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arXiv · 2101.08939

Hoare meets Heisenberg: A Lightweight Logic for Quantum Programs

Aarthi Sundaram·Robert Rand·Kartik Singhal·Youngchan Cho·Brad Lackey
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Abstract

We show that Gottesman's (1998) semantics for Clifford circuits based on the Heisenberg representation gives rise to a lightweight Hoare-like logic for efficiently characterizing 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 bipartition, (iii) checking the transversality of a gate with respect to a given stabilizer code, and (iv) computing post-measurement states for computational basis measurements. Further, this logic is extended to accommodate universal quantum computing by deriving Hoare triples for the $T$-gate, multiply-controlled unitaries such as the Toffoli gate, and some gate injection circuits that use associated magic states. A number of interesting results emerge from this logic, including a lower bound on the number of $T$ gates necessary to perform a multiply-controlled $Z$ gate.

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Aarthi Sundaram, Robert Rand, Kartik Singhal, Youngchan Cho, Brad Lackey. 2021-01-22. Hoare meets Heisenberg: A Lightweight Logic for Quantum Programs. https://doi.org/10.22331/q-2026-07-23-2172

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