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We present an efficient addition circuit, borrowing techniques from classical carry-lookahead arithmetic. Our quantum carry-lookahead () adder accepts two n-bit numbers and adds them in O (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 \ adder can be used within current modular multiplication circuits to reduce substantially the run-time of Shor's algorithm.
Draper et al. (Sat,) studied this question.