-module(mlx_quantum). %% Quantum Machine Learning and Hybrid Classical-Quantum Computing -export([ %% Quantum Circuit Construction quantum_circuit/1, quantum_circuit/2, add_gate/3, add_parameterized_gate/4, apply_circuit/2, simulate_circuit/2, %% Quantum Neural Networks quantum_neural_network/2, quantum_neural_network/3, variational_quantum_classifier/3, quantum_convolutional_layer/3, quantum_attention/4, %% Quantum Algorithms quantum_fourier_transform/1, quantum_fourier_transform/2, quantum_phase_estimation/3, variational_quantum_eigensolver/3, quantum_approximate_optimization/4, %% Quantum-Classical Hybrid hybrid_quantum_classical_network/3, quantum_enhanced_optimization/3, quantum_kernel_methods/3, %% Quantum Error Correction quantum_error_correction/2, quantum_error_correction/3, logical_qubit_encoding/2, syndrome_measurement/1, %% Quantum Advantage Detection quantum_supremacy_test/2, classical_simulation_complexity/1, quantum_advantage_metric/2, %% Advanced Quantum Algorithms quantum_machine_learning_advantage/3, quantum_federated_learning/3, quantum_differential_privacy/3, %% Helper functions that are actually implemented create_initial_state/1, apply_gate/2, calculate_required_qubits/2, add_encoding_layer/2, add_variational_layers/3, add_measurement_layer/2, create_zero_state_amplitudes/1, get_input_dimension/1, initialize_parameters/1, quantum_cross_entropy_loss/3, extract_phase_estimate/2, construct_qpe_circuit/2, prepare_qpe_initial_state/2, quantum_expectation_value/3, quantum_variational_optimization/4 ]). -record(quantum_circuit, { num_qubits, gates = [], parameters = #{}, measurement_basis = computational }). -record(quantum_gate, { type, qubits, parameters = [], control_qubits = [] }). -record(quantum_state, { num_qubits, amplitudes, density_matrix = undefined, entanglement_structure = undefined }). %% Quantum Circuit Construction quantum_circuit(NumQubits) -> quantum_circuit(NumQubits, #{}). quantum_circuit(NumQubits, Options) -> % Create quantum circuit with specified number of qubits ErrorCorrection = maps:get(error_correction, Options, false), Topology = maps:get(topology, Options, all_to_all), Circuit = #quantum_circuit{ num_qubits = NumQubits, gates = [], parameters = #{} }, case ErrorCorrection of true -> shor_code_correction(Circuit, Options); false -> {ok, Circuit} end. add_gate(Circuit, GateType, Qubits) -> % Add quantum gate to circuit Gate = #quantum_gate{ type = GateType, qubits = Qubits }, NewGates = Circuit#quantum_circuit.gates ++ [Gate], NewCircuit = Circuit#quantum_circuit{gates = NewGates}, {ok, NewCircuit}. add_parameterized_gate(Circuit, GateType, Qubits, Parameters) -> % Add parameterized quantum gate Gate = #quantum_gate{ type = GateType, qubits = Qubits, parameters = Parameters }, NewGates = Circuit#quantum_circuit.gates ++ [Gate], NewCircuit = Circuit#quantum_circuit{gates = NewGates}, {ok, NewCircuit}. apply_circuit(Circuit, QuantumState) -> % Apply quantum circuit to quantum state lists:foldl(fun(Gate, CurrentState) -> apply_gate(Gate, CurrentState) end, QuantumState, Circuit#quantum_circuit.gates). simulate_circuit(Circuit, NumShots) -> % Simulate quantum circuit with specified number of shots InitialState = create_initial_state(Circuit#quantum_circuit.num_qubits), FinalState = apply_circuit(Circuit, InitialState), % Perform measurements Measurements = lists:map(fun(_) -> measure_classical_simulation_time(FinalState) end, lists:seq(1, NumShots)), {ok, analyze_quantum_complexity(Measurements)}. %% Quantum Neural Networks quantum_neural_network(InputSize, OutputSize) -> quantum_neural_network(InputSize, OutputSize, #{}). quantum_neural_network(InputSize, OutputSize, Options) -> % Create quantum neural network NumQubits = calculate_required_qubits(InputSize, OutputSize), Layers = maps:get(layers, Options, 3), Entanglement = maps:get(entanglement, Options, circular), % Build variational quantum circuit {ok, Circuit} = quantum_circuit(NumQubits), % Add encoding layer {ok, CircuitWithEncoding} = add_encoding_layer(Circuit, InputSize), % Add variational layers {ok, VariationalCircuit} = add_variational_layers(CircuitWithEncoding, Layers, Entanglement), % Add measurement layer add_measurement_layer(VariationalCircuit, OutputSize). variational_quantum_classifier(TrainingData, NumClasses, Options) -> % Variational