We introduce a methodology for generating random multi-qubit stabilizer codes based on solving a constraint satisfaction problem (CSP) on random bipartite graphs. This framework allows us to enforce stabilizer commutation, X/Z balancing, finite rate, sparsity, and maximum-degree constraints simultaneously in a CSP that we can then solve numerically. Using a state-of-the-art CSP solver, we obtain convincing evidence for the existence of a satisfiability threshold. Furthermore, the extent of the satisfiable phase increases with the number of qubits. In that phase, finding sparse codes becomes an easy problem. Moreover, we observe that the sparse codes found in the satisfiable phase practically achieve the channel capacity for erasure noise. Our results show that intermediate-size finite-rate sparse quantum codes are easy to find, while also demonstrating a flexible methodology for generating good codes with custom properties. We therefore establish a complete and customizable pipeline for random quantum code discovery that can be geared towards near to mid-term quantum processor layouts.
翻译:我们引入了一种方法来随机生成基于随机双方图解限制满意度问题(CSP)的多位稳定器代码。 这个框架允许我们同时在 CSP 中执行稳定转换、 X/Z平衡、 有限率、 宽度和最大度限制, 然后我们就可以用数字解决。 使用最先进的 CSP 解答器, 我们获得可靠的证据来证明存在一个可测量性临界值。 此外, 与qubit 数量相比, 可比较性阶段的范围会随着可比较性阶段的增加而增加。 在这一阶段, 找到稀有的代码会成为一个容易的问题。 此外, 我们观察到在可作比较的阶段发现的稀有代码实际上达到了消除噪音的通道能力。 我们的结果显示, 中等规模的有限量的代码很容易找到, 同时展示了生成具有定制特性的好代码的灵活方法。 因此, 我们为随机量代码的发现建立了一个完整和可定制的管道, 可以适应近中期量子处理器布局。