We study classical polynomial-time approximation algorithms for the transverse-field Ising model (TFIM) Hamiltonian, allowing a mixture of ferromagnetic and anti-ferromagnetic interactions between pairs of qbits, alongside transverse field terms with arbitrary non-negative weights.
Our main results are a series of approximation algorithms (all approximation ratios with respect to the true quantum optimum): (i) a simple maximum of two product state rounding algorithm achieving an approximation ratio
γ≈0.71 , (ii) a strengthened rounding, inspired by the anticommutation property of the two
Xi,ZiZj observables achieving ratio
γ≈0.7860, and (iii) a further improvement by interpolation achieving ratio
γ≈0.8156. We also give an explicit (purely ferromagnetic) TFIM instance on three qbits for which every product state achieves at most
169/180≈0.9389 of the true optimum, yielding an upper bound for all algorithms producing product state approximations, even in the purely ferromagnetic case.
Working within the Stochastic Series Expansion (SSE) framework, we construct efficient quantum cluster algorithms for transverse field Ising antiferromagnets on the pyrochlore lattice and the planar pyrochlore lattice, for the fully frustrated square lattice Ising model in a transverse field (dual to the 2+1 dimensional odd Ising gauge theory), and for a transverse field Ising model with multi-spin interactions on the square lattice, which is dual to a 2+1 dimensional even Ising gauge theory (and reduces to the two dimensional quantum loop model in a certain limit). Our cluster algorithms use a microcanonical update procedure that generalizes and exploits the notion of "pre-marked motifs" introduced earlier in the context of a quantum cluster algorithm for triangular lattice transverse field Ising antiferromagnets. We demonstrate that the resulting algorithms are significantly more efficient than the standard link percolation based quantum cluster approach. We also introduce a new canonical update scheme that leads to a further improvement in measurement of some observables arising from its ability to make one-dimensional clusters in the "imaginary time" direction. Finally, we demonstrate that refinements in the choice of premarking strategies can lead to additional improvements in the efficiency of the microcanonical updates. As a first example of the physics that can be studied using these algorithmic developments, we obtain evidence for a power-law ordered intermediate-temperature phase associated with the two-step melting of long-range order in the fully frustrated square lattice transverse field Ising model.
In this thesis, the Density Matrix Renormalization Group algorithm is introduced. Some key concepts from quantum mechanics are discussed to aid in understanding the method and it is put into the context of the Renormalization Group. The algorithm is discussed from a mostly practical perspective. As a model to test the algorithm, the Ising Model with a Transverse Field was used. This model is interesting because it is one of the simplest one-dimensional models that displays a phase transition at a finite external field. It can also be solved analytically, which makes it possible to easily check the results of DMRG calculations. The DMRG algorithm was implemented from scratch in C++ for the Transverse Field Ising Model. Using this implementation, the errors in DMRG calculations were explored, as well as the phase transition in the Transverse Field Ising Chain.