Formalizing mathematics in a proof assistant, where a machine checks every definition, statement and proof, has set a new standard of rigor. Large language models are now capable of formalizing autonomously, even at the scale of whole textbooks. We bring this standard of rigor to physics, where theoretical arguments carry idealizations that are rarely stated fully, and any logical gaps could have a cascading effect on interdependent results. Recognizing the need to evaluate autoformalization systems for physics, we release AxQM, 1,019 kernel-checkable proof-synthesis tasks over 479 items drawn from the textbook Quantum Computation and Quantum Information by Nielsen and Chuang. The tasks are stated in a custom Lean library of finite-dimensional quantum mechanics. By task count, it is the largest proof-synthesis benchmark in physics by a factor of four. AxQM is derived from a near-complete formalization of the formal portions of the textbook, so every task is guaranteed a solution, which we keep private. Grading of the benchmark is done deterministically by the Lean kernel, which checks that the proof compiles, that no sorry appears in it or in any declaration it depends on, and that it introduces no new axioms.
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15 pages, 3 figures. Benchmark available at https://github.com/Axiomatic-AI/AxQM
Formalizing mathematics in a proof assistant, where a machine checks every definition, statement and proof, has set a new standard of rigor. Large language models are now capable of formalizing autonomously, even at the scale of whole textbooks. We bring this standard of rigor to physics, where theoretical arguments carry idealizations that are rarely stated fully, and any logical gaps could have a cascading effect on interdependent results. Recognizing the need to evaluate autoformalization systems for physics, we release AxQM, 1,019 kernel-checkable proof-synthesis tasks over 479 items drawn from the textbook Quantum Computation and Quantum Information by Nielsen and Chuang. The tasks are stated in a custom Lean library of finite-dimensional quantum mechanics. By task count, it is the largest proof-synthesis benchmark in physics by a factor of four. AxQM is derived from a near-complete formalization of the formal portions of the textbook, so every task is guaranteed a solution, which we keep private. Grading of the benchmark is done deterministically by the Lean kernel, which checks that the proof compiles, that no sorry appears in it or in any declaration it depends on, and that it introduces no new axioms.
Formal verification is becoming increasingly practical for quantum computing, yet the ability of AI agents to construct machine-checkable proofs in this domain remains unmeasured. We introduce Lean-QuantumAlg-Bench and Lean-QIT-Bench, two Lean 4 benchmarks containing 36 and 40 theorem-completion tasks for quantum algorithms and quantum information theory, respectively. Every task compiles in a fixed environment and is evaluated by deterministic proof checking and targeted semantic review, with difficulty weights assigned before model execution. We evaluate four models-GPT-5.5, Kimi K3, DeepSeek V4-Pro, and MiniMax M3-within a common theorem-proving framework under two settings: a task-only baseline and library-augmented deduction (LAD), which additionally provides access to a verified domain library. The highest difficulty-weighted scores are 60.4 out of 100 on the quantum-algorithm benchmark and 59.6 out of 100 on the quantum-information benchmark. LAD improves both score and completion rate in all eight model-benchmark comparisons, with gains of up to 15.9 points, providing evidence that verified libraries can strengthen domain-specific proof agents. The results reveal recurring weaknesses of agentic proving in areas such as quantum simulation, quantum learning, quantum information measures, and entanglement theory. Monetary and wall-clock costs per score point also vary substantially across models, highlighting important capability-efficiency trade-offs. We expect these benchmarks to establish a reproducible baseline for developing more capable and reliable proof agents, and to pave the way toward self-evolving AI scientists for advancing quantum information science.
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.
Quantum key distribution provides secure communication by relying on physics principles and mathematical reasoning. Security proofs across the field can contain gaps, often between individually valid theorems integrated under mismatched assumptions. We produce an end-to-end security theorem formalized in the Lean proof assistant, eliminating hidden hypotheses by building only on Lean's standard axioms. Our proof covers composable security of the entanglement-based qubit BB84 protocol against coherent attacks by the postselection technique, accounting for finite block size, and is written in a language that enforces local operations and classical communication to help ensure a faithful representation of the protocol. The proof itself was enabled by LLM-based autoformalization, and with the proof complete, we used the same pipeline as a discovery machine, reducing the finite-size penalty on the key rate. Our work sets a reproducible standard for security arguments in quantum cryptography and shows how formal proof engines can accelerate discovery across quantum information science.
Practical quantum programs interleave quantum and classical computation: classical data shape the quantum operations that are performed, and measurement outcomes shape the classical computation that follows. Formal verification of such programs in a proof assistant requires mechanizing a program logic that handles this interplay consistently at every level, from the mathematics of quantum mechanics, through program syntax and semantics, to specifications and proof rules. Such mechanizations, along with the program proofs built on them, typically require substantial manual effort.
We present QEDIA, a framework for verifying quantum programs with classical variables in Lean 4. At its foundation is an open library that formalizes the quantum Hoare logic QHL+ across all these levels and proves the logic sound for both partial and total correctness. We construct the library and the program proofs, under human supervision, using an LLM-based prover agent, whose general-purpose core we extend for quantum program verification. We illustrate QEDIA with verified programs including the quantum Fourier transform, phase estimation, repeat-until-success circuits, and Hamiltonian simulation.
We build a team of specialized large language-model agents and present an agent-driven workflow for research-level formalization in theoretical physics, with the autoformalization of the fundamental theorem of matrix-product states as a demonstration. The agents, coordinated through a structured mathematical blueprint and periodic human review, orchestrated and executed the full formalization autonomously. For some statements, the agents were able to explore new proof routes that are not part of the standard literature. Along the way the agents produced extensive tensor-network and quantum-information libraries not previously available in Mathlib, Lean's mathematical library. As a physical application, the formalization also extends towards symmetry-protected topological phases in one dimension. We find that the main bottleneck in large-scale autoformalization is enforcing mathematical intent and we provide a detailed study of the full process and various subtleties involved. We release the codebase as the library \href{https://github.com/LionSR/TNLean}{TNLean}, together with a \nChapters{}-chapter \href{https://lionsr.github.io/TNLean/blueprint/}{blueprint} of the formalization effort.
Quantum information theory is built on entropic quantities; among them, the sandwiched Rényi relative entropy is a fundamental divergence with various applications, and its data processing inequality (DPI) under quantum channels is a cornerstone result. In this work, we present a Lean 4 library for quantum information, designed as a reusable formal infrastructure for theoretical analysis. As a central demonstration of the library, we formalize the DPI for the sandwiched Rényi relative entropy for positive semidefinite operators on finite-dimensional quantum systems. The library provides a basis-independent operator-theoretic framework for finite-dimensional quantum mechanics compatible with the standard mathematical library Mathlib, including reusable interfaces for finite-dimensional systems, states, channels, tensor products, partial traces, Choi operators, Kraus representations, and Stinespring representations. It also builds infrastructure for noncommutative trace inequalities, including operator monotonicity and convexity via the real continuous functional calculus, block-operator positivity, Hilbert-Schmidt operator spaces, Jensen's operator inequality, generalized perspectives, operator power means, and Lieb-Ando trace inequalities. On top of this framework, we formalize entropy-specific ingredients for the DPI: variational formulas for the sandwiched quasi-entropy via Young and reverse-Young inequalities, tensor-product compatibility of real powers, and Haar measures on unitary groups. Together, these components yield a Lean formalization of the DPI, give strong subadditivity as a corollary, and provide the last missing component needed to complete the Lean formalization of the generalized quantum Stein's lemma. More broadly, the development provides machine-checkable foundations for future formalized and AI-assisted research in quantum information theory.