基于杯积改进的横向非 Clifford 门
Improved Transversal Non-Clifford Gates from Cup Products
- UC Berkeley（加州大学伯克利分校）
- Tel Aviv University（特拉维夫大学）
- IBM Quantum（IBM量子）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文通过杯积变换构造支持横向非 Clifford 门的量子 LDPC 码，实现常数重量稳定子下多项式距离与接近线性逻辑量子比特数，降低魔法态开销指数。
AI中文摘要：
在量子容错计算中，获得低开销的非 Clifford 门协议是一个重大挑战。为此，我们构造了具有低重量稳定子的量子码，这些码支持横向（即低深度）实现非 Clifford 的 $C^{r-1}Z$ 门，对于每个常数 $r\geq 3$。特别地，我们获得了长度为 $n$ 的量子 LDPC 码（具有常数重量稳定子），其多项式距离 $d\geq n^{(1-ε)/r}$，支持在接近线性的数量 $k\geq n^{1-ε}$ 个不相交的逻辑量子比特元组上横向执行 $C^{r-1}Z$ 门，对于任意小的 $ε>0$。我们的构造是首个具有常数重量稳定子且满足 $dk\gg n$ 的构造，并因此实现了任意小的魔法态开销指数 $γ=\log(n/k)/\log(d)>0$。相比之下，先前的类似构造至少需要多对数重量的稳定子。我们还展示了如何获得线性数量的 $k=Ω(n)$ 个逻辑 $C^{r-1}Z$ 门，尽管稳定子重量和物理电路深度为 $n^ε$。我们证明我们的横向门还支持对特定逻辑量子比特的寻址（即定位）。为了获得我们的码，我们开发了一种基于杯积的一般变换，该变换将满足乘法性质的经典码映射到支持横向 $C^{r-1}Z$ 的量子码。我们将此变换应用于一类新的经典 Tanner 码，这些码由代数码的穿孔张量积构造而成。
英文摘要：
It is a major challenge in quantum fault-tolerance to obtain low-overhead protocols for performing non-Clifford gates. In this vein, we construct quantum codes with low-weight stabilizers that support transversal (i.e. low-depth) implementations of the non-Clifford $C^{r-1}Z$ gate, for every constant $r\geq 3$. In particular, we obtain length-$n$ quantum LDPC codes (with constant-weight stabilizers) of polynomial distance $d\geq n^{(1-ε)/r}$ supporting transversal $C^{r-1}Z$ gates on a close-to-linear number $k\geq n^{1-ε}$ of disjoint tuples of logical qubits, for arbitrarily small $ε>0$. Our construction is the first with constant-weight stabilizers that obtains $dk\gg n$, and as a consequence achieves arbitrarily small magic state overhead exponent $γ=\log(n/k)/\log(d)>0$. Comparable prior constructions instead required at least polylogarithmic stabilizer weight. We also show how to obtain linearly many $k=Ω(n)$ logical $C^{r-1}Z$ gates, though with stabilizer weight and physical circuit depth $n^ε$. We show that our transversal gates also support addressing (i.e. targeting) of specific logical qubits. To obtain our codes, we develop a general transformation based on cup products that maps classical codes satisfying a multiplication property to quantum codes with transversal $C^{r-1}Z$. We apply this transformation to a new family of classical Tanner codes that we construct from punctured tensor products of algebraic codes.
