Jeremy Ahrens Huang | Searcharxiv

arXiv2026

Nonuniform QCPH Collapse Implies QCPH Collapse

Despite the importance of the non-uniform Polynomial-Time Hierarchy (PH/poly) in classical complexity understanding the collapse conditions of the Polynomial-Time Hierarchy (PH), a quantum equivalent of PH/poly has yet to be studied in the literature. We introduce the non-uniform computational Quantum Polynomial-Time Hierarchy (QCPH/mpoly), the quantum equivalent of PH/poly, and show that it collapses if and only if QCPH, the quantum equivalent of PH introduced by Gharibian et al. (comput. complex. 2022), also collapses. We also show that QCPH collapses if coQCMA is contained in QCMA/mpoly. These results are analogous to those of Yap (TCS 1983) commonly used to invoke the collapse of PH in classical complexity. QCPH/mpoly is analogous to QCPH with non-uniform quantum verifier circuits.

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arXiv2026

On the (Classical and Quantum) Fine-Grained Complexity of Approximate CVP and Max-Cut

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 $c_0 \varepsilon<ς 0$, 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.

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