Efficient Classical Simulation of Spin Networks

In general, quantum systems are believed to be exponentially hard to simulate using classical computers. It is in these hard cases where we hope to find quantum algorithms that provide speed up over classical algorithms. In the paradigm of quantum adiabatic computation, instances of spin networks with 2-local interactions could hopefuly efficiently compute certain problems in NP-complete [4]. Thus, we are interested in the adiabatic evolution of spin networks. There are analytical solutions to specific Hamiltonians for 1D spin chains [5]. However, analytical solutions to networks of higher dimensionality are unknown. The dynamics of Cayley trees (three binary trees connected at the root) at zero temperature are unknown. The running time of the adiabatic evolution of Cayley trees could provide an insight into the dynamics of more complicated spin networks. Matrix Product States (MPS) define a wavefunction anzatz that approximates slightly entangled quantum systems using poly(n) parameters. The MPS representation is exponentially smaller than the exact representation, which involves 0( 2 n) parameters. The MPS Algorithm evolves states in the MPS representation [3, 8, 2, 6, 10, 7]. We present an extension to the DMRG algorithm that computes an approximation to the adiabatic evolution of Cayley trees with rotationally-symmetric 2-local Hamiltonians in time polynomial in the depth of the tree. This algorithm takes advantage of the symmetry of the Hamiltonian to evolve the state of a Cayley tree exponentially faster than using the standard DMRG algorithm. In this thesis, we study the time-evolution of two local Hamiltonians in a spin chain and a Cayley tree. The numerical results of the modified MPS algorithm can provide an estimate on the entropy of entanglement present in ground states of Cayley trees. Furthermore, the study of the Cayley tree explores the dynamics of fractional-dimensional spin networks. Thesis Supervisor: Edward Farhi Title: Professor