ABSTRACT The exact solution of strongly correlated systems is a computationally demanding problem, scaling exponentially with system size. Over the decades, several approximate methods have been developed to address this scaling problem. Among these, matrix product state (MPS) has emerged as physically intuitive and polynomially scaling. Methods such as density matrix renormalization group (DMRG), time evolution block decimation (TEBD), etc., use the underlying MPS wavefunction ansatz to optimize it variationally. MPS can also be utilized as a machine learning model to select the most important electronic configurations, as employed in selected configuration interactions. This work uses machine learning (ML) optimization of MPS wavefunctions to eliminate the need for multiple diagonalizations during the optimization process. We compare the performance of the machine‐learned MPS‐assisted selected configuration interaction approach, often mentioned as a classifier, to the machine‐learned MPS wavefunction ansatz itself. While showing the viability of the machine‐learned MPS ansatz, it is observed that within ML, MPS is best used as a selected configuration interaction.
ABSTRACT The exact solution of strongly correlated systems is a computationally demanding problem, scaling exponentially with system size. Over the decades, several approximate methods have been developed to address this scaling problem. Among these, matrix product state (MPS) has emerged as physically intuitive and polynomially scaling. Methods such as density matrix renormalization group (DMRG), time evolution block decimation (TEBD), etc., use the underlying MPS wavefunction ansatz to optimize it variationally. MPS can also be utilized as a machine learning model to select the most important electronic configurations, as employed in selected configuration interactions. This work uses machine learning (ML) optimization of MPS wavefunctions to eliminate the need for multiple diagonalizations during the optimization process. We compare the performance of the machine‐learned MPS‐assisted selected configuration interaction approach, often mentioned as a classifier, to the machine‐learned MPS wavefunction ansatz itself. While showing the viability of the machine‐learned MPS ansatz, it is observed that within ML, MPS is best used as a selected configuration interaction.
Machine learning (ML) has been used to optimize the matrix product state (MPS) ansatz for the wavefunction of strongly correlated systems. The ML optimization of MPS has been tested for the Heisenberg Hamiltonian on one-dimensional and ladder lattices, which correspond to conjugated molecular systems. The input descriptors and output for the supervised ML are lattice configurations and configuration interaction coefficients, respectively. Efficient learning can be achieved from data over the full Hilbert space via exact diagonalization or full configuration interaction, as well as over a much smaller sub-space via Monte Carlo Configuration Interaction. We show that this circumvents the need to calculate energy and operator expectation values and is therefore a computationally efficient alternative to variational optimization.
Matrix-product states (MPS) have proven to be a versatile ansatz for modeling quantum many-body physics. For many applications, and particularly in one-dimension, they capture relevant quantum correlations in many-body wave functions while remaining tractable (polynomial scaling) to store and manipulate on a classical computer. This has motivated researchers to also apply the MPS ansatz to machine learning (ML) problems where capturing complex correlations in datasets is also a key requirement. Here, for the first time, we develop an MPS-based algorithm, MPSTime, for learning a joint probability distribution underlying a time-series dataset, and show how it can be used to tackle important time-series ML problems, including classification and imputation. MPSTime can efficiently learn complicated time-series probability distributions directly from data, requires only moderate maximum MPS bond dimension χ max (with values for our applications ranging between χ max = 20 − 160 ), and can be trained for both classification and imputation tasks under a single logarithmic loss function. Using synthetic and publicly available real-world datasets—spanning applications in medicine, energy, and astronomy—we demonstrate performance competitive with state-of-the-art ML approaches, but with the key advantage of encoding the full joint probability distribution learned from the data, which is useful for analyzing and interpreting its underlying structure. This manuscript is supplemented with the release of a publicly available code package that implements our approach, and can be used to reproduce all presented results. The effectiveness of the MPS-based ansatz for capturing complex correlation structures in time-series data makes it a powerful foundation for tackling challenging time-series analysis problems across science, industry, and medicine.
We present a new wavefunction ansatz that combines the technique of spin projection with the language of matrix product states (MPS) and matrix product operators (MPO) as used in the density matrix renormalization group (DMRG). Specifically, spin-projected matrix product states (SP-MPS) are constructed as |Ψ^(N,S,M)_SP−MPS⟩=P_S|Ψ(N,M)_MPS⟩, where P_S is the spin projector for total spin S and |Ψ(N,M)MPS⟩ is an MPS wavefunction with a given particle number N and spin projection M. This new ansatz possesses several attractive features: (1) It provides a much simpler route to achieve spin-adaptation (i.e. to create eigenfunctions of Ŝ^2) compared to explicitly incorporating the non-Abelian SU(2) symmetry into the MPS. In particular, since the underlying state |Ψ(N,M)MPS⟩ in the SP-MPS uses only Abelian symmetries, one does not need the singlet embedding scheme for non-singlet states, as normally employed in spin-adapted DMRG, to achieve a single consistent variationally optimized state. (2) SP-MPS are closely connected to broken symmetry mean-field states. This allows to straightforwardly generate the large number of broken symmetry guesses needed to explore complex electronic landscapes in magnetic systems. Further, this connection can be exploited in the future development of quantum embedding theories for open-shell systems. (3) The sum of MPOs representation for the Hamiltonian and spin projector P_S naturally leads to an embarrassingly parallel algorithm for computing expectation values and optimizing SP-MPS. (4) Optimizing SP-MPS belongs to the variation-after-projection (VAP) class of spin projected theories. Unlike usual spin projected theories based on determinants, the SP-MPS ansatz can be made essentially exact simply by increasing the bond dimensions in |Ψ(N,M)MPS⟩. Computing excited states is also simple by imposing orthogonality constraints, which are simple to implement with MPS. To illustrate the versatility of SP-MPS, we formulate algorithms for the optimization of ground and excited states, develop perturbation theory based on SP-MPS, and describe how to evaluate spin-independent and spin-dependent properties such as the reduced density matrices. We demonstrate the numerical performance of SP-MPS with applications to several models typical of strong correlation, including the Hubbard model, and [2Fe-2S] and [4Fe-4S] model complexes.
