To compute approximate solutions for combinatorial optimization problems, we describe variational methods based on the product state (PS) and matrix product state (MPS) ansätze. We perform variational energy minimization with respect to a quantum annealing Hamiltonian and utilize randomness by embedding the approaches in the metaheuristic iterated local search (ILS). The resulting quantum-inspired ILS algorithms are benchmarked on maximum cut problems of up to 50000 variables. We show that they can outperform traditional (M)PS methods, classical ILS, the quantum approximate optimization algorithm and other variational quantum-inspired solvers.To compute approximate solutions for combinatorial optimization problems, we describe variational methods based on the product state (PS) and matrix product state (MPS) ansätze. We perform variational energy minimization with respect to a quantum annealing Hamiltonian and utilize randomness by embedding the approaches in the metaheuristic iterated local search (ILS). The resulting quantum-inspired ILS algorithms are benchmarked on maximum cut problems of up to 50000 variables. We show that they can outperform traditional (M)PS methods, classical ILS, the quantum approximate optimization algorithm and other variational quantum-inspired solvers.

quant-pharXiv:2512.20613Physical Review ResearchDec 24, 20250 citations

Matrix Product State Representations for Machine Learning

Eric M. Howard, Iftekher S. Chowdhury, Ian Nagle

Lecture notes in networks and systemsJan 1, 20211 citation

Clustering using matrix product states

Xiao Shi, Yun Shang, Chu Guo

The matrix product state has been demonstrated to be able to explore the most relevant portion of the exponentially large quantum Hilbert space and find accurate solutions for one-dimensional interacting quantum many-body systems. Inspired by this success, here we propose a clustering algorithm based on the matrix product state, which first maps the classical data into quantum states represented as matrix product states, and then minimizes the loss function using a variational matrix product states algorithm in the enlarged space. We demonstrate this algorithm by applying it to several commonly used machine learning data sets, showing that this algorithm could reach higher learning precision and that it is less likely to be trapped in local minima compared to the standard K-means algorithm. We also show that this algorithm can achieve state-of-the-art learning precision on popular computer vision data sets when used in combination with better initialization schemes.The matrix product state has been demonstrated to be able to explore the most relevant portion of the exponentially large quantum Hilbert space and find accurate solutions for one-dimensional interacting quantum many-body systems. Inspired by this success, here we propose a clustering algorithm based on the matrix product state, which first maps the classical data into quantum states represented as matrix product states, and then minimizes the loss function using a variational matrix product states algorithm in the enlarged space. We demonstrate this algorithm by applying it to several commonly used machine learning data sets, showing that this algorithm could reach higher learning precision and that it is less likely to be trapped in local minima compared to the standard K-means algorithm. We also show that this algorithm can achieve state-of-the-art learning precision on popular computer vision data sets when used in combination with better initialization schemes.

Physical Review AMay 18, 20228 citations

Continuous matrix product states for nonrelativistic quantum fields: A lattice algorithm for inhomogeneous systems

