We determine the dynamical structure factors of two gapped correlated electron systems, namely the Ising model in a strong transverse field and the two-leg spin-1/2 Heisenberg ladder in the limit of strong rung coupling. We consider the low-temperature limit, employing a variety of analytical and numerical techniques. The coherent modes of single-particle excitations, which are delta functions at zero temperature, are shown to broaden asymmetrically in energy with increasing temperature. Firstly, we apply a low-temperature “resummation” inspired by the Dyson equation to a linked-cluster expansion of the two-leg Heisenberg ladder. We include matrix elements to second order in the interaction between states containing up to two particles. A low-frequency response similar to the “Villain mode” is also observed. Next, we apply a cumulant expansion technique to the transverse field Ising model. We resolve the issue of negative spectral weight caused by double pole in the leading self-energy diagram by including a resummation of terms obtained from the six-point function, demonstrating that the perturbation series in 2n-spin correlation functions can be extended to higher orders. The result generalises to higher dimensions and the analytic calculation is compared to a numerical Pade approximant. We outline the extension of this method to the strong coupling ladder. Finally, we compare the previous results to numerical data obtained by full diagonalisation of finite chains and numerical evaluation of the Pfaffian, a method specific to the transverse field Ising chain. The latter method is used for a phenomenological study of the asymmetric broadening as well as an evaluation of fitting functions for the broadened lineshapes.

<p>The recent groundbreaking demonstrations of quantum supremacy in noisy intermediate scale quantum (NISQ) computing era has triggered an intense activity in establishing finer boundaries between classical and quantum computing. In this dissertation, we use established techniques based on quantum Monte Carlo (QMC) to map quantum problems into probabilistic networks where the fundamental unit of computation, p-bit, is inherently probabilistic and can be tuned to fluctuate between ‘0’ and ‘1’ with desired probability. We can view this mapped network as a Boltzmann machine whose states each represent a Feynman path leading from an initial configuration of q-bits to a final configuration. Each such path, in general, has a complex amplitude, ψ which can be associated with a complex energy. The real part of this energy can be used to generate samples of Feynman paths in the usual way, while the imaginary part is accounted for by treating the samples as complex entities, unlike ordinary Boltzmann machines where samples are positive. This mapping of a quantum circuit onto a Boltzmann machine with complex energies should be particularly useful in view of the advent of special-purpose hardware accelerators known as Ising Machines which can obtain a very large number of samples per second through massively parallel operation. We also demonstrate this acceleration using a recently used quantum problem and speeding its QMC simulation by a factor of ∼ 1000× compared to a highly optimized CPU program. Although this speed-up has been demonstrated using a graph colored architecture in FPGA, we project another ∼ 100× improvement with an architecture that utilizes clockless analog circuits. We believe that this will contribute significantly to the growing efforts to push the boundaries of the simulability of quantum circuits with classical/probabilistic resources and comparing them with NISQ-era quantum computers. </p>