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Paper 1
On the (Classical and Quantum) Fine-Grained Complexity of Approximate CVP and Max-Cut
Jeremy Ahrens Huang, Young Kun Ko, Chunhao Wang
- Year
- 2024
- Journal
- arXiv preprint
- DOI
- arXiv:2411.04124
- arXiv
- 2411.04124
We show a linear-size reduction from gap Max-2-Lin(2) (a generalization of the gap $\mathrm{Max}$-$\mathrm{Cut}$ problem) to $γ\text{-}\mathrm{CVP}_p$ for $γ= \mathrm{O}(1)$ and finite $p\geq 1$, as well as a no-go theorem against poly-sized non-adaptive quantum reductions from $k$-SAT to $\mathrm{CVP}_2$. This implies three headline results: (i) Faster algorithms for $γ\text{-}\mathrm{CVP}$ are also faster algorithms for Max-2-Lin(2) and Max-Cut. Depending on the approximation regime, even a $2^{0.78n}$-time or $2^{0.3n}$-time algorithm would improve upon the state-of-the-art algorithm such as Williams' 2004 algorithm [Theoretical Computer Science 2005] or Arora et al.'s 2010 algorithm [Journal of the ACM 2015]. This provides evidence that $γ\text{-}\mathrm{CVP}$ for $γ=\mathrm{O}(1)$ requires exponential time, improving upon the previous lower-bound for $γ<3$ by Bennett et al. [arxiv:1704.03928]. (ii) A new almost $2^{(1/2+\varepsilon/4ς+o(1))n}$-time classical algorithm and a new almost $2^{(1/3+\varepsilon/6ς+o(1))n}$-time quantum algorithm for $(1-\varepsilon,1-ς)$-gap Max-2-Lin(2). This algorithm is faster than the algorithm of Arora et al., as well as the algorithm of Williams, and the algorithm of Manurangsi and Trevisan [arxiv:1807.09898] when $c_0 \varepsilon<ς<c_1 \varepsilon$ for some constants $c_0, c_1$. (iii) If the Quantum Strong Exponential Time Hypothesis (QSETH) can be used to show a $2^{δn}$-time lower-bound for Max-Cut, Max-2-Lin(2), or $\mathrm{CVP}_2$ for any constant $δ>0$, it must be via an adaptive quantum reduction unless $\mathrm{NP} \subseteq \mathrm{pr}\text{-}\mathrm{QSZK}$. This illuminates some difficulties in characterizing the hardness of approximate CSPs and shows that the post-quantum security of lattice-based cryptography likely cannot be supported by QSETH.
Open paperPaper 2
Quantum Computing Approach for Energy Optimization in a Prosumer Community
Carlo Mastroianni, Luigi Scarcello, Jacopo Settino
- Year
- 2022
- Journal
- arXiv preprint
- DOI
- arXiv:2209.04411
- arXiv
- 2209.04411
This paper presents a quantum approach for the formulation and solution of the prosumer problem, i.e., the problem of minimizing the energy cost incurred by a number of users in an energy community, while addressing the constraints given by the balance of energy and the user requirements. As the problem is NP-complete, a hybrid quantum/classical algorithm could help to acquire a significant speedup, which is particularly useful when the problem size is large. This work describes the steps through which the problem can be transformed, reformulated and given as an input to Quantum Approximate Optimization Algorithm (QAOA), and reports some experimental results, in terms of the quality of the solution and time to achieve it, obtained with a quantum simulator, when varying the number of constraints and, correspondingly, the number of qubits.
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