How to Fault-Tolerantly Realize Any Quantum Circuit with Local Operations

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 three-dimensional quantum fault-tolerance architecture involving O(n^{3/2}log^{3}⁡n) qubits and a quasi-two-dimensional 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 quasipolylogarithmic 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.