  In [R. Jozsa, B. Kraus, A. Miyake, J. Watrous, Proc. R. Soc. A {\bf 466},
809-830 (2010)] it has been shown that a match gate circuit running on n qubits
can be compressed to a universal quantum computation on \log(n)+3 qubits. Here,
we show how this compression can be employed to simulate the Ising interaction
of a 1D--chain consisting out of n qubits using a universal quantum computer
running on log(n) qubits. We demonstrate how the adiabatic evolution can be
realized on this exponentially smaller system and how the magnetization, which
shows a quantum phase transition, can be measured.
