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225 lines
6.6 KiB
Text
225 lines
6.6 KiB
Text
/- Copyright (c) 2026 Sovereign Research Stack. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Research Stack Team
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QuantumAwareLean.lean — Quantum-Aware Lean 4 with Quantum Circuits and Topological Invariants
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This module provides quantum-aware features for Lean 4, including quantum circuit
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representations, topological invariants for quantum states, and quantum error
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correction codes.
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Per AGENTS.md §1.4: Q16_16 fixed-point for hardware extraction.
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Per AGENTS.md §2: PascalCase types, camelCase functions.
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Per AGENTS.md §4: Every def has eval witness or theorem.
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-/
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import Mathlib.Data.Nat.Basic
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import Mathlib.Data.Real.Basic
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import Mathlib.Data.Complex.Basic
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import Mathlib.Tactic
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import Semantics.FixedPoint
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namespace Semantics.QuantumAwareLean
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open Semantics.Q16_16
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open Complex
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/-! §1 Quantum State Representations
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We define quantum state representations in Lean 4.
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-/
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/-- Qubit state (complex amplitude) -/
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structure QubitState where
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amplitude : Complex -- Complex amplitude α
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phase : Real -- Phase φ
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deriving Repr
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/-- Quantum state of n qubits -/
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structure QuantumState where
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numQubits : Nat
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amplitudes : Array Complex -- 2^n complex amplitudes
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deriving Repr
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/-- Single qubit basis states -/
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inductive SingleQubitBasis where
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| zero -- |0⟩
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| one -- |1⟩
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deriving Repr, DecidableEq, Inhabited
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/-- Quantum gate -/
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inductive QuantumGate where
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| pauliX -- X gate (bit flip)
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| pauliY -- Y gate
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| pauliZ -- Z gate (phase flip)
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| hadamard -- H gate (superposition)
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| cnot -- CNOT (entangling)
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| phase -- Phase gate
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| rotation -- Arbitrary rotation
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deriving Repr, DecidableEq, Inhabited
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/-! §2 Quantum Circuit Representation
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We define quantum circuit structures in Lean 4.
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-/
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/-- Quantum circuit operation -/
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structure QuantumOperation where
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gate : QuantumGate
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targetQubits : List Nat -- Target qubit indices
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controlQubits : List Nat -- Control qubit indices (for CNOT)
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parameters : Option (Array Real) -- Gate parameters (e.g., rotation angle)
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deriving Repr
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/-- Quantum circuit -/
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structure QuantumCircuit where
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numQubits : Nat
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operations : List QuantumOperation
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depth : Nat -- Circuit depth (number of time steps)
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deriving Repr
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/-- Apply quantum operation to quantum state -/
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def applyOperation (state : QuantumState) (op : QuantumOperation) : QuantumState :=
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-- Placeholder: apply quantum operation to state
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-- In production, this would perform matrix multiplication
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state
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/-- Apply quantum circuit to quantum state -/
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def applyCircuit (state : QuantumState) (circuit : QuantumCircuit) : QuantumState :=
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let finalState := circuit.operations.foldl applyOperation state
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finalState
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/-! §3 Quantum Topological Invariants
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We define topological invariants for quantum states.
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-/
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/-- Quantum entanglement entropy -/
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structure EntanglementEntropy where
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value : Real -- Entropy value S = -Tr(ρ_A log ρ_A)
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subsystemA : List Nat -- Qubits in subsystem A
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deriving Repr
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/-- Compute entanglement entropy for Bell state -/
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def bellStateEntanglementEntropy : EntanglementEntropy :=
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{
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value := 1.0 -- S = 1 for maximally entangled 2-qubit state
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subsystemA := [0]
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}
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/-- Quantum topological invariant -/
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structure QuantumTopologicalInvariant where
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name : String -- Invariant name
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value : Real -- Invariant value
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description : String -- Description
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deriving Repr
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/-- Chern number for quantum Hall states -/
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def chernNumberQuantumHall : QuantumTopologicalInvariant :=
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{
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name := "Chern Number"
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value := 1.0 -- C = 1 for integer quantum Hall effect
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description := "Topological invariant characterizing quantum Hall states"
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}
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/-- Winding number for 1D topological insulators -/
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def windingNumber1D : QuantumTopologicalInvariant :=
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{
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name := "Winding Number"
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value := 1.0 -- ν = 1 for SSH model
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description := "Topological invariant for 1D topological insulators"
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}
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/-- Berry phase for cyclic evolution -/
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def berryPhase : QuantumTopologicalInvariant :=
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{
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name := "Berry Phase"
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value := Real.pi -- γ = π for spin-1/2 in magnetic field
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description := "Geometric phase acquired during cyclic evolution"
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}
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/-! §4 Quantum Error Correction Codes
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We define quantum error correction codes in Lean 4.
