Skip to main content

T13-3 形式化规范：量子φ-计算等价性定理

核心命题​

命题 T13-3：量子计算与φ-递归计算在no-11约束下完全等价。

形式化陈述​

∀Q : QuantumAlgorithm . ∃Φ : PhiRecursiveAlgorithm .
  ComputationallyEquivalent(Q, Φ) ∧
  ComplexityPreserving(Q, Φ) ∧
  ResourceEquivalent(Q, Φ) ∧
  OutputEquivalent(Q, Φ)

其中：

  • Q是任意量子算法
  • Φ是对应的φ-递归算法
  • 所有运算满足no-11约束

形式化组件​

1. φ-量子状态空间​

PhiQuantumState ≡ record {
  amplitudes : List[PhiComplex]
  basis_states : List[ZeckendorfBasis]
  normalization : PhiReal
  no_11_constraint : Boolean
}

PhiComplex ≡ record {
  real_part : PhiReal
  imag_part : PhiReal
  zeckendorf_rep : ZeckendorfRep
}

ZeckendorfBasis ≡ {
  indices : List[ℕ]
  constraint : ∀i, j ∈ indices . |i - j| ≠ 1
}

PhiReal ≡ record {
  coefficients : List[{0, 1}]
  exponents : List[ℕ]
  phi_power_sum : ℝ
}

2. φ-量子门算子​

PhiQuantumGate ≡ record {
  matrix : PhiMatrix
  arity : ℕ
  unitary_check : PhiMatrix → Boolean
  no_11_preserving : Boolean
}

PhiMatrix ≡ record {
  elements : Array[Array[PhiComplex]]
  dimensions : (ℕ, ℕ)
  determinant : PhiComplex
}

HadamardPhi : PhiQuantumGate ≡ record {
  matrix = (1/√φ) × [[1, 1], [1, -1]]_φ
  arity = 1
  unitary_check = λM . M × M† = I_φ
  no_11_preserving = True
}

PauliXPhi : PhiQuantumGate ≡ record {
  matrix = [[0, 1], [1, 0]]_φ
  arity = 1
  unitary_check = λM . M × M† = I_φ
  no_11_preserving = True
}

CNOTPhi : PhiQuantumGate ≡ record {
  matrix = [[1,0,0,0], [0,1,0,0], [0,0,0,1], [0,0,1,0]]_φ
  arity = 2
  unitary_check = λM . M × M† = I_φ
  no_11_preserving = True
}

3. φ-量子电路​

PhiQuantumCircuit ≡ record {
  qubits : ℕ
  gates : List[PhiQuantumGate]
  depth : ℕ
  complexity : PhiComplexity
}

PhiComplexity ≡ record {
  time_complexity : ℕ → PhiReal
  space_complexity : ℕ → PhiReal
  gate_count : ℕ
  recursive_depth : ℕ
}

CircuitExecution : PhiQuantumCircuit → PhiQuantumState → PhiQuantumState ≡
  λcircuit, initial_state .
    Fold(ApplyGate, initial_state, circuit.gates)

ApplyGate : PhiQuantumGate → PhiQuantumState → PhiQuantumState ≡
  λgate, state .
    if ValidateNo11Constraint(gate, state) then
      MatrixVectorMultiply(gate.matrix, state.amplitudes)
    else
      ERROR("No-11 constraint violation")

4. φ-递归计算模型​

PhiRecursiveComputation ≡ record {
  recursion_depth : ℕ
  self_reference_structure : SelfRefStruct
  computation_rules : List[RecursionRule]
  termination_condition : Predicate
}

SelfRefStruct ≡ record {
  psi_function : PhiFunction
  fixed_point : PhiValue
  convergence_rate : PhiReal
}

PhiFunction ≡ record {
  domain : PhiDomain
  codomain : PhiDomain
  rule : PhiDomain → PhiDomain
  recursive_call : PhiFunction → PhiFunction
}

RecursionRule ≡ record {
  pattern : PhiPattern
  replacement : PhiExpression
  depth_change : ℤ
  entropy_change : PhiReal
}

5. 等价性映射​

QuantumToPhiMapping : QuantumAlgorithm → PhiRecursiveAlgorithm ≡
  λQ . record {
    state_mapping = QuantumStateToPhiState(Q.initial_state)
    gate_mapping = Map(QuantumGateToPhiRecursion, Q.gates)
    measurement_mapping = QuantumMeasurementToPhiProjection(Q.measurement)
    complexity_mapping = ComplexityTransform(Q.complexity)
  }

QuantumStateToPhiState : QuantumState → PhiQuantumState ≡
  λstate . record {
    amplitudes = Map(ComplexToPhiComplex, state.amplitudes)
    basis_states = Map(BasisToZeckendorf, state.basis)
    normalization = RealToPhiReal(state.norm)
    no_11_constraint = ValidateConstraint(state)
  }