Martin Ganahl, Guifré Vidal

By combining the continuous matrix product state (cMPS) representation for quantum fields in the continuum with standard optimization techniques for matrix product states (MPS) on the lattice, we obtain an approximation $|\mathrm{\ensuremath{\Psi}}\ensuremath{\rangle}$, directly in the continuum, of the ground state of nonrelativistic quantum field theories. This construction works both for translation-invariant systems and in the more challenging context of inhomogeneous systems, as we demonstrate for an interacting bosonic field in a periodic potential. Given the continuum Hamiltonian $H$ , we consider a sequence of discretized Hamiltonians ${{H({\ensuremath{\epsilon}}_{\ensuremath{\alpha}})}}_{\ensuremath{\alpha}=1,2,\ensuremath{\cdots},p}$ on increasingly finer lattices with lattice spacing ${\ensuremath{\epsilon}}_{1}>{\ensuremath{\epsilon}}_{2}>\ensuremath{\cdots}>{\ensuremath{\epsilon}}_{p}$. We first use energy minimization to optimize an MPS approximation $|\mathrm{\ensuremath{\Psi}}({\ensuremath{\epsilon}}_{1})\ensuremath{\rangle}$ for the ground state of $H({\ensuremath{\epsilon}}_{1})$. Given the MPS $|\mathrm{\ensuremath{\Psi}}({\ensuremath{\epsilon}}_{\ensuremath{\alpha}})\ensuremath{\rangle}$ optimized for the ground state of $H({\ensuremath{\epsilon}}_{\ensuremath{\alpha}})$, we use it to initialize the energy minimization for Hamiltonian $H({\ensuremath{\epsilon}}_{\ensuremath{\alpha}+1})$, resulting in the optimized MPS $|\mathrm{\ensuremath{\Psi}}({\ensuremath{\epsilon}}_{\ensuremath{\alpha}+1})\ensuremath{\rangle}$. By iteration we produce an optimized MPS $|\mathrm{\ensuremath{\Psi}}({\ensuremath{\epsilon}}_{p})\ensuremath{\rangle}$ for the ground state of $H({\ensuremath{\epsilon}}_{p})$, from which we finally extract the cMPS approximation $|\mathrm{\ensuremath{\Psi}}\ensuremath{\rangle}$ for the ground state of $H$ . Two key ingredients of our proposal are as follows: (i) a procedure to discretize $H$ into a lattice model where each site contains a two-dimensional vector space (spanned by vacuum $|0\ensuremath{\rangle}$ and one boson $|1\ensuremath{\rangle}$ states), and (ii) a procedure to map MPS representations from a coarser lattice to a finer lattice.By combining the continuous matrix product state (cMPS) representation for quantum fields in the continuum with standard optimization techniques for matrix product states (MPS) on the lattice, we obtain an approximation $|\mathrm{\ensuremath{\Psi}}\ensuremath{\rangle}$, directly in the continuum, of the ground state of nonrelativistic quantum field theories. This construction works both for translation-invariant systems and in the more challenging context of inhomogeneous systems, as we demonstrate for an interacting bosonic field in a periodic potential. Given the continuum Hamiltonian $H$, we consider a sequence of discretized Hamiltonians ${{H({\ensuremath{\epsilon}}_{\ensuremath{\alpha}})}}_{\ensuremath{\alpha}=1,2,\ensuremath{\cdots},p}$ on increasingly finer lattices with lattice spacing ${\ensuremath{\epsilon}}_{1}>{\ensuremath{\epsilon}}_{2}>\ensuremath{\cdots}>{\ensuremath{\epsilon}}_{p}$. We first use energy minimization to optimize an MPS approximation $|\mathrm{\ensuremath{\Psi}}({\ensuremath{\epsilon}}_{1})\ensuremath{\rangle}$ for the ground state of $H({\ensuremath{\epsilon}}_{1})$. Given the MPS $|\mathrm{\ensuremath{\Psi}}({\ensuremath{\epsilon}}_{\ensuremath{\alpha}})\ensuremath{\rangle}$ optimized for the ground state of $H({\ensuremath{\epsilon}}_{\ensuremath{\alpha}})$, we use it to initialize the energy minimization for Hamiltonian $H({\ensuremath{\epsilon}}_{\ensuremath{\alpha}+1})$, resulting in the optimized MPS $|\mathrm{\ensuremath{\Psi}}({\ensuremath{\epsilon}}_{\ensuremath{\alpha}+1})\ensuremath{\rangle}$. By iteration we produce an optimized MPS $|\mathrm{\ensuremath{\Psi}}({\ensuremath{\epsilon}}_{p})\ensuremath{\rangle}$ for the ground state of $H({\ensuremath{\epsilon}}_{p})$, from which we finally extract the cMPS approximation $|\mathrm{\ensuremath{\Psi}}\ensuremath{\rangle}$ for the ground state of $H$. Two key ingredients of our proposal are as follows: (i) a procedure to discretize $H$ into a lattice model where each site contains a two-dimensional vector space (spanned by vacuum $|0\ensuremath{\rangle}$ and one boson $|1\ensuremath{\rangle}$ states), and (ii) a procedure to map MPS representations from a coarser lattice to a finer lattice.

arXiv:1801.02219Physical review. B./Physical review. BNov 5, 201820 citations

Quantum encoding of functions and images with matrix product states

Josh Green, Jingbo B. Wang

Physical Review AFeb 3, 20260 citations

H

, we consider a sequence of discretized Hamiltonians ${{H({\ensuremath{\epsilon}}_{\ensuremath{\alpha}})}}_{\ensuremath{\alpha}=1,2,\ensuremath{\cdots},p}$ on increasingly finer lattices with lattice spacing ${\ensuremath{\epsilon}}_{1}>{\ensuremath{\epsilon}}_{2}>\ensuremath{\cdots}>{\ensuremath{\epsilon}}_{p}$. We first use energy minimization to optimize an MPS approximation $|\mathrm{\ensuremath{\Psi}}({\ensuremath{\epsilon}}_{1})\ensuremath{\rangle}$ for the ground state of $H({\ensuremath{\epsilon}}_{1})$. Given the MPS $|\mathrm{\ensuremath{\Psi}}({\ensuremath{\epsilon}}_{\ensuremath{\alpha}})\ensuremath{\rangle}$ optimized for the ground state of $H({\ensuremath{\epsilon}}_{\ensuremath{\alpha}})$, we use it to initialize the energy minimization for Hamiltonian $H({\ensuremath{\epsilon}}_{\ensuremath{\alpha}+1})$, resulting in the optimized MPS $|\mathrm{\ensuremath{\Psi}}({\ensuremath{\epsilon}}_{\ensuremath{\alpha}+1})\ensuremath{\rangle}$. By iteration we produce an optimized MPS $|\mathrm{\ensuremath{\Psi}}({\ensuremath{\epsilon}}_{p})\ensuremath{\rangle}$ for the ground state of $H({\ensuremath{\epsilon}}_{p})$, from which we finally extract the cMPS approximation $|\mathrm{\ensuremath{\Psi}}\ensuremath{\rangle}$ for the ground state of

H

. Two key ingredients of our proposal are as follows: (i) a procedure to discretize

H

into a lattice model where each site contains a two-dimensional vector space (spanned by vacuum $|0\ensuremath{\rangle}$ and one boson $|1\ensuremath{\rangle}$ states), and (ii) a procedure to map MPS representations from a coarser lattice to a finer lattice.