7 research outputs found
Quantum algorithms for computing general discrete logarithms and orders with tradeoffs
We generalize our earlier works on computing short discrete logarithms with tradeoffs, and bridge them with Seifert\u27s work on computing orders with tradeoffs, and with Shor\u27s groundbreaking works on computing orders and general discrete logarithms. In particular, we enable tradeoffs when computing general discrete logarithms.
Compared to Shor\u27s algorithm, this yields a reduction by up to a factor of two in the number of group operations evaluated quantumly in each run, at the expense of having to perform multiple runs. Unlike Shor\u27s algorithm, our algorithm does not require the group order to be known. It simultaneously computes both the order and the logarithm.
We analyze the probability distributions induced by our algorithm, and by Shor\u27s and Seifert\u27s order finding algorithms, describe how these algorithms may be simulated when the solution is known, and estimate the number of runs required for a given minimum success probability when making different tradeoffs
Revisiting Shor's quantum algorithm for computing general discrete logarithms
We heuristically demonstrate that Shor's algorithm for computing general
discrete logarithms, modified to allow the semi-classical Fourier transform to
be used with control qubit recycling, achieves a success probability of
approximately 60% to 82% in a single run. By slightly increasing the number of
group operations that are evaluated quantumly, and by performing a limited
search in the classical post-processing, we furthermore show how the algorithm
can be modified to achieve a success probability exceeding 99% in a single run.
We provide concrete heuristic estimates of the success probability of the
modified algorithm, as a function of the group order, the size of the search
space in the classical post-processing, and the additional number of group
operations evaluated quantumly. In analogy with our earlier works, we show how
the modified quantum algorithm may be simulated classically when the logarithm
and group order are both known. Furthermore, we show how slightly better
tradeoffs may be achieved, compared to our earlier works, if the group order is
known when computing the logarithm.Comment: The pre-print has been extended to show how slightly better tradeoffs
may be achieved, compared to our earlier works, if the group order is known.
A minor issue with an integration limit, that lead us to give a rough success
probability estimate of 60% to 70%, as opposed to 60% to 82%, has been
corrected. The heuristic and results reported in the original pre-print are
otherwise unaffecte
Large-Scale Simulation of Shor's Quantum Factoring Algorithm
Shor's factoring algorithm is one of the most anticipated applications of
quantum computing. However, the limited capabilities of today's quantum
computers only permit a study of Shor's algorithm for very small numbers. Here
we show how large GPU-based supercomputers can be used to assess the
performance of Shor's algorithm for numbers that are out of reach for current
and near-term quantum hardware. First, we study Shor's original factoring
algorithm. While theoretical bounds suggest success probabilities of only 3-4
%, we find average success probabilities above 50 %, due to a high frequency of
"lucky" cases, defined as successful factorizations despite unmet sufficient
conditions. Second, we investigate a powerful post-processing procedure, by
which the success probability can be brought arbitrarily close to one, with
only a single run of Shor's quantum algorithm. Finally, we study the
effectiveness of this post-processing procedure in the presence of typical
errors in quantum processing hardware. We find that the quantum factoring
algorithm exhibits a particular form of universality and resilience against the
different types of errors. The largest semiprime that we have factored by
executing Shor's algorithm on a GPU-based supercomputer, without exploiting
prior knowledge of the solution, is 549755813701 = 712321 * 771781. We put
forward the challenge of factoring, without oversimplification, a non-trivial
semiprime larger than this number on any quantum computing device.Comment: differs from the published version in formatting and style; open
source code available at https://jugit.fz-juelich.de/qip/shorgp