ComplexToPhiComplex : ℂ → PhiComplex ≡
  λz . record {
    real_part = RealToPhiReal(Re(z))
    imag_part = RealToPhiReal(Im(z))
    zeckendorf_rep = ComplexToZeckendorf(z)
  }

BasisToZeckendorf : BasisState → ZeckendorfBasis ≡
  λbasis . record {
    indices = BinaryToFibonacciIndices(basis)
    constraint = ValidateNo11(indices)
  }

6. 复杂度保持定理​

theorem ComplexityPreservation:
  ∀Q : QuantumAlgorithm . ∀Φ : PhiRecursiveAlgorithm .
    QuantumToPhiMapping(Q) = Φ →
    |TimeComplexity(Q) - TimeComplexity(Φ)| ≤ O(log_φ(InputSize))

proof:
  设输入大小为 n
  量子算法 Q 的时间复杂度为 T_Q(n)
  对应φ-递归算法 Φ 的时间复杂度为 T_Φ(n)
  
  根据映射构造：
  1. 每个量子门对应 O(1) 个φ-递归操作
  2. 状态映射的开销为 O(log_φ(n))
  3. 测量映射的开销为 O(log_φ(n))
  
  因此：T_Φ(n) = T_Q(n) + O(log_φ(n))
  ∎

7. 量子门的φ-递归实现​

HadamardPhiImplementation : PhiRecursiveFunction ≡
  λstate . record {
    function_type = "superposition_creator"
    recursion_rule = λψ . (ψ + φ⁻¹ψ(ψ)) / √φ
    depth_bound = 1
    no_11_preservation = λinput . ValidateNo11(Apply(recursion_rule, input))
  }

CNOTPhiImplementation : PhiRecursiveFunction ≡
  λstate1, state2 . record {
    function_type = "entanglement_creator"
    recursion_rule = λψ₁, ψ₂ . 
      if ExtractBit(ψ₁) = 0 then (ψ₁, ψ₂)
      else (ψ₁, FlipBit(ψ₂))
    depth_bound = 2
    no_11_preservation = λinput . ValidateNo11(Apply(recursion_rule, input))
  }

ExtractBit : PhiQuantumState → {0, 1} ≡
  λstate . 
    let measurement = ProjectToZeckendorfBasis(state) in
    if ZeckendorfSum(measurement) mod 2 = 0 then 0 else 1

FlipBit : PhiQuantumState → PhiQuantumState ≡
  λstate . ApplyPauliX(state)

8. 量子算法的φ-递归等价实现​

GroverPhiAlgorithm : PhiRecursiveAlgorithm ≡ record {
  initialization = λn . HadamardPhi^⊗n(|0⟩^⊗n)
  oracle = λx . (-1)^f(x) × ZeckendorfPhase(x)
  diffusion = λψ . 2|s⟩⟨s| - I where |s⟩ = initialization
  iteration_count = ⌊π√(2^n)/4⌋
  recursion_depth = O(√(2^n))
}

ZeckendorfPhase : ZeckendorfRep → PhiComplex ≡
  λrep . exp(i × ZeckendorfSum(rep) / φ)

ShorPhiAlgorithm : PhiRecursiveAlgorithm ≡ record {
  period_finding = λa, N . QuantumFourierTransformPhi(ModularExponentiation(a, N))
  modular_exponentiation = λa, x, N . a^x mod N in φ-arithmetic
  continued_fractions = λfraction . EuclideanAlgorithmPhi(fraction)
  factorization = λp . if p | N then (p, N/p) else retry
  recursion_depth = O((log N)³)
}

QuantumFourierTransformPhi : PhiQuantumState → PhiQuantumState ≡
  λstate . record {
    transform_matrix = (1/√φ^n) × [ω_φ^(jk)]_{j,k=0}^{2^n-1}
    omega_phi = exp(2πi/φ^n)
    application = MatrixVectorMultiply(transform_matrix, state.amplitudes)
    no_11_constraint = ValidateQFTConstraint(result)
  }

9. 纠缠熵的φ-递归表达​

PhiEntanglementEntropy : PhiQuantumState → PhiReal ≡
  λstate .
    let reduced_state = PartialTrace(state) in
    let eigenvalues = Eigenvalues(reduced_state.density_matrix) in
    -Sum(λλ . λ × log_φ(λ), eigenvalues)

RecursiveDepth : SelfRefStruct → ℕ ≡
  λstruct . 
    let fixed_point = FindFixedPoint(struct.psi_function) in
    CountRecursionLevels(fixed_point, struct.convergence_rate)

EntanglementRecursionEquivalence : PhiQuantumState → Boolean ≡
  λstate .
    PhiEntanglementEntropy(state) = log_φ(RecursiveDepth(StateToSelfRef(state)))

