Quantum Fourier Addition, Simplified to Toffoli Addition

TL;DR

This paper demonstrates a systematic translation of quantum Fourier transform-based addition circuits into Toffoli-based adders, revealing similar fault-tolerance costs and enabling automated circuit optimization.

Contribution

It introduces a method to convert QFT-addition circuits into Toffoli-based adders, showing their equivalence in fault-tolerance costs and providing new circuit identities for optimization.

Findings

QFT-addition can be translated into Toffoli-addition.

Fault-tolerance costs are similar for both adder types.

New circuit identities enable automated optimization.

Abstract

Quantum addition circuits are considered being of two types: 1) Toffolli-adder circuits which use only classical reversible gates (CNOT and Toffoli), and 2) QFT-adder circuits based on the quantum Fourier transformation. We present the first systematic translation of the QFT-addition circuit into a Toffoli-based adder. This result shows that QFT-addition has fundamentally the same fault-tolerance cost (e.g. T-count) as the most cost-efficient Toffoli-adder: instead of using approximate decompositions of the gates from the QFT circuit, it is more efficient to merge gates. In order to achieve this, we formulated novel circuit identities for multi-controlled gates and apply the identities algorithmically. The employed techniques can be used to automate quantum circuit optimisation heuristics.