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Dynamical electronic- and vibrational-structure theories have received a growing interest in the last years due to their ability to simulate spectra recorded with ultrafast experimental techniques. The exact time evolution of a molecular system can, in principle, be obtained from the time-dependent version of full configuration interaction. Such an approach is, however, limited to few-atom systems due to the exponential increase of its cost with the system dimension. In the present work, we overcome this unfavorable scaling by employing the time-dependent density matrix renormalization group (TD-DMRG) which parametrizes the time-dependent wavefunction as a matrix product state. The time-dependent Schroedinger equation is then integrated with a sweep-based algorithm, as in standard time-independent DMRG. Unlike other TD-DMRG approaches, the one presented here leads to a set of coupled equations that can be integrated exactly. The resulting theory enables us to study real- and imaginary-time evolutions of Hamiltonians comprising more than 20 degrees of freedom that are challenging for current state-of-the-art quantum dynamics algorithms. We apply our algorithm to the simulation of quantum dynamics of models of increasing complexity, ranging from simple excitonic Hamiltonians to more complex ab-initio vibronic ones.
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Configuration-interaction-type calculations on electronic and vibrational structure are often the method of choice for the reliable approximation of many-particle wave functions and energies. The exponential scaling, however, limits their application range. An efficient approximation to the full configuration interaction solution can be obtained with the density matrix renormalization group (DMRG) algorithm without a restriction to a predefined excitation level. In a standard DMRG implementation, however, excited states are calculated with a ground-state optimization in the space orthogonal to all lower lying wave function solutions. A trivial parallelization is therefore not possible and the calculation of highly excited states becomes prohibitively expensive, especially in regions with a high density of states. Here, we introduce two variants of the density matrix renormalization group algorithm that allow us to target directly specific energy regions and therefore highly excited states. The first one, based on shift-and-invert techniques, is particularly efficient for low-lying states, but is not stable in regions with a high density of states. The second one, based on the folded auxiliary operator, is less efficient, but more accurate in targeting high-energy states. We apply the algorithm to the solution of the nuclear Schroedinger equation, but emphasize that it can be applied to the diagonalization of general Hamiltonians as well, such as the electronic Coulomb Hamiltonian to address X-ray spectra. In combination with several root-homing algorithms and a stochastic sampling of the determinant space, excited states of interest can be adequately tracked and analyzed during the optimization. We demonstrate that we can accurately calculate prominent spectral features of large molecules such as the sarcosyn-glycine dipeptide.
Matrix product state has become the algorithm of choice when studying one-dimensional interacting quantum many-body systems, which demonstrates to be able to explore the most relevant portion of the exponentially large quantum Hilbert space and find accurate solutions. Here we propose a quantum inspired K-means clustering algorithm which first maps the classical data into quantum states represented as matrix product states, and then minimize the loss function using the variational matrix product states method in the enlarged space. We demonstrate the performance of this algorithm by applying it to several commonly used machine learning datasets and show that this algorithm could reach higher prediction accuracies and that it is less likely to be trapped in local minima compared to the classical K-means algorithm.
Motivated by the existence of exact many-body quantum scars in the AKLT chain, we explore the connection between Matrix Product State (MPS) wavefunctions and many-body quantum scarred Hamiltonians. We provide a method to systematically search for and construct parent Hamiltonians with towers of exact eigenstates composed of quasiparticles on top of an MPS wavefunction. These exact eigenstates have low entanglement in spite of being in the middle of the spectrum, thus violating the strong Eigenstate Thermalization Hypothesis (ETH). Using our approach, we recover the AKLT chain starting from the MPS of its ground state, and we derive the most general nearest-neighbor Hamiltonian that shares the AKLT quasiparticle tower of exact eigenstates. We further apply this formalism to other simple MPS wavefunctions, and derive new families of Hamiltonians that exhibit AKLT-like quantum scars. As a consequence, we also construct a scar-preserving deformation that connects the AKLT chain to the integrable spin-1 pure biquadratic model. Finally, we also derive other families of Hamiltonians that exhibit new types of exact quantum scars, including a $U(1)$-invariant perturbed Potts model.
The similarities between Hartree-Fock (HF) theory and the density-matrix renormalization group (DMRG) are explored. Both methods can be formulated as the variational optimization of a wave-function ansatz. Linearization of the time-dependent variational principle near a variational minimum allows to derive the random phase approximation (RPA). We show that the non-redundant parametrization of the matrix product state (MPS) tangent space [J. Haegeman et al., Phys. Rev. Lett. 107, 070601 (2011)] leads to the Thouless theorem for MPS, i.e. an explicit non-redundant parametrization of the entire MPS manifold, starting from a specific MPS reference. Excitation operators are identified, which extends the analogy between HF and DMRG to the Tamm-Dancoff approximation (TDA), the configuration interaction (CI) expansion, and coupled cluster theory. For a small one-dimensional Hubbard chain, we use a CI-MPS ansatz with single and double excitations to improve on the ground state and to calculate low-lying excitation energies. For a symmetry-broken ground state of this model, we show that RPA-MPS allows to retrieve the Goldstone mode. We also discuss calculations of the RPA-MPS correlation energy. With the long-range quantum chemical Pariser-Parr-Pople Hamiltonian, low-lying TDA-MPS and RPA-MPS excitation energies for polyenes are obtained.
We demonstrate that the optimal states in lossy quantum interferometry may be efficiently simulated using low rank matrix product states. We argue that this should be expected in all realistic quantum metrological protocols with uncorrelated noise and is related to the elusive nature of the Heisenberg precision scaling in presence of decoherence.
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