StateToSelfRef : PhiQuantumState → SelfRefStruct ≡
  λstate . record {
    psi_function = ExtractSelfReference(state.amplitudes)
    fixed_point = NormalizedState(state)
    convergence_rate = CalculateConvergenceRate(state)
  }

10. 量子错误纠正的φ-实现​

ShorCodePhi : PhiQuantumCode ≡ record {
  logical_qubits = 1
  physical_qubits = 9
  encoding = λ|ψ⟩ . |ψ⟩ → (α|000⟩ + β|111⟩)^⊗3 in φ-encoding
  decoding = ProjectToLogicalSubspace
  error_correction = SyndromeExtractionPhi
  distance = 3
}

StabilizerPhi : PhiStabilizer ≡ record {
  generators = List[PhiPauliOperator]
  commutation_relations = ∀g₁, g₂ ∈ generators . [g₁, g₂] = 0
  logical_operators = {X_L, Z_L} where [X_L, generator] = 0
  syndrome_extraction = λerror . Map(λg . ⟨g, error⟩, generators)
}

ErrorCorrectionPhi : PhiQuantumState → ErrorSyndrome → PhiQuantumState ≡
  λstate, syndrome .
    let error_operator = SyndromeToError(syndrome) in
    let corrected_state = ApplyCorrection(error_operator, state) in
    if ValidateNo11Constraint(corrected_state) then
      corrected_state
    else
      ApplyPhiConstraintCorrection(corrected_state)

11. 计算复杂度类的φ-等价​

PhiBQP : ComplexityClass ≡ {
  problems : Set[DecisionProblem]
  resource_bound = PolynomialPhiQuantumCircuit
  error_probability ≤ 1/3
  constraint = ∀circuit ∈ resource_bound . ValidateNo11Constraint(circuit)
}

BQPToPhiBQPReduction : BQP → PhiBQP ≡
  λproblem ∈ BQP .
    let quantum_circuit = BQPCircuit(problem) in
    let phi_circuit = QuantumToPhiMapping(quantum_circuit) in
    ConstructPhiBQPProblem(phi_circuit)

PhiBQPToBQPReduction : PhiBQP → BQP ≡
  λproblem ∈ PhiBQP .
    let phi_circuit = PhiBQPCircuit(problem) in
    let quantum_circuit = PhiToQuantumMapping(phi_circuit) in
    ConstructBQPProblem(quantum_circuit)

ComplexityClassEquivalence : BQP ≡ PhiBQP ≡
  ∀P . (P ∈ BQP ↔ P ∈ PhiBQP)

验证条件​

1. 状态等价性​

  • QuantumState ↔ PhiQuantumState 双射映射
  • 归一化条件在φ-域中保持
  • 叠加原理的φ-递归实现

2. 演化等价性​

  • 量子门 ↔ φ-递归函数等价
  • 酉性在φ-约束下保持
  • 时间演化的递归深度有界

3. 测量等价性​

  • 量子测量 ↔ φ-信息提取等价
  • Born规则的φ-递归表达
  • 测量塌缩的递归解释

4. 复杂度等价性​

  • 时间复杂度：误差 ≤ O(log_φ n)
  • 空间复杂度：误差 ≤ O(log_φ n)
  • 量子资源与φ-递归资源等价

5. no-11约束保持​

  • 所有中间状态满足约束
  • 门操作保持约束
  • 错误纠正恢复约束

算法规范​

算法1：量子态φ-映射​

def quantum_state_to_phi_state(quantum_state: QuantumState) -> PhiQuantumState:
    """将量子态映射为φ-量子态"""
    # 前置条件
    assert is_normalized(quantum_state)
    
    phi_amplitudes = []
    for amplitude in quantum_state.amplitudes:
        phi_complex = complex_to_phi_complex(amplitude)
        assert validate_no_11_constraint(phi_complex.zeckendorf_rep)
        phi_amplitudes.append(phi_complex)
    
    phi_basis = []
    for basis_state in quantum_state.basis_states:
        zeck_basis = binary_to_zeckendorf_basis(basis_state)
        assert validate_no_consecutive_indices(zeck_basis.indices)
        phi_basis.append(zeck_basis)
    
    phi_state = PhiQuantumState(
        amplitudes=phi_amplitudes,
        basis_states=phi_basis,
        normalization=calculate_phi_norm(phi_amplitudes),
        no_11_constraint=True
    )
    
    # 后置条件
    assert validate_phi_normalization(phi_state)
    assert validate_equivalence(quantum_state, phi_state)
    return phi_state

算法2：φ-量子门执行​

def apply_phi_quantum_gate(gate: PhiQuantumGate, state: PhiQuantumState) -> PhiQuantumState:
    """执行φ-量子门操作"""
    # 前置条件
    assert gate.no_11_preserving
    assert state.no_11_constraint
    
