A logarithmic-depth quantum carry-lookahead adder

We present an efficient addition circuit, borrowing techniques from the classical carry-lookahead arithmetic circuit. Our quantum carrylookahead (qcla) adder accepts two n-bit numbers and adds them in O(log n) depth using O(n) ancillary qubits. We present both in-place and out-of-place versions, as well as versions that add modulo 2n and modulo 2n − 1. Previously, the linear-depth ripple-carry addition circuit has been the method of choice. Our work reduces the cost of addition dramatically with only a slight increase in the number of required qubits. The qcla adder can be used within current modular multiplication circuits to reduce substantially the run-time of Shor’s algorithm.