We ask whether the knowledge of a single eigenstate of a local Hamiltonian is sufficient to uniquely determine the Hamiltonian. We present evidence that the answer is yes" for generic local Hamiltonians, given either the ground state or an excited eigenstate. In fact, knowing only the two-point equal-time correlation functions of local observables with respect to the eigenstate should generically be sufficient to exactly recover the Hamiltonian for finite-size systems, with numerical algorithms that run in a time that is polynomial in the system size. We also investigate the large-system limit, the sensitivity of the reconstruction to error, and the case when correlation functions are only known for observables on a fixed sub-region. Numerical demonstrations support the results for finite one-dimensional spin chains (though caution must be taken when extrapolating to infinite-size systems in higher dimensions). For the purpose of our analysis, we define the k -correlation spectrum" of a state, which reveals properties of local correlations in the state and may be of independent interest.
Recovering an unknown Hamiltonian from measurements is an increasingly important task for certification of noisy quantum devices and simulators. Recent works have succeeded in recovering the Hamiltonian of an isolated quantum system with local interactions from long-ranged correlators of a single eigenstate. Here, we show that such Hamiltonians can be recovered from local observables alone, using computational and measurement resources scaling linearly with the system size. In fact, to recover the Hamiltonian acting on each finite spatial domain, only observables within that domain are required. The observables can be measured in a Gibbs state as well as a single eigenstate; furthermore, they can be measured in a state evolved by the Hamiltonian for a long time, allowing us to recover a large family of time-dependent Hamiltonians. We derive an estimate for the statistical recovery error due to approximation of expectation values using a finite number of samples, which agrees well with numerical simulations.
Abstract Local Hamiltonians arise naturally in physical systems. Despite their seemingly ‘simple’ local structures, exotic features such as non-local correlations and topological orders exhibit in eigenstates of these systems. Previous studies for recovering local Hamiltonians from measurements on an eigenstate ψ require information of nonlocal correlation functions. In this work, we argue that local measurements on ψ is enough to recover the Hamiltonian in most of the cases. Specially, we develop an algorithm to demonstrate the observation. Our algorithm is tested numerically for randomly generated local Hamiltonians of different system sizes and returns promising reconstructions with desired accuracy. Additionally, for random generated Hamiltonians (not necessarily local), our algorithm also provides precise estimations.
This paper addresses the so-called inverse problem which consists in searching for (possibly multiple) parent target Hamiltonian(s), given a single quantum state as input. Starting from ${\mathrm{\ensuremath{\Psi}}}_{0}$, an eigenstate of a given local Hamiltonian
H0, we ask whether or not there exists another parent Hamiltonian
HP for ${\mathrm{\ensuremath{\Psi}}}_{0}$, with the same local form as
H0. Focusing on one-dimensional quantum disordered systems, we extend the recent results obtained for Bose-glass ground states [M. Dupont and N. Laflorencie, Phys. Rev. B 99, 020202(R) (2019)] to Anderson localization, and the many-body localization (MBL) physics occurring at high energy. We generically find that any localized eigenstate is a very good approximation for an eigenstate of a distinct parent Hamiltonian, with an energy variance ${\ensuremath{\sigma}}_{\mathrm{P}}^{2}(L)={\ensuremath{\langle}{\mathcal{H}}_{\mathrm{P}}^{2}\ensuremath{\rangle}}_{{\mathrm{\ensuremath{\Psi}}}_{0}}\ensuremath{-}{\ensuremath{\langle}{\mathcal{H}}_{\mathrm{P}}\ensuremath{\rangle}}_{{\mathrm{\ensuremath{\Psi}}}_{0}}^{2}$ vanishing as a power law of system size
L. This decay is microscopically related to a chain-breaking mechanism, also signaled by bottlenecks of vanishing entanglement entropy. A similar phenomenology is observed for both Anderson and MBL. In contrast, delocalized ergodic many-body eigenstates uniquely encode the Hamiltonian in the sense that ${\ensuremath{\sigma}}_{\mathrm{P}}^{2}(L)$ remains finite at the thermodynamic limit, i.e., $L\ensuremath{\rightarrow}+\ensuremath{\infty}$. As a direct consequence, the ergodic-MBL transition can be very well captured from the scaling of ${\ensuremath{\sigma}}_{\mathrm{P}}^{2}(L)$.
Abstract Recent works have shown that generic local Hamiltonians can be efficiently inferred from local measurements performed on their eigenstates or thermal states. Realistic quantum systems are often affected by dissipation and decoherence due to coupling to an external environment. This raises the question whether the steady states of such open quantum systems contain sufficient information allowing for full and efficient reconstruction of the system’s dynamics. We find that such a reconstruction is possible for generic local Markovian dynamics. We propose a recovery method that uses only local measurements; for systems with finite-range interactions, the method recovers the Lindbladian acting on each spatial domain using only observables within that domain. We numerically study the accuracy of the reconstruction as a function of the number of measurements, type of open-system dynamics and system size. Interestingly, we show that couplings to external environments can in fact facilitate the reconstruction of Hamiltonians composed of commuting terms.
Computing ground states of local Hamiltonians is a fundamental problem in condensed matter physics. The problem is known to be QMA-complete, even for one-dimensional Hamiltonians [1]. This means that we do not even expect that there is a sub-exponential size description of the ground state that allows efficient computation of local observables such as the energy. In sharp contrast, the heuristic density matrix renormalization group (DMRG) algorithm invented two decades ago [5] has been remarkably successful in practice on one-dimensional problems. The situation is reminiscent of the unexplained success of the simplex algorithm before the advent of ellipsoid and interior-point methods. Is there a principled explanation for this, in the form of a large class of one-dimensional Hamiltonians whose ground states can be provably efficiently approximated? Here we give such an algorithm for gapped one-dimensional Hamiltonians: our algorithm outputs an (inverse-polynomial) approximation to the ground state, expressed as a matrix product state (MPS) of polynomial bond dimension. The running time of the algorithm is polynomial in the number of qudits n and the approximation quality δ, for a fixed local dimension d and gap Δ > 0.
Physical properties of the ground and excited states of a
k-local Hamiltonian are largely determined by the
k-particle reduced density matrices (
k-RDMs), or simply the
k-matrix for fermionic systems---they are at least enough for the calculation of the ground-state and excited-state energies. Moreover, for a nondegenerate ground state of a
k-local Hamiltonian, even the state itself is completely determined by its
k-RDMs, and therefore contains no genuine
>k-particle correlations, as they can be inferred from
k-particle correlation functions. It is natural to ask whether a similar result holds for nondegenerate excited states. In fact, for fermionic systems, it has been conjectured that any nondegenerate excited state of a 2-local Hamiltonian is simultaneously a unique ground state of another 2-local Hamiltonian, hence is uniquely determined by its 2-matrix. And a weaker version of this conjecture states that any nondegenerate excited state of a 2-local Hamiltonian is uniquely determined by its 2-matrix among all the pure
n-particle states. We construct explicit counterexamples to show that both conjectures are false. We further show that any nondegenerate excited state of a
k-local Hamiltonian is a unique ground state of another
2k-local Hamiltonian, hence is uniquely determined by its
2k-RDMs (or
2k-matrix). These results set up a solid framework for the study of excited-state properties of many-body systems.