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

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.