Compressed simulation of thermal and excited states of the one-dimensionalXYmodel

Abstract

Since several years, the preparation and manipulation of a small number of quantum systems in a controlled and coherent way is feasible in many experiments. In fact, these experiments are nowadays commonly used for quantum simulation and quantum computation. As recently shown, such a system can, however, also be utilized to simulate specific behaviors of exponentially larger systems. That is, certain quantum computations can be performed by an exponentially smaller quantum computer. This compressed quantum computation can be employed to observe, for instance, the quantum phase transition of the one-dimensional (1D) $XY$ model using very few qubits. We extend here this notion to simulate the behavior of thermal as well as excited states of the 1D $XY$ model. In particular, we consider the 1D $XY$ model of a spin chain of $n$ qubits and derive a quantum circuit processing only $log(n)$ qubits which simulates the original system. We demonstrate how the behavior of thermal as well as any eigenstate of the system can be efficiently simulated in this compressed fashion and present a quantum circuit on $log(n)$ qubits to measure the magnetization, the number of kinks, and correlations occurring in the thermal as well as any excited state of the original systems. Moreover, we derive compressed circuits to study time evolutions.