We show that Gottesman's semantics (GROUP22, 1998) for Clifford circuits based on the Heisenberg representation can be treated as a type system that can efficiently characterize a common subset of quantum programs. Our applications include (i) certifying whether auxiliary qubits can be safely disposed of, (ii) determining if a system is separable across a given bi-partition, (iii) checking the transversality of a gate with respect to a given stabilizer code, and (iv) typing post-measurement states for computational basis measurements. Further, this type system is extended to accommodate universal quantum computing by deriving types for the $T$-gate, multiply-controlled unitaries such as the Toffoli gate, and some gate injection circuits that use associated magic states. These types allow us to prove a lower bound on the number of $T$ gates necessary to perform a multiply-controlled $Z$ gate.
翻译:我们显示,戈特斯曼基于海森堡代表制的克里福德电路的语义学(GROUP22, 1998年)可以被视为一种能够有效地描述量子程序共同子集的类型系统。我们的应用程序包括:(一) 证明辅助方位是否能够安全处置,(二) 确定一个系统是否可跨越给定的双向分离,(三) 检查一个门与给定稳定器代码的交叉性,以及(四) 输入测距后状态以进行计算测量。此外,这一类型系统通过为“$-T”门、“托夫利门”等倍控单子以及一些使用相关魔法状态的门式注射电路的衍生型号来适应通用量计算。这些型号让我们能够证明在“$-T”门上存在较低的约束,这是执行“倍控值$-Z”门所必需的。