Approximating the Ground State of a Many-Particle Quantum System with Semi-Definite Relaxations

Numerical methods represent an essential ingredient for the study of many-body systems. A paradigmatic problem is to compute the ground state of a many-particle quantum system whose interactions are described by a Hamiltonian H. The goal is to identify the quantum state that minimizes the energy. In general, an analytic solution to the problem is out of reach as there are many parameters involved in the optimization. The common approach, then, consists of variational methods in which one first defines a physically motivated ansatz for the ground state that depends on much fewer parameters and then optimizes over these parameters. These methods provide the searched solution if the ground state happens to be part of the variational ansatz or a hopefully good upper bound to the ground-state energy otherwise. Density-matrix-renormalization-group (DMRG) algorithms [1] are one of the most popular examples of this variational approach.

During the last decade, different relaxations of the above Hamiltonian minimization problem have been proposed. Interestingly, they provide lower bound to the ground-state energy, complementing the upper bounds that are obtainable using variational methods. These algorithms can be understood as lower levels of a general hierarchy of semi-definite programming (SDP) relaxations for non-commutative polynomial optimization developed recently in [2]. Recent applications of this approach to Hamiltonian problems involving coupled fermions can be found in [3,4] for one-dimensional systems. The goal of this activity is to test the ultimate power of these SDP relaxations. In particular, we plan to tackle the much more interesting and challenging case of two-dimensional systems, considering interacting fermions or more exotic systems as in [5].

Our method hinges on a conversion from a computationally hard problem to a semidefinite programming (SDP) relaxation that is easily solved on a cluster. The conversion is a difficult problem that requires symbolic calculations. We identified two libraries, SymPy for Python and SymbolicC++ for C++. We developed code in these two languages to solve the problem of translation, and we made it available under GNU Public License version 3. The two implementations are hosted at the following links:

These implementations bypass any other supporting libraries that are commonly used in converting and formatting optimization problems. The problem domains where polynomial optimization problems of noncommuting variables arrive from are quantum physics and quantum correlations. Such problems are replete with sparse structures in the constraints, and to ensure high efficiency, we had to address this sparsity. Thus the conversion engine does not use third-party helper libraries, apart from the ones used for noncommuting symbolic operations. We achieved a scalability up to a hundred noncommuting variables in problems where the number of constraints increases quadratically in the number of variables. This is the size of the problems we would like to solve.

This work is supported by the European Commission Seventh Framework Programme under Grant Agreement Number FP7-601138 PERICLES, by the Red Española de Supercomputación grants number FI-2013-1-0008 and  FI-2013-3-0004, and by the Swedish National Infrastructure for Computingproject number SNIC 2014/2-7.

[1] S. White, Phys. Rev. Lett. 69, 2863 (1992).
[2] S. Pironio, M. Navascues, A. Acin, SIAM J. Optim. 20, 5, 2157 (2010).
[3] T. Baumgratz, M. B. Plenio, New J. Phys. 14, 023027 (2012).
[4] T. Barthel, R. Hübener, Phys. Rev. Lett. 108, 200404 (2012).
[5] A. E. Feiguin, E. Rezayi, C. Nayak, S. Das Sarma, Phys. Rev. Lett. 100, 166803 (2008).