in the error scaling, which should be of independent interest.
We show that, in certain circumstances, exact excitation energies appear as locally site-independent (or flat) modes if one records the excitation spectrum of the effective Hamiltonian while sweeping through the lattice in the variational matrix-product-state formulation of the density matrix renormalization group, a remarkable property since the effective Hamiltonian is only constructed to target the ground state. Conversely, modes that are very flat over several consecutive iterations are systematically found to correspond to faithful excitations. We suggest to use this property to extract accurate information about excited states using the standard ground-state algorithm. The results are spectacular for critical systems, for which the low-energy conformal tower of states can be obtained very accurately at essentially no additional cost, as demonstrated by confirming the predictions of boundary conformal field theory for two simple minimal models: the transverse-field Ising model and the critical three-state Potts model. This approach is also very efficient to detect the quasidegenerate low-energy excitations in topological phases and to identify localized excitations in systems with impurities. Finally, using the variance of the Hamiltonian as a criterion, we assess the accuracy of the resulting matrix-product-state representations of the excited states.
A key feature of ground states of gapped local 1D Hamiltonians is their relatively low entanglement --- they are well approximated by matrix product states (MPS) with bond dimension scaling polynomially in the length N of the chain, while general states require a bond dimension scaling exponentially. We show that the bond dimension of these MPS approximations can be improved to a constant, independent of the chain length, if we relax our notion of approximation to be more local: for all length- k segments of the chain, the reduced density matrices of our approximations are ϵ -close to those of the exact state. If the state is a ground state of a gapped local Hamiltonian, the bond dimension of the approximation scales like (k/ϵ)1+o(1) , and at the expense of worse but still poly (k,1/ϵ) scaling of the bond dimension, we give an alternate construction with the additional features that it can be generated by a constant-depth quantum circuit with nearest-neighbor gates, and that it applies generally for any state with exponentially decaying correlations. For a completely general state, we give an approximation with bond dimension exp⁡(O(k/ϵ)) , which is exponentially worse, but still independent of N . Then, we consider the prospect of designing an algorithm to find a local approximation for ground states of gapped local 1D Hamiltonians. When the Hamiltonian is translationally invariant, we show that the ability to find O(1) -accurate local approximations to the ground state in T(N) time implies the ability to estimate the ground state energy to O(1) precision in O(T(N)log⁡(N)) time.
I revisit the infinite-size variant of the Density Matrix Renormalization Group (iDMRG) algorithm for obtaining a fixed-point translationally invariant matrix product wavefunction in the context of one-dimensional quantum systems. A crucial ingredient of this algorithm is an efficient transformation for obtaining the matrix elements of the wavefunction as the lattice size is increased, and I introduce here a versatile transformation that is demonstrated to be much more effective than previous versions. The resulting algorithm has a surprisingly close relationship to Vidal's Time Evolving Block Decimation for infinite systems, but allows much faster convergence. Access to a translationally invariant matrix product state allows the calculation of correlation functions based on the transfer matrix, which directly gives the spectrum of all correlation lengths. I also show some advantages of the Matrix Product Operator (MPO) technique for constructing expectation values of higher moments, such as the exact variance $$.
We introduce an error measure for matrix-product states without requiring the relatively costly two-site density-matrix renormalization group (2DMRG). This error measure is based on an approximation of the full variance $\ensuremath{\langle}\ensuremath{\psi}|{(\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}\ensuremath{-}E)}^{2}|\ensuremath{\psi}\ensuremath{\rangle}$. When applied to a series of matrix-product states at different bond dimensions obtained from a single-site density-matrix renormalization group (1DMRG) calculation, it allows for the extrapolation of observables towards the zero-error case representing the exact ground state of the system. The calculation of the error measure is split into a sequential part of cost equivalent to two calculations of $\ensuremath{\langle}\ensuremath{\psi}|\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}|\ensuremath{\psi}\ensuremath{\rangle}$ and a trivially parallelized part scaling like a single operator application in 2DMRG. The reliability of this error measure is demonstrated by four examples: the L=30,S=1/2L=30,S=1/2L=30,S=1/2 Heisenberg chain, the L=50L=50L=50 Hubbard chain, an electronic model with long-range Coulomb-like interactions, and the Hubbard model on a cylinder with a size of $10\ifmmode\times\else\texttimes\fi{}4$. Extrapolation in this error measure is shown to be on par with extrapolation in the 2DMRG truncation error or the full variance $\ensuremath{\langle}\ensuremath{\psi}|{(\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}\ensuremath{-}E)}^{2}|\ensuremath{\psi}\ensuremath{\rangle}$ at a fraction of the computational effort.