  We show how to realize a general quantum circuit involving gates between
arbitrary pairs of qubits by means of geometrically local quantum operations
and efficient classical computation. We prove that circuit-level local
stochastic noise modeling an imperfect implementation of our derived schemes is
equivalent to local stochastic noise in the original circuit. Our constructions
incur a constant-factor increase in the quantum circuit depth and a polynomial
overhead in the number of qubits: To execute an arbitrary quantum circuit on
$n$ qubits, we give a 3D quantum fault-tolerance architecture involving
$O(n^{3/2} \log^3 n)$ qubits, and a quasi-2D architecture using $O(n^2 \log^3
n)$ qubits. Applied to recent fault-tolerance constructions, this gives a
fault-tolerance threshold theorem for universal quantum computations with local
operations, a polynomial qubit overhead and a quasi-polylogarithmic depth
overhead. More generally, our transformation dispenses with the need for
considering the locality of operations when designing schemes for
fault-tolerant quantum information processing.
