发表机构
QodeX Quantum（QodeX量子）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究置换不变码间的误差透明切换，证明切换酉算子存在当且仅当两码由距离三的码路径连接，并在13量子比特上构造出达到机器精度的集体切换算子。
AI 中文摘要
码切换将编码量子比特在两个码之间移动，这两个码的横向门合起来是通用的，但切换操作本身通常不是横向的，并且可能将单个故障扩散为不可纠正的误差。我们研究在同一组量子比特上的两个置换不变码之间的切换，其中一个具有横向Clifford门，另一个具有来自非Clifford二元二十面体群的横向门。两者都与横向$X$、$Z$和$F = HS^\dagger$兼容，且具有此性质的编码器构成一个小线性空间。我们证明其距离至少为三的码至多由两个二次方程切出，且所有这些码具有相同的误差几何，即它们的单量子比特误差态具有相同的内积。由此可知，一个与量子比特置换以及每一时刻共享的横向门对易、并且能容忍在未知时间发生的单个单量子比特故障的切换酉算子，存在当且仅当这两个码由一条距离为三的码的连续路径连接。这样的切换可以做到误差透明，使得任何时刻的故障在到达目标码时表现为相同的单量子比特误差。对于实编码器，距离为三的码是一个二次方程的解，在十三个量子比特上，它们构成一个显式的圆，穿过两个Clifford码和一个二十面体码。我们沿此圆构造切换酉算子作为集体算子，将其合成为物理控制留作开放问题，并验证其每次故障的恢复不保真度达到机器精度，而两种自然替代方案的平均不保真度分别为$0.13$和$0.85$。对最多29个奇数量子比特数的调查列出了此类切换存在的位置。
英文摘要
Code switching moves an encoded qubit between two codes whose transversal gates together are universal, but the switching operation itself is usually not transversal and can spread a single fault into an uncorrectable error. We study switching between two permutation-invariant codes on the same qubits, one with transversal Clifford gates and one with transversal gates from the non-Clifford binary icosahedral group. Both are compatible with transversal $X$, $Z$, and $F = HS^\dagger$, and the encoders with this property form a small linear space. We show that its codes of distance at least three are cut out by at most two quadratic equations, and that all of them have the same error geometry in the sense that their single-qubit error states have the same inner products. It follows that a switching unitary commuting with qubit permutations and with the shared transversal gates at every instant, and tolerating one single-qubit fault at an unknown time, exists if and only if the two codes are joined by a continuous path of distance-three codes. Such a switch can be made error transparent, so that a fault at any moment reaches the target code as the same single-qubit error. For real encoders, the distance-three codes are the solutions of a single quadratic equation, and on thirteen qubits, they form an explicit circle through two Clifford codes and an icosahedral code. We construct the switching unitary along this circle as a collective operator, leaving its synthesis into physical control open, and verify that its recovery infidelity per fault is at machine precision, whereas two natural alternatives leave average infidelities of $0.13$ and $0.85$. A survey over odd qubit numbers up to $29$ lists where such switches exist.
Comments17 pages, 2 figures
