• Corpus ID: 249063174

Perturbation Theory and the Sum of Squares

  title={Perturbation Theory and the Sum of Squares},
  author={Matthew B. Hastings},
The sum-of-squares (SoS) hierarchy is a powerful technique based on semi-definite programming that can be used for both classical and quantum optimization problems. This hierarchy goes under several names; in particular, in quantum chemistry it is called the reduced density matrix (RDM) method. We consider the ability of this hierarchy to reproduce weak coupling perturbation theory for three different kinds of systems: spin (or qubit) systems, bosonic systems (the anharmonic oscil-lator), and… 
1 Citations



Optimizing strongly interacting fermionic Hamiltonians

Among other results, this work gives an efficient classical certification algorithm for upper-bounding the largest eigenvalue in the q=4 SYK model, and an efficient quantum Certification algorithm for lower- bounding this largest eigensvalue; both algorithms achieve constant-factor approximations with high probability.


Abstract : A new approach is presented to the many-particle problem in quantum mechanics by proposing a method of finding natural orbitals and natural geminals of a system without prior knowledge of

Uncertainty relations and reduced density matrices: Mapping many-body quantum mechanics onto four particles

For the description of ground-state correlation phenomena an accurate mapping of many-body quantum mechanics onto four particles is developed. The energy for a quantum system with no more than

Quantum marginal problem and N-representability

A fermionic version of the quantum marginal problem was known from the early sixties as N-representability problem. In 1995 it was mentioned by the National Research Council of the USA as one of ten

The role of model systems in the few‐body reduction of the N‐fermion problem

The problem of upper and lower ground state energy bounds for many-fermion systems is considered from the viewpoint of reduced density matrices. Model density matrices are used for upper bounds to,

Variational calculations of fermion second-order reduced density matrices by semidefinite programming algorithm

The ground-state fermion second-order reduced density matrix (2-RDM) is determined variationally using itself as a basic variable using the positive semidefiniteness conditions, P, Q, and G conditions that are described in terms of the 2-R DM.

Gapless spin-fluid ground state in a random quantum Heisenberg magnet.

  • SachdevYe
  • Physics
    Physical review letters
  • 1993
The spin-S quantum Heisenberg magnet with Gaussian-random, infinite-range exchange interactions is examined with generalizing to SU(M) symmetry and studying the large M limit to find the spin-fluid phase to be generically gapless.

The algebraic degree of semidefinite programming

Given a generic semidefinite program, specified by matrices with rational entries, each coordinate of its optimal solution is an algebraic number. We study the degree of the minimal polynomials of

Maximizing quadratic programs: extending Grothendieck's inequality

  • M. CharikarA. Wirth
  • Computer Science
    45th Annual IEEE Symposium on Foundations of Computer Science
  • 2004
The approximation algorithm for this type of quadratic programming problem uses the canonical semidefinite relaxation and returns a solution whose ratio to the optimum is in /spl Omega/(1/ logn), which can be seen as an extension to that of maximizing x/sup T/Ay.

A positivstellensatz for non-commutative polynomials

A non-commutative polynomial which is positive on a bounded semi-algebraic set of operators has a weighted sum of squares representation. This Positivstellensatz parallels similar results in the