近似量子码距离的难度超越 $\sqrt{N}$

中文摘要 AI 辅助

我们研究了稳定子码最小距离近似问题的计算复杂性。在经典情形下，最小距离问题即使在块长度的线性加性间隙内近似也是NP-hard的。然而，对于量子码，Kapshikar和Kundu以及Grigorescu、Jha和Samperton的归约仅达到块长度$N$的加性$O(\sqrt{N})$间隙。我们填补了这一间隙，并在另一个方向上，启动了该问题的细粒度研究。我们证明，除非$\mathsf{NP} \subseteq \mathsf{coRP}$，否则不存在随机化算法能在加性$\alpha N$（其中$\alpha > 0$为常数）内近似距离。在细粒度设置中，对于每个$\varepsilon > 0$，除非SETH失效，否则不存在随机化$2^{(1-\varepsilon)\kappa}\\,\mathrm{poly}(N)$时间算法能计算具有$\kappa$个逻辑量子比特的码的距离；并且除非非均匀Gap-ETH失效，否则在块长度、逻辑量子比特数和间隙同时为线性的实例上，不存在随机化$2^{o(N)}$时间算法能在线性加性间隙内近似距离。

英文摘要

We study the computational complexity of approximating the minimum distance of a stabilizer code. Classically, the minimum distance problem is NP-hard to approximate even to within an additive gap linear in the block length. For quantum codes, however, the reductions of Kapshikar and Kundu and of Grigorescu, Jha and Samperton reach only an additive $O(\sqrt{N})$ gap in the block length $N$. We close this gap and, in a separate direction, initiate the fine-grained study of the problem. We show that no randomized algorithm approximates the distance to within an additive $αN$, for a constant $α> 0$, unless $\mathsf{NP} \subseteq \mathsf{coRP}$. In the fine-grained setting, for every $\varepsilon > 0$, no randomized $2^{(1-\varepsilon)κ}\,\mathrm{poly}(N)$-time algorithm computes the distance of a code with $κ$ logical qubits unless SETH falls; and no randomized $2^{o(N)}$-time algorithm approximates the distance to within a linear additive gap, on instances where block length, number of logical qubits, and gap are simultaneously linear, unless non-uniform Gap-ETH falls.

发表机构

• University of Ottawa（渥太华大学）

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