  Quantum computers hold great promise, but it remains a challenge to find
efficient quantum circuits that solve interesting computational problems. We
show that finding optimal quantum circuits is essentially equivalent to finding
the shortest path between two points in a certain curved geometry. By recasting
the problem of finding quantum circuits as a geometric problem, we open up the
possibility of using the mathematical techniques of Riemannian geometry to
suggest new quantum algorithms, or to prove limitations on the power of quantum
computers.
