度量自对偶完备化与量子及图态距离的最优加性硬度

中文摘要 AI 辅助

我们证明，对于某个常数 $c > 0$，量子码距离在加性误差 $c N$ 内近似是 NP 难的，其中 $N$ 是量子比特数。我们的归约是确定性的。这改进了先前的平方根加性间隙至 $\Omega(N)$，并解决了 Kapshikar 和 Kundu 明确提出的线性间隙问题。我们的结果适用于具有相同 $X$- 和 $Z$-校验空间的 CSS 码，且具有恒定速率和恒定相对距离。对于每个固定的 $\lambda>1$，存在常数 $c>0$，使得即使每个非恒等稳定子权重大于量子距离的 $\lambda$ 倍，硬度仍然成立。我们还改进了 Grigorescu、Jha 和 Samperton 关于 $N$ 个顶点上图态距离的硬度间隙，从立方根改进至 $\Omega(N)$，解决了他们明确提出的开放问题。由于两个距离至多为 $N$，两个硬度结果都是渐近最优的。我们的图态距离硬度结果适用于平衡二分图，其二进制邻接矩阵是自身的逆（模 2）。我们上述两个硬度界限的主要技术是经典的：我们展示了如何将任何长度为 $m$ 的码 $C$ 转换为长度为 $N=\Theta(m)$ 的自对偶码 $A(C)$，同时精确地将原始陪集度量加倍。该转换是确定性的且高效的。我们称之为 $C$ 的度量自对偶完备化。它带有一个线性嵌入 $\tau: \mathbb F_2^m \hookrightarrow \mathbb F_2^N$。该嵌入将 $\mathbb F_2^m$ 中向量之间的所有汉明距离以及相应陪集之间的所有成对距离加倍。该嵌入还保证 $A(C)$ 中权重至多 $2m$ 的所有码字恰好是 $\tau(C)$。

英文摘要

We prove that the quantum code distance is NP-hard to approximate within an additive error of $c N$, for some constant $c >0$, where $N$ is the number of qubits. Our reductions are deterministic. This improves the previous square-root additive gap to $Ω(N)$ and resolves the explicitly stated linear-gap question of Kapshikar and Kundu. Our result holds for CSS codes with identical $X$- and $Z$-check spaces, and with a constant rate and constant relative distance. For every fixed $λ>1$, there is a constant $c>0$ such that hardness still holds even when every nonidentity stabilizer has weight greater than $λ$ times the quantum distance. We also improve the hardness gap of graph state distance on $N$ vertices of Grigorescu, Jha, and Samperton from cube-root to $Ω(N)$, resolving their explicitly stated open question. Both hardness results are asymptotically optimal since both distances are at most $N$. Our graph state distance hardness result holds for balanced bipartite graphs with a binary adjacency matrix that is its own inverse (mod 2). Our main technique for both hardness bounds above is classical: we show how to convert any code $C$ of length $m$ into a self-dual code $A(C)$ of length $N=Θ(m)$ while exactly doubling the original coset metric. The conversion is deterministic and efficient. We call it the metric self-dual completion of $C$. It comes with a linear embedding $τ: \mathbb F_2^m \hookrightarrow \mathbb F_2^N$. The embedding doubles all Hamming distances between vectors in $\mathbb F_2^m$ and all pairwise distances between corresponding cosets. The embedding also guarantees that all codewords of $A(C)$ of weight at most $2m$ are exactly $τ(C)$.

发表机构

• UCLA（加州大学洛杉矶分校）

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