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 arguably 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 leading order requires 8L^4 2-qubit gates arranged in a circuit of depth 32L^3 --- identical to leading order to that possible using an architecture that can interact arbitrary pairs of qubits.

Design and synthesis of efficient circuits for quantum computers

“…Some of the above proposals for quantum addition circuits emphasize on minimizing the number of required qubits, while other methods try to minimize the circuit depth. Other efforts concentrate on building architectures restricted on the condition of local communications between the qubits either in 1D-NTC (1-Dimension, linear Nearest-neighbour, Two qubit gates, Concurrent execution) such as those of Fowler-Devitt-Hollenberg (FDH) [130] and Kutin's [131], or in 2D-NTC such as those of Choi-VanMeter (CV) [132] and Pham-Svore (PS) [133].…”

“…Notable exceptions of the above top-down trend that builds a complete modular exponentiation circuit from the quantum equivalent of classical binary adder are circuits that use the Draper's QFT adder [23] like Beauregard's circuit [24], Fowler-Devitt-Hollenberg circuit (FDH) [130], and Kutin's first circuit of [131]. These three circuits implement the addition of two integers by converting one of them in the Fourier domain using QFT and then converting the sum back to the binary representation.…”

“…An inspection of Figure 4.7 shows that the the case of the ΦADDC is handled with the topology "scc" of Figure 5.7. This scheme is slightly different form the one proposed in [130] as we want at the end to retain the qubits order. Thus, the depth of a localized ΦADDC becomes 2n instead of n.…”

“…Figure 5.10 is the localized version of the ΦADD of Figure 4.9 (see also [130]). Controlled rotation gates c-R k are applied between pairs of qubits a j and b l for k = 1 .…”

“…Finally, a localized version of the QFT [130] is shown in Figure 5.11. Hadamard gates are included in this design and qubits labeling is attached after each SWAP gate to help keep track the operation.…”