线性PCPPs的近最优高速率手术

中文摘要 AI 辅助

量子低密度奇偶校验（LDPC）码手术的一个核心问题是设计辅助系统，该系统能够并行测量大量逻辑算符，同时保持稀疏性和容错距离。我们引入了一种非常通用的方法，用于设计“高速率”的手术辅助装置，即并行测量的算符数量与辅助系统规模相比很大。给定任意初始的$[[ n, k, d ]]$量子LDPC Calderbank-Shor-Steane（CSS）码，以及任意大小为$\mu\leq n$且包含$t \leq k$个逻辑量子比特的子码，该方法产生一个稀疏的高速率手术辅助装置，其大小为$\mu(\log\mu)^{O(\log\log \mu)}=\mu^{1+o(1)}$，能够测量子码中的所有$t$个逻辑算符。空间开销在渐近意义下最优，直到次多项式因子，因为下界为$\Omega(\mu)$。该辅助装置在$\mu$的多项式时间内生成。当使用这些手术辅助装置进行逻辑测量时，若执行$d$轮，现象学容错距离至少为$d$。\n\n我们的主要结果来自于将手术辅助装置与相对余循环扩张（relative cosystolic expansion）联系起来，并关联到线性概率可检验邻近性证明（PCPPs），后者允许随机验证者仅通过预言机访问输入和声称的证明，来概率性地验证线性电路的输入。

英文摘要

A central problem for surgery with quantum Low-Density Parity Check (LDPC) codes is the design of auxiliary systems which measure large sets of logical operators in parallel, while preserving sparsity and fault distance. We introduce a very general method for designing surgery gadgets which are 'high rate', meaning that the number of operators measured in parallel is large in comparison to the size of the auxiliary system. Given an arbitrary initial $[[ n, k, d ]]$ quantum LDPC Calderbank-Shor-Steane (CSS) code, and an arbitrary subcode of size $μ\leq n$ that contains $t \leq k$ logical qubits, the method produces a sparse high-rate surgery gadget with size $μ(\logμ)^{O(\log\log μ)}=μ^{1+o(1)}$ which measures all $t$ logicals in the subcode. The space overhead is asymptotically optimal up to subpolynomial factors, as the lower bound is $Ω(μ)$. This gadget is produced in time polynomial in $μ$. When logical measurement is performed using these surgery gadgets, the phenomenological fault distance is at least $d$ when performed for $d$ rounds. Our main result comes from relating surgery gadgets with relative cosystolic expansion to linear Probabilistically Checkable Proofs of Proximity (PCPPs), which allow a randomised verifier to probabilistically verify the input to a linear circuit, using only oracle access to the input and a claimed proof.

发表机构

• Xanadu

机构由 AI 辅助整理，请以论文原文为准。

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