Clifford group lies at the core of quantum computation—it underlies quantum error correction, its elements can be used to perform magic state distillation and they form randomized benchmarking protocols, Clifford group is used to study quantum entanglement, and more. The ability to utilize Clifford group elements in practice relies heavily on the efficiency of their circuit-level implementation. Finding short circuits is a hard problem; despite Clifford group being finite, its size grows quickly with the number of qubits n, limiting known optimal implementations to n = 4 qubits. For n = 6, the number of Clifford group elements is about 2.1 × 10²³. In this paper, we report a set of algorithms, along with their C++ implementation, that implicitly synthesize optimal circuits for all 6-qubit Clifford group elements by storing a subset of the latter in a database of size 2.1TB (1kB = 1024B). We demonstrate how to extract arbitrary optimal 6-qubit Clifford circuit in 0.0009358 and 0.0006274 s using consumer- and enterprise-grade computers (hardware) respectively, while relying on this database. We use this implementation to establish a new example of quantum advantage by Clifford circuits over CNOT gate circuits and find optimal Clifford 2-designs for up to 4 qubits.

克利福德小组是量子计算的核心——它是量子纠错的基础，它的元素可用于执行魔法状态蒸馏，它们形成随机基准测试协议，克利福德小组用于研究量子纠缠等等。在实践中使用 Clifford 群元素的能力在很大程度上取决于其电路级实现的效率。寻找短路是一个难题。尽管 Clifford 群是有限的，但它的大小会随着量子比特n的数量快速增长，将已知的最佳实现限制为n  = 4 量子比特。对于n  = 6，克利福德群元素的数量约为 2.1 × 10 ²³。在本文中，我们报告了一组算法，以及它们的 C ++实现，通过将后者的子集存储在大小为 2.1TB (1kB = 1024B) 的数据库中，隐式合成所有 6 量子比特 Clifford 组元素的最佳电路。我们演示了如何分别使用消费级和企业级计算机（硬件）在 0.0009358 和 0.0006274 秒内提取任意最佳 6 量子比特 Clifford 电路，同时依赖此数据库。我们使用此实现来建立 Clifford 电路相对于 CNOT 门电路的量子优势的新示例，并找到最多 4 个量子位的最佳 Clifford 2 设计。