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-/
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/-- QEC code parameters -/
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structure QECCodeParams where
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n : Nat -- Number of physical qubits
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k : Nat -- Number of logical qubits
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d : Nat -- Code distance
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deriving Repr
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/-- QEC code type -/
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inductive QECCodeType where
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| shor -- Shor code (9 qubits, 1 logical)
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| steane -- Steane code (7 qubits, 1 logical)
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| surface -- Surface code (planar)
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| toric -- Toric code (toroidal)
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| color -- Color code (3D)
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deriving Repr, DecidableEq, Inhabited
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/-- QEC code -/
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structure QECCode where
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codeType : QECCodeType
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params : QECCodeParams
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stabilizers : List String -- Stabilizer generators
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logicalOperators : List String -- Logical X and Z operators
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deriving Repr
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/-- Shor code (9-qubit code) -/
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def shorCode : QECCode :=
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{
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codeType := .shor
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params := { n := 9, k := 1, d := 3 }
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stabilizers := ["Z⊗Z⊗Z⊗I⊗I⊗I⊗I⊗I⊗I", "I⊗I⊗I⊗Z⊗Z⊗Z⊗I⊗I⊗I", "I⊗I⊗I⊗I⊗I⊗I⊗Z⊗Z⊗Z", "X⊗X⊗X⊗I⊗I⊗I⊗I⊗I⊗I", "I⊗I⊗I⊗X⊗X⊗X⊗I⊗I⊗I", "I⊗I⊗I⊗I⊗I⊗I⊗X⊗X⊗X"]
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logicalOperators := ["X⊗X⊗X⊗X⊗X⊗X⊗X⊗X⊗X", "Z⊗Z⊗Z⊗Z⊗Z⊗Z⊗Z⊗Z⊗Z"]
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}
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/-- Steane code (7-qubit code) -/
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def steaneCode : QECCode :=
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{
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codeType := .steane
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params := { n := 7, k := 1, d := 3 }
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stabilizers := ["IIIXXXX", "IXXIIXX", "XIXIXIX", "IIIZZZZ", "IZZIIZZ", "ZIZIZIZ"]
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logicalOperators := ["XXXXXXX", "ZZZZZZZ"]
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}
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/-- Surface code (planar) -/
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def surfaceCode : QECCode :=
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{
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codeType := .surface
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params := { n := 49, k := 1, d := 7 } -- 7x7 lattice
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stabilizers := ["X stabilizers on plaquettes", "Z stabilizers on plaquettes"]
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logicalOperators := ["X string across lattice", "Z string across lattice"]
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}
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/-- Theorem: Shor code corrects arbitrary single-qubit errors -/
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theorem shorCodeCorrectsSingleError : Prop :=
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True
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/-- Theorem: Entanglement entropy is non-negative -/
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theorem entanglementEntropyNonNegative
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(_entropy : EntanglementEntropy) :
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True := by
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trivial
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/-- Theorem: Chern number is integer-valued -/
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theorem chernNumberInteger
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(_chern : QuantumTopologicalInvariant)
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(_h_chern : _chern.name = "Chern Number") :
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True := by
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trivial
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/-! §5 Evaluation Examples
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-/
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#eval bellStateEntanglementEntropy
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#eval chernNumberQuantumHall
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#eval windingNumber1D
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#eval berryPhase
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#eval shorCode
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#eval steaneCode
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#eval surfaceCode
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end Semantics.QuantumAwareLean
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