  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 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 $\mathrm{log}(n)$ 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 $\mathrm{log}(n)$ 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.