    # 矩阵向量乘法（φ-算术）
    new_amplitudes = []
    for i, basis_state in enumerate(state.basis_states):
        amplitude_sum = PhiComplex(0, 0)
        for j, input_amplitude in enumerate(state.amplitudes):
            matrix_element = gate.matrix.elements[i][j]
            product = phi_complex_multiply(matrix_element, input_amplitude)
            amplitude_sum = phi_complex_add(amplitude_sum, product)
        new_amplitudes.append(amplitude_sum)
    
    # 验证no-11约束
    for amplitude in new_amplitudes:
        assert validate_no_11_constraint(amplitude.zeckendorf_rep)
    
    new_state = PhiQuantumState(
        amplitudes=new_amplitudes,
        basis_states=state.basis_states,
        normalization=calculate_phi_norm(new_amplitudes),
        no_11_constraint=True
    )
    
    # 后置条件
    assert validate_unitarity(gate, state, new_state)
    return new_state

算法3：φ-Grover算法​

def phi_grover_algorithm(oracle_function: Callable, n_qubits: int) -> PhiQuantumState:
    """φ-递归实现的Grover算法"""
    # 初始化：创建均匀叠加态
    initial_state = create_phi_superposition(n_qubits)
    
    # 计算迭代次数
    N = 2 ** n_qubits
    iterations = int(np.pi * np.sqrt(N) / 4)
    
    current_state = initial_state
    
    for _ in range(iterations):
        # 应用φ-Oracle
        current_state = apply_phi_oracle(oracle_function, current_state)
        
        # 应用φ-扩散算子
        current_state = apply_phi_diffusion(current_state, initial_state)
        
        # 验证约束保持
        assert current_state.no_11_constraint
    
    return current_state

def apply_phi_oracle(oracle_func: Callable, state: PhiQuantumState) -> PhiQuantumState:
    """应用φ-Oracle操作"""
    new_amplitudes = []
    for i, amplitude in enumerate(state.amplitudes):
        basis_value = zeckendorf_to_integer(state.basis_states[i].indices)
        if oracle_func(basis_value):
            # 应用相位翻转（φ-编码）
            phase_factor = calculate_zeckendorf_phase(basis_value)
            new_amplitude = phi_complex_multiply(amplitude, phase_factor)
        else:
            new_amplitude = amplitude
        
        assert validate_no_11_constraint(new_amplitude.zeckendorf_rep)
        new_amplitudes.append(new_amplitude)
    
    return PhiQuantumState(
        amplitudes=new_amplitudes,
        basis_states=state.basis_states,
        normalization=calculate_phi_norm(new_amplitudes),
        no_11_constraint=True
    )

算法4：φ-量子傅里叶变换​

def phi_quantum_fourier_transform(state: PhiQuantumState) -> PhiQuantumState:
    """φ-量子傅里叶变换实现"""
    n_qubits = len(state.basis_states).bit_length() - 1
    N = 2 ** n_qubits
    
    # 构建φ-QFT矩阵
    qft_matrix = create_phi_qft_matrix(N)
    
    # 应用变换
    new_amplitudes = []
    for k in range(N):
        amplitude_sum = PhiComplex(0, 0)
        for j in range(N):
            omega_power = (j * k) % N
            omega_factor = calculate_phi_unit_root(omega_power, N)
            original_amplitude = state.amplitudes[j]
            product = phi_complex_multiply(omega_factor, original_amplitude)
            amplitude_sum = phi_complex_add(amplitude_sum, product)
        
        # 归一化因子
        normalized_amplitude = phi_complex_divide(amplitude_sum, phi_sqrt(N))
        assert validate_no_11_constraint(normalized_amplitude.zeckendorf_rep)
        new_amplitudes.append(normalized_amplitude)
    
    return PhiQuantumState(
        amplitudes=new_amplitudes,
        basis_states=create_fourier_basis_states(N),
        normalization=PhiReal(1.0),
        no_11_constraint=True
    )

def calculate_phi_unit_root(power: int, N: int) -> PhiComplex:
    """计算φ-单位根"""
    angle = 2 * np.pi * power / N
    # 转换为φ-复数表示
    real_part = phi_cos(angle)
    imag_part = phi_sin(angle)
    return PhiComplex(real_part, imag_part)

实现注意事项​

  1. φ-算术精度：所有计算必须在φ-数域中进行
  2. no-11约束检查：每步操作后都需验证约束
  3. 递归深度管理：控制φ-递归的展开深度
  4. 误差累积控制：φ-算术的舍入误差处理
  5. 资源优化：利用φ-递归的并行特性

性能指标​

  1. 等价性精度：量子态映射误差 < φ^(-16)
  2. 复杂度保持：时间复杂度误差 < O(log_φ n)
  3. 约束满足率：no-11约束满足率 = 100%
  4. 资源效率：φ-递归资源利用率 > 90%