Quantum Computing in Lean

A Lean 4 formalization of basic quantum computing definitions.

It is a Lake/mathlib4 project and starts with a small core API for finite-dimensional complex matrices, state vectors, common qubit states and one-, two-, and three-qubit gates, scoped quantum notation, projections, and measurement probabilities.

Highlights

• Thin wrappers and helper lemmas over finite-dimensional complex matrices and vectors, including notation, normalization, and unitarity facts.
• Named one- and two-qubit states, one-, two-, and three-qubit gates, and projectors, with verified gate actions and standard decomposition identities.
• Scoped Dirac ket notation and common gate aliases.
• Computational, generalized, projective, and partial-trace measurement APIs.
• Formalized examples including Hadamard-based uniform random-number generation and variations of the no-cloning theorem.

Layout

The diagram below shows the public module structure at a glance. The text list that follows keeps the individual files easy to scan.

Core

• QuantumComputing/Matrix.lean: thin wrappers and helper lemmas over mathlib matrices and vectors, including adjoints, projections, traces, and Kronecker products.
• QuantumComputing/States.lean: named qubit states, including basis states and Bell states.
• QuantumComputing/State.lean: pure-state and density-matrix wrappers.
• QuantumComputing/Notation.lean: scoped Dirac ket notation, scalar notation, and gate aliases.

Gates

• QuantumComputing/Gates.lean: public gate entry point.
• QuantumComputing/Gates/Basic.lean: named gates such as X, H, CNOT, CZ, SWAP, and TOFFOLI.
• QuantumComputing/Gates/Projectors.lean: named projectors such as P0, P1, PPlus, and PMinus.
• QuantumComputing/Gates/Properties.lean: unitarity, adjoint, and involution facts for named gates.
• QuantumComputing/Gates/Actions.lean: gate actions on named states.
• QuantumComputing/Gates/Decompositions.lean: standard gate decomposition identities.

Measurement

• QuantumComputing/Measurement.lean: public measurement entry point.
• QuantumComputing/Measurement/Computational.lean: computational-basis measurement definitions and probability facts.
• QuantumComputing/Measurement/Generalized.lean: generalized measurements and typed operators.
• QuantumComputing/Measurement/Projective.lean: projective measurements and the bridge to generalized measurements.
• QuantumComputing/Measurement/PartialTrace.lean: partial trace and partial measurement probability facts.
• QuantumComputing/Measurement/Examples.lean: concrete examples for named states.

Theorems

• QuantumComputing/Theorems.lean: public theorem entry point.
• QuantumComputing/Theorems/RandomNumberGenerator.lean: Hadamard-based random-number generation.
• QuantumComputing/Theorems/NoCloning.lean: no-cloning results.

Requirements

This project is pinned to Lean v4.29.1 and mathlib4 v4.29.1. Lake reads the Lean version from lean-toolchain and the mathlib dependency from lakefile.lean.

Build

lake exe cache get
lake build

Using the Library

The top-level module imports the public quantum computing API:

import QuantumComputing

#check QuantumComputing.ketPlus
#check QuantumComputing.H
#check QuantumComputing.Theorems.NoCloning.no_cloning_1

Scoped Dirac ket notation, scalar notation, and gate aliases are available as an opt-in surface:

import QuantumComputing

open scoped QuantumComputing

#check |+⟩
#check |0⟩⊗[3]
#check √2⁻¹
#check QuantumComputing.CX
#check QuantumComputing.CCX

You can also import focused modules when working on a smaller part of the library:

import QuantumComputing.Gates
import QuantumComputing.Notation
import QuantumComputing.Measurement
import QuantumComputing.Theorems.NoCloning

Editor Notes

VS Code's Lean extension treats ⟨ and ⟩ as bracket pairs. Dirac ket notation such as |+⟩ intentionally has a closing ⟩ without a matching ⟨, so VS Code may color the ⟩ as an unmatched bracket even though Lean accepts the file.

To avoid that false editor warning, override Lean's workspace bracket list and omit the ["⟨", "⟩"] pair. A compact .vscode/settings.json workaround is:

{
  "[lean4]": {
    "editor.language.brackets": [
      ["(", ")"],
      ["`(", ")"],
      ["``(", ")"],
      ["[", "]"],
      ["#[", "]"],
      ["@[", "]"],
      ["%[", "]"],
      ["{", "}"]
    ]
  }
}

Reload the VS Code window after changing this setting.

Lean 3 History

The main branch is the Lean 4 version of this project. The previous Lean 3 project is preserved for reference on the historical lean3 branch.

This Lean 4 version is a redesign rather than a line-by-line translation. See docs/porting-from-lean3.md for the porting coverage matrix, intentionally unported Lean 3 infrastructure, and future cleanup ideas.

References

• Quantum Programming tutorial
• An Introduction to Quantum Computing, Without the Physics
• Verified Quantum Computing

Related Projects

• Lean Theorem Prover
• Lean Community
• QWIRE: a quantum circuit language and formal verification tool by Robert Rand et al.