A logarithmic-depth quantum carry-lookahead adder

TL;DR

This paper introduces a quantum carry-lookahead adder that performs addition in logarithmic depth, significantly improving efficiency over traditional linear-depth methods, and is suitable for quantum algorithms like Shor's.

Contribution

The paper develops a quantum carry-lookahead adder with O(log n) depth, reducing addition complexity while maintaining linear qubit usage, and provides multiple versions including modular addition.

Findings

Achieves O(log n) depth addition circuit

Reduces runtime of Shor's algorithm

Supports in-place, out-of-place, and modular addition

Abstract

We present an efficient addition circuit, borrowing techniques from the classical carry-lookahead arithmetic circuit. Our quantum carry-lookahead (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 2^n and modulo 2^n - 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.