Abstract Quantum embedding theories are powerful tools for approximately solving large‐scale, strongly correlated quantum many‐body problems. The main idea of quantum embedding is to glue together a highly accurate quantum theory at the local scale and a less accurate quantum theory at the global scale. We introduce the first quantum embedding theory that is also variational, in that it is guaranteed to provide a one‐sided bound for the exact ground‐state energy. Our method, which we call the variational embedding method, provides a lower bound for this quantity. The method relaxes the representability conditions for quantum marginals to a set of linear and semidefinite constraints that operate at both local and global scales, resulting in a semidefinite program (SDP) to be solved numerically. The accuracy of the method can be systematically improved. The method is versatile and can be applied, in particular, to quantum many‐body problems for both quantum spin systems and fermionic systems, such as those arising from electronic structure calculations. We describe how the proper notion of quantum marginal, sufficiently general to accommodate both of these settings, should be phrased in terms of certain algebras of operators. We also investigate the duality theory for our SDPs, which offers valuable perspective on our method as an embedding theory. As a byproduct of this investigation, we describe a formulation for efficiently implementing the variational embedding method via a partial dualization procedure and the solution of quantum analogues of the Kantorovich problem from optimal transport theory. © 2021 Wiley Periodicals LLC.Abstract Quantum embedding theories are powerful tools for approximately solving large‐scale, strongly correlated quantum many‐body problems. The main idea of quantum embedding is to glue together a highly accurate quantum theory at the local scale and a less accurate quantum theory at the global scale. We introduce the first quantum embedding theory that is also variational, in that it is guaranteed to provide a one‐sided bound for the exact ground‐state energy. Our method, which we call the variational embedding method, provides a lower bound for this quantity. The method relaxes the representability conditions for quantum marginals to a set of linear and semidefinite constraints that operate at both local and global scales, resulting in a semidefinite program (SDP) to be solved numerically. The accuracy of the method can be systematically improved. The method is versatile and can be applied, in particular, to quantum many‐body problems for both quantum spin systems and fermionic systems, such as those arising from electronic structure calculations. We describe how the proper notion of quantum marginal, sufficiently general to accommodate both of these settings, should be phrased in terms of certain algebras of operators. We also investigate the duality theory for our SDPs, which offers valuable perspective on our method as an embedding theory. As a byproduct of this investigation, we describe a formulation for efficiently implementing the variational embedding method via a partial dualization procedure and the solution of quantum analogues of the Kantorovich problem from optimal transport theory. © 2021 Wiley Periodicals LLC.

arXiv:1910.00560Communications on Pure and Applied MathematicsFeb 27, 202110 citations

Variational Optimization of Many-Body Wave Functions

Ryo Maezono

Springer eBooksNov 8, 20250 citations

Stochastic Variational Approach to Quantum-Mechanical Few-Body Problems

Y. Suzuki, Kálmán Varga

Springer eBooksJan 1, 1998464 citations

Beyond binarity: Semidefinite programming for ternary quadratic problems

Frank de Meijer, Veronica Piccialli, Renata Sotirov +1 more

We study the ternary quadratic problem (TQP), a quadratic optimization problem with linear constraints where the variables take values in $\{0, \pm 1\}$ . While semidefinite programming (SDP) techniques are well established for $\{0,1\}$ - and $\{\pm 1\}$ -valued quadratic problems, no dedicated integer semidefinite programming framework exists for the ternary case. In this paper, we introduce a ternary SDP formulation for the TQP that forms the basis of an exact solution approach. We derive new theoretical insights in rank-one ternary positive semidefinite matrices, which lead to a basic SDP relaxation that is further strengthened by valid triangle, RLT, split and $k$ -gonal inequalities. These are embedded in a tailored branch-and-bound algorithm that iteratively solves strengthened SDPs, separates violated inequalities, applies a ternary branching strategy and computes high-quality feasible solutions. We test our algorithm on TQP variations motivated by practice, including unconstrained, linearly constrained and quadratic ratio problems. Computational results on these instances demonstrate the effectiveness of the proposed algorithm.We study the ternary quadratic problem (TQP), a quadratic optimization problem with linear constraints where the variables take values in $\{0, \pm 1\}$. While semidefinite programming (SDP) techniques are well established for $\{0,1\}$- and $\{\pm 1\}$-valued quadratic problems, no dedicated integer semidefinite programming framework exists for the ternary case. In this paper, we introduce a ternary SDP formulation for the TQP that forms the basis of an exact solution approach. We derive new theoretical insights in rank-one ternary positive semidefinite matrices, which lead to a basic SDP relaxation that is further strengthened by valid triangle, RLT, split and $k$-gonal inequalities. These are embedded in a tailored branch-and-bound algorithm that iteratively solves strengthened SDPs, separates violated inequalities, applies a ternary branching strategy and computes high-quality feasible solutions. We test our algorithm on TQP variations motivated by practice, including unconstrained, linearly constrained and quadratic ratio problems. Computational results on these instances demonstrate the effectiveness of the proposed algorithm.

arXiv:2603.28979Mar 30, 20260 citations

We study the ternary quadratic problem (TQP), a quadratic optimization problem with linear constraints where the variables take values in

\{0, \pm 1\}

. While semidefinite programming (SDP) techniques are well established for

\{0,1\}

- and

\{\pm 1\}

-valued quadratic problems, no dedicated integer semidefinite programming framework exists for the ternary case. In this paper, we introduce a ternary SDP formulation for the TQP that forms the basis of an exact solution approach. We derive new theoretical insights in rank-one ternary positive semidefinite matrices, which lead to a basic SDP relaxation that is further strengthened by valid triangle, RLT, split and

k

-gonal inequalities. These are embedded in a tailored branch-and-bound algorithm that iteratively solves strengthened SDPs, separates violated inequalities, applies a ternary branching strategy and computes high-quality feasible solutions. We test our algorithm on TQP variations motivated by practice, including unconstrained, linearly constrained and quadratic ratio problems. Computational results on these instances demonstrate the effectiveness of the proposed algorithm.