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