Truncated phase-based quantum arithmetic: error propagation and resource reduction

TL;DR

This paper introduces a modified quantum Fourier adder that reduces hardware complexity by eliminating small-angle rotations, with analytical and simulation-based validation, significantly improving resource efficiency for large-scale quantum algorithms like Shor's.

Contribution

It presents a novel truncation method for phase rotations in quantum adders, reducing resource requirements while maintaining fidelity, applicable to large algorithms such as RSA-2048 factoring.

Findings

Quantum Fourier adder can be truncated to π/64 rotations.

Resource savings in qubits and magic states are significant.

The approach is competitive with Toffoli-based methods.

Abstract

There are two important, and potentially interconnecting, avenues to the realisation of large-scale quantum algorithms: improvement of the hardware, and reduction of resource requirements demanded by algorithm components. In focusing on the latter, one crucial subroutine to many sought-after applications is the quantum adder. A variety of different implementations exist with idiosyncratic pros and cons. One of these, the Draper quantum Fourier adder, offers the lowest qubit count of any adder, but requires a substantial number of gates as well as extremely fine rotations. In this work, we present a modification of the Draper adder which eliminates small-angle rotations to highly coarse levels, matched with some strategic corrections. This reduces hardware requirements without sacrificing the qubit saving. We show that the inherited loss of fidelity is directly given by the rate of carry and borrow bits in the computation. We derive formulae to predict this, complemented by complete gate-level matrix product state simulations of the circuit. Moreover, we analytically describe the effects of possible stochastic control error. We present an in-depth analysis of this approach in the context of Shor's algorithm, focusing on the factoring of RSA-2048. Surprisingly, we find that each of the $7\times 10^7$ quantum Fourier transforms may be truncated down to $\pi/64$, with additive rotations left only slightly finer. This result is much more efficient than previously realised. We quantify savings both in terms of logical resources and raw magic states, demonstrating that phase adders can be competitive with Toffoli-based constructions.