Quantum information theory (QIT) characterizes the capabilities and fundamental limits of quantum information processing, underpinning quantum communication, computation, and error correction. Formalizing its coding theorems requires connecting finite-block protocols, analytic inequalities, and asymptotic limits within a unified machine-checked framework. Existing developments, however, lack a reusable operational layer that defines codes, error criteria, achievable rates, and capacities independently of their information-theoretic characterizations. In this work, we present LeanQIT, a Lean 4 library for finite-dimensional QIT. It provides composable, kernel-checked interfaces for quantum states and channels, source and channel codes, finite-block performance criteria, hypothesis testing, one-shot quantities, and asymptotic rate constructions. Using this infrastructure, we formalize Schumacher's quantum source-coding theorem, the Holevo--Schumacher--Westmoreland classical-capacity theorem, and the entanglement-assisted classical-capacity theorem together with its strong converse. By separating operational definitions from analytic characterizations and exposing reusable achievability, converse, and asymptotic components, Lean-QIT provides a machine-readable foundation for formal QIT and a compositional knowledge substrate for emerging AI-assisted formalization, automated proof search, and agentic reasoning in quantum information and computation.
Chengkai Zhu, Ziao Tang, Guocheng Zhen et al.· 2 citations
Logical entangling gates are a major source of physical spacetime overhead in fault-tolerant quantum computation. Phantom codes reduce this cost by implementing every ordered in-block logical CNOT through physical qubit permutations and Pauli-frame updates. Whether this mechanism can coexist with the low-weight stabilizer structure of qLDPC codes is a central question for low-overhead fault-tolerant architectures. We give a deterministic answer within binary CSS hypergraph product (HGP) codes. Up to natural equivalences, the simplex-repetition family is the unique HGP family satisfying the phantom condition. We then evaluate this family under circuit-level noise in logical GHZ-state preparation and Trotterized many-body quantum simulation. The codes retain low-weight stabilizer checks and yield concrete advantages over rotated surface-code baselines in both benchmarks. Reconfigurable neutral-atom arrays offer a natural setting for this approach, supporting nonlocal qLDPC operations while enabling in-block logical CNOTs without additional physical operations. Together, these results make precise how permutation-based logical entangling constrains code design within the HGP framework, demonstrate the circuit-level benefits of the unique family, and guide the search for phantom qLDPC families with better asymptotic parameters for low-overhead fault tolerance on neutral-atom hardware.