Implementation of Shor's Algorithm on a Linear Nearest Neighbour Qubit Array

Abstract

Shor's algorithm, which given appropriate hardware can factorise an integer $N$ in a time polynomial in its binary length $L$, has arguable spurred the race to build a practical quantum computer. Several different quantum circuits implementing Shor's algorithm have been designed, but each tacitly assumes that arbitrary pairs of qubits within the computer can be interacted. While some quantum computer architectures possess this property, many promising proposals are best suited to realising a single line of qubits with nearest neighbour interactions only. In light of this, we present a circuit implementing Shor's factorisation algorithm designed for such a linear nearest neighbour architecture. Despite the interaction restrictions, the circuit requires just $2L+4$ qubits and to first order requires $8L^{4}$ gates arranged in a circuit of depth $32L^{3}$ -- identical to first order to that possible using an architecture that can interact arbitrary pairs of qubits.