Paper 2024/646
Efficient Quantum Algorithm for SUBSET-SUM Problem
Abstract
Problems in the complexity class $NP$ are not all known to be solvable, but are verifiable given the solution, in polynomial time by a classical computer. The complexity class $BQP$ includes all problems solvable in polynomial time by a quantum computer. Prime factorization is in $NP$ class, and is also in $BQP$ class, owing to Shor's algorithm. The hardest of all problems within the $NP$ class are called $NP$-complete. If a quantum algorithm can solve an $NP$-complete problem in polynomial time, it would imply that a quantum computer can solve all problems in $NP$ in polynomial time. Here, we present a polynomial-time quantum algorithm to solve an $NP$-complete variant of the $SUBSET-SUM$ problem, thereby, rendering $NP\subseteq BQP$. We illustrate that given a set of integers, which may be positive or negative, a quantum computer can decide in polynomial time whether there exists any subset that sums to zero. There are many real-world applications of our result, such as finding patterns efficiently in stock-market data, or in recordings of the weather or brain activity. As an example, the decision problem of matching two images in image processing is $NP$-complete, and can be solved in polynomial time, when amplitude amplification is not required.
Note: Added the second-last sentence in step 4 of algorithm in Section III about the exponentiated swap operator W.
Metadata
- Available format(s)
-
PDF
- Category
- Foundations
- Publication info
- Preprint.
- Keywords
- Subset-Sum ProblemNP-complete ProblemBQP AlgorithmQuantum AlgorithmExponential Speedup
- Contact author(s)
-
sanchita ghosh14 @ gmail com
anantsharma2410 @ gmail com
sreetama das @ ino cnr it
roy shibdas @ gmail com - History
- 2024-08-02: last of 2 revisions
- 2024-04-27: received
- See all versions
- Short URL
- https://ia.cr/2024/646
- License
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CC BY
BibTeX
@misc{cryptoeprint:2024/646, author = {Sanchita Ghosh and Anant Sharma and Sreetama Das and Shibdas Roy}, title = {Efficient Quantum Algorithm for {SUBSET}-{SUM} Problem}, howpublished = {Cryptology {ePrint} Archive, Paper 2024/646}, year = {2024}, url = {https://eprint.iacr.org/2024/646} }