🤖 AI Summary
This work addresses four fundamental challenges in quantum program verification: safe ancilla qubit elimination, bipartite separability checking, transversality verification of gate operations, and post-measurement state derivation. We propose the first verifiable type system grounded in the Heisenberg representation and stabilizer theory. Methodologically, we elevate Gottesman semantics to a static type framework—enabling efficient characterization of Clifford circuits—and extend it to support T gates, Toffoli gates, and magic-state injection circuits, thereby facilitating Hoare triple derivation and T-count lower-bound proofs. Our contributions are threefold: (1) full automation of the four core verification tasks; (2) a rigorous proof of the tight T-count lower bound for multi-controlled Z gates; and (3) a quantum program type system that jointly achieves high expressiveness and polynomial-time efficiency, establishing the first verifiable and scalable static analysis foundation for general quantum programs.
📝 Abstract
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.
