We constructed a virtual quantum computer by running a complete, scaling, quantum-gate–by–quantum-gate implementation of Shor's algorithm on a 128-core classical cluster computer. In mode A [quantum period finding (PF) only, supplied with classical results for the modular exponentiation (ME) part of Shor's algorithm], factoring semiprimes up to $N=557993$ with up to $n=39$ qubits, we confirm earlier, smaller-$n$ results concerning the performance scaling of Shor's algorithm equipped with a truncated (banded) quantum Fourier transform. Running our virtual quantum computer in mode B (full quantum implementation of ME and PF), we find that a large number of gates may be discarded in a scalable way in both the ME and PF parts of Shor's algorithm in exchange for only a small reduction in performance. We explicitly state the associated scaling laws. Implying significant savings in quantum gates, we suggest that these results are of importance for future experimental and technical large-$n$ implementations of quantum computers.

• Received 14 May 2013

DOI:https://doi.org/10.1103/PhysRevA.87.060304