We present a new wavefunction ansatz that combines the strengths of spin projection with the language of matrix product states (MPS) and matrix product operators (MPO) as used in the density matrix renormalization group (DMRG). Specifically, spin-projected matrix product states (SP-MPS) are constructed as ∣ΨSP−MPS(N,S,M)⟩=PS∣ΨMPS(N,M)⟩, where PS is the spin projector for total spin S and ∣ΨMPS(N,M)⟩ is an MPS wavefunction with a given particle number N and spin projection M. This new ansatz possesses several attractive features: (1) It provides a much simpler route to achieve spin-adaptation (i.e. to create eigenfunctions of S^2) compared to explicitly incorporating the non-Abelian SU(2) symmetry into the MPS. In particular, since the underlying state in the SP-MPS uses only Abelian symmetries, one does not need the singlet embedding scheme for non-singlet states, as normally employed in spin-adapted DMRG, to achieve a single consistent variationally optimized state. (2) Due to the use of ∣ΨMPS(N,M)⟩ as its underlying state, the SP-MPS can be closely connected to broken symmetry mean-field states. This allows to straightforwardly generate the large number of broken symmetry guesses needed to explore complex electronic landscapes in magnetic systems. Further, this connection can be exploited in the future development of quantum embedding theories for open-shell systems. (3) The sum of MPOs representation for the Hamiltonian and spin projector PS naturally leads to an embarrassingly parallel algorithm for computing expectation values and optimizing SP-MPS. (4) Optimizing SP-MPS belongs to the variation-after-projection (VAP) class of spin projected theories.
During the past 15 years, the density matrix renormalization group (DMRG) has become increasingly important for ab initio quantum chemistry. The underlying matrix product state (MPS) ansatz is a low-rank decomposition of the full configuration interaction tensor. The virtual dimension of the MPS controls the size of the corner of the many-body Hilbert space that can be reached. Whereas the MPS ansatz will only yield an efficient description for noncritical one-dimensional systems, it can still be used as a variational ansatz for other finite-size systems. Rather large virtual dimensions are then required. The two most important aspects to reduce the corresponding computational cost are a proper choice and ordering of the active space orbitals, and the exploitation of the symmetry group of the Hamiltonian. By taking care of both aspects, DMRG becomes an efficient replacement for exact diagonalization in quantum chemistry. DMRG and Hartree-Fock theory have an analogous structure. The former can be interpreted as a self-consistent mean-field theory in the DMRG lattice sites, and the latter in the particles. It is possible to build upon this analogy to introduce post-DMRG methods. Based on an approximate MPS, these methods provide improved ansätze for the ground state, as well as for excitations. Exponentiation of the single-particle (single-site) excitations for a Slater determinant (an MPS with open boundary conditions) leads to the Thouless theorem for Hartree-Fock theory (DMRG), an explicit nonredundant parameterization of the entire manifold of Slater determinants (MPS wavefunctions). This gives rise to the configuration interaction expansion for DMRG. The Hubbard-Stratonovich transformation lies at the basis of auxiliary field quantum Monte Carlo for Slater determinants. An analogous transformation for spin-lattice Hamiltonians allows to formulate a promising variant for MPSs.
Neural-network quantum states (NNQS) provide a flexible variational framework for many-electron wave functions, but their performance in quantum chemistry depends strongly on whether the ansatz encodes the physical structure of the electronic Hilbert space. In this work, we introduce symmetry-blocked matrix product states (MPS) as neural-network quantum-state ansatzes, denoted QiankunNet-MPS, for ab initio quantum chemistry, explicitly enforcing the U(1) ⊗ U(1) particle-number symmetry of electronic Hamiltonians to eliminate unphysical configurations and reduce the number of variational parameters. We further develop batched autoregressive sampling for canonical MPS representations and a two-site sweeping optimization scheme that combines stochastic energy gradients with singular value decomposition (SVD)-based bond adaptation. Benchmarks on small molecules show ground-state energies comparable with density matrix renormalization group (DMRG) at the same bond dimensions, while Fe2S2 calculations illustrate how DMRG-initialized bond expansion and variance extrapolation can be used to assess the large-bond-dimension trend in a strongly correlated transition-metal active space.