Matchgates are an especially multiflorous class of two-qubit nearestneighbour quantum gates, defined by a set of algebraic constraints. They occurfor example in the theory of perfect matchings of graphs, non-interactingfermions, and one-dimensional spin chains. We show that the computational powerof circuits of matchgates is equivalent to that of space-bounded quantumcomputation with unitary gates, with space restricted to being logarithmic inthe width of the matchgate circuit. In particular, for the conventional settingof polynomial-sized (logarithmic-space generated) families of matchgatecircuits, known to be classically simulatable, we characterise their power ascoinciding with polynomial-time and logarithmic-space bounded universal unitaryquantum computation.
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