Abstract

There have been several efforts to apply quantum SAT solving methods to factor large integers. While these methods may provide insight into quantum SAT solving, to date they have not led to a convincing path to integer factorization that is competitive with the best known classical method, the Number Field Sieve. Many of the techniques tried involved directly encoding multiplication to SAT or an equivalent NP-hard problem and looking for satisfying assignments of the variables representing the prime factors. The main challenge in these cases is that, to compete with the Number Field Sieve, the quantum SAT solver would need to be superpolynomially faster than classical SAT solvers. In this paper the use of SAT solvers is restricted to a smaller task related to factoring: finding smooth numbers, which is an essential step of the Number Field Sieve. We present a SAT circuit that can be given to quantum SAT solvers such as annealers in order to perform this step of factoring. If quantum SAT solvers achieve any asymptotic speedup over classical brute-force search for smooth numbers, then our factoring algorithm is faster than the classical NFS.

Highlights

  • There have been several efforts to apply quantum SAT solving methods to factor large integers

  • If the quantum SAT solving heuristic achieves a speed-up over classical circuit-SAT solving algorithms, we show that this leads to a factoring algorithm that is asymptotically faster than the regular Number Field Sieve (NFS)

  • While there is no convincing evidence to date that non-fault-tolerant quantum SAT solvers will provide an asymptotic speed-up over classical SAT solvers, with this approach we at least avoid the situation where the quantum SAT solver must outperform classical SAT solvers by a superpolynomial factor in order to compete with the NFS

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Summary

OPEN On speeding up factoring with quantum SAT solvers

There have been several efforts to apply quantum SAT solving methods to factor large integers. If quantum SAT solvers achieve any asymptotic speedup over classical brute-force search for smooth numbers, our factoring algorithm is faster than the classical NFS. If the quantum SAT solving heuristic achieves a speed-up over classical circuit-SAT solving algorithms, we show that this leads to a factoring algorithm that is asymptotically faster than the regular NFS. We found one circuit implementing the Elliptic Curve Method (ECM)[6] which, if used as a subroutine of the NFS, could result in a speedup for factoring integers since SAT solvers can use this circuit to find smooth numbers asymptotically as fast as brute-force search. We present a circuit that, when used as a NFS subroutine, yields an algorithm with the same asymptotic runtime as the classical NFS, and faster if quantum SAT solvers achieve any non-trivial speedup. We discuss the results of this paper as well as future work

Previous work
Circuits for smoothness detection
In the limit
NFS runs in
Conclusion
Author contributions
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