Cavity approach to variational quantum mechanics

Abstract

A local and distributive algorithm is proposed to find an optimal trial wave-function minimizing the Hamiltonian expectation in a quantum system. To this end, the quantum state of the system is connected to the Gibbs state of a classical system with the set of couplings playing the role of variational parameters. The average energy is written within the replica-symmetric approximation and the optimal parameters are obtained by a heuristic message passing algorithm based on the Bethe approximation. The performance of this approximate algorithm depends on the structure and quality of the trial wave-functions; starting from a classical system of isolated elements, i.e. mean-field approximation, and improving on that by considering the higher order many-body interactions. The method is applied to some disordered quantum Ising models in transverse fields and the results are compared with the exact ones for small systems.

16 pages, 4 figures; one appendix and two figures added

References in corpus (10)

• Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems
• Computational complexity and fundamental limitations to fermionic quantum Monte Carlo simulations
• The power of quantum systems on a line
• Quantum Graphical Models and Belief Propagation
• Quantum Belief Propagation
• On the path integral representation for quantum spin models and its application to the quantum cavity method and to Monte Carlo simulations
• Cavity method for quantum spin glasses on the Bethe lattice
• Belief propagation algorithm for computing correlation functions in finite-temperature quantum many-body systems on loopy graphs
• The stochastic matching problem
• Stochastic optimization by message passing

Cited by in corpus (6)

• The Quantum Adiabatic Algorithm applied to random optimization problems: the quantum spin glass perspective
• Sign problem in the Bethe approximation
• Low-temperature excitations within the Bethe approximation
• Quantum Cluster Variational Method and Message Passing Algorithms Revisited
• Bethe free-energy approximations for disordered quantum systems
• Multilayer wave functions: A recursive coupling of local excitations