Scaling laws for Shor's algorithm with a banded quantum Fourier transform

TL;DR

This paper analyzes how limiting the quantum Fourier transform's bandwidth in Shor's algorithm affects its success rate, deriving scaling laws and demonstrating that a bandwidth of 8 qubits suffices for factoring RSA-2048 with high probability.

Contribution

The study introduces analytical and numerical scaling laws for the performance of a banded quantum Fourier transform in Shor's algorithm, revealing how success probability depends on bandwidth and problem size.

Findings

Performance scales exponentially with bandwidth and problem size.

Analytical predictions match numerical simulations closely.

Bandwidth of 8 qubits enables successful RSA-2048 factoring with high probability.

Abstract

We investigate the performance of a streamlined version of Shor's algorithm in which the quantum Fourier transform is replaced by a banded version that for each qubit retains only coupling to its $b$ nearest neighbors. Defining the performance $P(n,b)$ of the $n$-qubit algorithm for bandwidth $b$ as the ratio of the success rates of Shor's algorithm equipped with the banded and the full bandwidth ($b=n-1$) versions of the quantum Fourier transform, our numerical simulations show that $P(n,b) \approx \exp[-\varphi_{max}^2 (n,b)/100]$ for $n < n_t(b)$ (non-exponential regime) and $P(n,b) \approx 2^{-\xi_b (n-8)}$ for $n>n_t(b)$ (exponential regime), where $n_{t}(b)$, the location of the transition, is approximately given by $n_{t}(b)\approx b+5.9 + \sqrt{7.7(b+2)-47}$ for $b\gtrsim 8$, $\varphi_{max} (n,b) = 2\pi[2^{-b-1} (n-b-2) + 2^{-n}]$, and $\xi_b\approx 1.1 \times 2^{-2b}$. Analytically we obtain $P(n,b) \approx \exp[-\varphi_{max}^2 (n,b)/64]$ for $n<n_t(b)$ and $P(n,b) \approx 2^{-\xi_b^{(a)} n}$ for $n>n_t(b)$, where $\xi_{b}^{(a)} \approx \frac{\pi^2}{12 \ln(2)} \times 2^{-2b} \approx 1.19 \times 2^{-2b}$. Thus, our analytical results predict the $\varphi_{max}^2$ scaling ($n<n_t$) and the $2^{-2b}$ scaling ($n>n_t$) of the data perfectly. In addition, in the large-$n$ regime, the prefactor in $\xi_b^{(a)}$ is close to the results of our numerical simulations and, in the low-$n$ regime, the numerical scaling factor in our analytical result is within a factor 2 of its numerical value. As an example we show that $b=8$ is sufficient for factoring RSA-2048 with a 95% success rate.