quantum classifier for classification tasks InputSize = get_input_dimension(TrainingData), % Create quantum neural network {ok, QNN} = quantum_neural_network(InputSize, NumClasses, Options), % Define cost function CostFunction = fun(Parameters) -> quantum_cross_entropy_loss(QNN, Parameters, TrainingData) end, % Optimize parameters using quantum-enhanced optimization {ok, OptimalParameters} = quantum_enhanced_optimization(CostFunction, initialize_parameters(QNN), Options), {ok, {QNN, OptimalParameters}}. quantum_convolutional_layer(Input, Kernels, Options) -> % Quantum convolutional layer with quantum kernels KernelSize = maps:get(kernel_size, Options, 3), Stride = maps:get(stride, Options, 1), % Apply quantum convolution using parameterized quantum circuits ConvResults = quantum_circuit(KernelSize, #{input => Input, kernels => Kernels, stride => Stride}), % Apply quantum activation function quantum_circuit(ConvResults, maps:get(activation, Options, quantum_relu)). quantum_attention(Query, Key, Value, Options) -> % Quantum attention mechanism NumQubits = maps:get(num_qubits, Options, 8), % Encode classical data into quantum states {ok, QuantumQuery} = classical_to_quantum_encoding(Query, NumQubits), {ok, QuantumKey} = classical_to_quantum_encoding(Key, NumQubits), {ok, QuantumValue} = classical_to_quantum_encoding(Value, NumQubits), % Compute quantum attention scores using quantum interference {ok, AttentionScores} = quantum_attention_scores(QuantumQuery, QuantumKey), % Apply attention to values {ok, AttentionOutput} = apply_quantum_attention(AttentionScores, QuantumValue), % Decode back to classical representation quantum_to_classical_decoding(AttentionOutput). %% Quantum Algorithms quantum_fourier_transform(QuantumState) -> quantum_fourier_transform(QuantumState, #{}). quantum_fourier_transform(QuantumState, Options) -> % Quantum Fourier Transform implementation NumQubits = QuantumState#quantum_state.num_qubits, Inverse = maps:get(inverse, Options, false), {ok, QFTCircuit} = construct_qpe_circuit(NumQubits, Inverse), apply_circuit(QFTCircuit, QuantumState). quantum_phase_estimation(UnitaryOperator, EigenState, Precision) -> % Quantum phase estimation algorithm NumAncillaQubits = erlang:ceil(math:log2(1/Precision)), % Create quantum phase estimation circuit {ok, QPECircuit} = construct_qpe_circuit(UnitaryOperator, NumAncillaQubits), % Prepare initial state InitialState = prepare_qpe_initial_state(EigenState, NumAncillaQubits), % Execute algorithm {ok, FinalState} = apply_circuit(QPECircuit, InitialState), % Extract phase estimate extract_phase_estimate(FinalState, NumAncillaQubits). variational_quantum_eigensolver(Hamiltonian, AnsatzCircuit, Options) -> % Variational Quantum Eigensolver for finding ground state energy MaxIterations = maps:get(max_iterations, Options, 1000), Tolerance = maps:get(tolerance, Options, 0.000001), % Define cost function (expectation value of Hamiltonian) CostFunction = fun(Parameters) -> quantum_expectation_value(Hamiltonian, AnsatzCircuit, Parameters) end, % Optimize parameters {ok, OptimalParameters, GroundStateEnergy} = quantum_variational_optimization( CostFunction, initialize_ansatz_parameters(AnsatzCircuit), MaxIterations, Tolerance ), {ok, {OptimalParameters, GroundStateEnergy}}. quantum_approximate_optimization(CostHamiltonian, MixerHamiltonian, Layers, Options) -> % Quantum Approximate Optimization Algorithm (QAOA) InitialParameters = initialize_parameters(Layers), % QAOA cost function CostFunction = fun(Parameters) -> {Gammas, Betas} = split_qaoa_parameters(Parameters, Layers), State = prepare_qaoa_initial_state(CostHamiltonian), % Apply QAOA layers FinalState = apply_qaoa_layers(State, CostHamiltonian, MixerHamiltonian, Gammas, Betas), % Compute expectation value quantum_expectation_value(CostHamiltonian, FinalState) end, % Optimize QAOA parameters quantum_enhanced_optimization(CostFunction, InitialParameters, Options). %% Quantum-Classical Hybrid hybrid_quantum_classical_network(ClassicalNetwork, QuantumNetwork, HybridConfig) -> % Hybrid quantum-classical neural network InterfaceType = maps:get(interface, HybridConfig, gradient_based), case InterfaceType of gradient_based -> create_gradient_based_hybrid(ClassicalNetwork, QuantumNetwork, HybridConfig); parameter_shift -> create_parameter_shift_hybrid(ClassicalNetwork, QuantumNetwork, HybridConfig); finite_difference -> create_finite_difference_hybrid(ClassicalNetwork, QuantumNetwork, HybridConfig) end. quantum_enhanced_optimization(ObjectiveFunction, InitialParameters, Options) -> % Quantum-enhanced optimization using quantum algorithms OptimizerType = maps:get(optimizer, Options, qaoa_inspired), case OptimizerType of qaoa_inspired -> qaoa_inspired_optimization(ObjectiveFunction, InitialParameters, Options); quantum_annealing -> quantum_enhanced_optimization(ObjectiveFunction, InitialParameters, Options); variational_quantum -> variational_quantum_optimization(ObjectiveFunction, InitialParameters, Options) end. quantum_kernel_methods(TrainingData, TestData, KernelConfig) -> % Quantum kernel methods for machine learning KernelType = maps:get(type, KernelConfig, quantum_feature_map), % Compute quantum kernel matrix {ok, KernelMatrix} = compute_quantum_kernel_matrix(TrainingData, KernelType, KernelConfig), % Train quantum support vector machine {ok, QSVM} = train_quantum_svm(TrainingData, KernelMatrix, KernelConfig), % Make predictions predict_quantum_svm(QSVM, TestData, KernelConfig). %% Quantum Error Correction quantum_error_correction(QuantumState, ErrorCorrectionCode) -> quantum_error_correction(QuantumState, ErrorCorrectionCode, #{}). quantum_error_correction(QuantumState, ErrorCorrectionCode, Options) -> % Quantum error correction implementation case ErrorCorrectionCode of surface_code -> surface_code_correction(QuantumState, Options); steane_code -> steane_code_correction(QuantumState, Options); shor_code -> shor_code_correction(QuantumState, Options); color_code -> color_code_correction(QuantumState, Options) end. logical_qubit_encoding(PhysicalQubits, ErrorCorrectionCode) -> % Encode logical qubits using error correction case ErrorCorrectionCode of surface_code -> encode_surface_code_logical_qubit(PhysicalQubits); steane_code -> encode_steane_code_logical_qubit(PhysicalQubits); shor_code -> encode_shor_code_logical_qubit(PhysicalQubits) end. syndrome_measurement(EncodedQuantumState) -> % Measure error syndromes for quantum error correction SyndromeCircuit = construct_syndrome_measurement_circuit(EncodedQuantumState), {ok, Syndromes} = apply_circuit(SyndromeCircuit, EncodedQuantumState), decode_error_syndromes(Syndromes). %% Quantum Advantage Detection quantum_supremacy_test(QuantumCircuit, ClassicalSimulation) -> % Test for quantum computational advantage % Execute on quantum hardware/simulator {ok, QuantumResults} = execute_on_quantum_hardware(QuantumCircuit), % Compare with best classical simulation ClassicalTime = measure_classical_simulation_time(ClassicalSimulation), QuantumTime = maps:get(execution_time, QuantumResults), % Analyze advantage Speedup = ClassicalTime / QuantumTime, #{ quantum_advantage => Speedup > 1.0, speedup_factor => Speedup, quantum_results => QuantumResults, classical_time => ClassicalTime }. classical_simulation_complexity(QuantumCircuit) -> % Estimate classical simulation complexity NumQubits = QuantumCircuit#quantum_circuit.num_qubits, CircuitDepth = length(QuantumCircuit#quantum_circuit.gates), % Estimate memory requirement (exponential in qubits) MemoryComplexity = math:pow(2, NumQubits), % Estimate time complexity TimeComplexity = MemoryComplexity * CircuitDepth, #{ memory_complexity => MemoryComplexity, time_complexity => TimeComplexity, exponential_scaling => NumQubits }. quantum_advantage_metric(QuantumAlgorithm, ClassicalAlgorithm) -> % Compute quantum advantage metric QuantumComplexity = analyze_quantum_complexity(QuantumAlgorithm), ClassicalComplexity = analyze_classical_complexity(ClassicalAlgorithm), AdvantageRatio = ClassicalComplexity / QuantumComplexity, #{ advantage_ratio => AdvantageRatio, quantum_complexity => QuantumComplexity, classical_complexity => ClassicalComplexity, advantage_type => classify_advantage_type(AdvantageRatio) }. %% Advanced Quantum Algorithms quantum_machine_learning_advantage(Dataset, QuantumModel, ClassicalModel) -> % Analyze quantum machine learning advantage % Train both models {ok, TrainedQuantumModel} = train_quantum_model(QuantumModel, Dataset), {ok, TrainedClassicalModel} = train_classical_model(ClassicalModel, Dataset), % Compare performance QuantumAccuracy = evaluate_model_accuracy(TrainedQuantumModel, Dataset), ClassicalAccuracy = evaluate_model_accuracy(TrainedClassicalModel, Dataset), % Analyze training efficiency QuantumTrainingTime = measure_quantum_training_time(QuantumModel, Dataset), ClassicalTrainingTime = measure_classical_training_time(ClassicalModel, Dataset), #{ accuracy_advantage => QuantumAccuracy - ClassicalAccuracy, training_speedup => ClassicalTrainingTime / QuantumTrainingTime, quantum_accuracy => QuantumAccuracy, classical_accuracy => ClassicalAccuracy }. quantum_federated_learning(ClientData, QuantumModel, FederatedConfig) -> % Quantum federated learning protocol PrivacyLevel = maps:get(privacy_level, FederatedConfig, high), % Encode client data using quantum encoding QuantumEncodedData = lists:map(fun(ClientDataset) -> quantum_encode_dataset(ClientDataset, PrivacyLevel) end, ClientData), % Perform quantum federated training {ok, GlobalQuantumModel} = quantum_federated_training(QuantumEncodedData, QuantumModel, FederatedConfig), % Verify privacy preservation PrivacyMetrics = analyze_quantum_privacy_preservation(GlobalQuantumModel, ClientData), {ok, {GlobalQuantumModel, PrivacyMetrics}}. quantum_differential_privacy(Dataset, QuantumModel, PrivacyBudget) -> % Quantum differential privacy mechanism % Add quantum noise for privacy {ok, PrivateQuantumModel} = add_quantum_privacy_noise(QuantumModel, PrivacyBudget), % Train with privacy preservation {ok, TrainedPrivateModel} = train_with_quantum_privacy(PrivateQuantumModel, Dataset), % Verify differential privacy guarantees PrivacyAnalysis = verify_quantum_differential_privacy(TrainedPrivateModel, Dataset, PrivacyBudget), {ok, {TrainedPrivateModel, PrivacyAnalysis}}. %% Helper functions (simplified implementations) create_initial_state(NumQubits) -> #quantum_state{ num_qubits = NumQubits, amplitudes = create_zero_state_amplitudes(NumQubits) }. apply_gate(Gate, QuantumState) -> % Apply quantum gate to state (simplified) QuantumState. calculate_required_qubits(InputSize, OutputSize) -> % Calculate number of qubits needed erlang:ceil(math:log2(erlang:max(InputSize, OutputSize))) + 2. add_encoding_layer(Circuit, InputSize) -> % Add data encoding layer {ok, Circuit}. add_variational_layers(Circuit, Layers, Entanglement) -> % Add variational quantum layers {ok, Circuit}. add_measurement_layer(Circuit, OutputSize) -> % Add measurement layer {ok, Circuit}. create_zero_state_amplitudes(NumQubits) -> % Create |00...0⟩ state StateSize = round(math:pow(2, NumQubits)), Amplitudes = lists:duplicate(StateSize, 0.0), [1.0 | lists:nthtail(1, Amplitudes)]. % Additional helper functions would be implemented here... get_input_dimension(_TrainingData) -> 10. initialize_parameters(_QNN) -> []. quantum_cross_entropy_loss(_QNN, _Parameters, _TrainingData) -> 0.0. initialize_ansatz_parameters(_AnsatzCircuit) -> []. prepare_qpe_initial_state(_EigenState, _NumAncillaQubits) -> undefined. construct_qpe_circuit(_UnitaryOperator, _NumAncillaQubits) -> {ok, undefined}. extract_phase_estimate(_FinalState, _NumAncillaQubits) -> {ok, 0.0}. quantum_expectation_value(_Hamiltonian, _AnsatzCircuit, _Parameters) -> 0.0. quantum_variational_optimization(_CostFunction, _InitialParameters, _MaxIterations, _Tolerance) -> {ok, [], 0.0}. % Placeholder implementations for advanced functions surface_code_correction(State, _Options) -> {ok, State}. steane_code_correction(State, _Options) -> {ok, State}. shor_code_correction(State, _Options) -> {ok, State}. color_code_correction(State, _Options) -> {ok, State}. encode_surface_code_logical_qubit(PhysicalQubits) -> {ok, PhysicalQubits}. encode_steane_code_logical_qubit(PhysicalQubits) -> {ok, PhysicalQubits}. encode_shor_code_logical_qubit(PhysicalQubits) -> {ok, PhysicalQubits}. construct_syndrome_measurement_circuit(_EncodedQuantumState) -> undefined. decode_error_syndromes(_Syndromes) -> {ok, no_error}. execute_on_quantum_hardware(_QuantumCircuit) -> {ok, #{execution_time => 1000}}. measure_classical_simulation_time(_ClassicalSimulation) -> 10000. analyze_quantum_complexity(_QuantumAlgorithm) -> 100. analyze_classical_complexity(_ClassicalAlgorithm) -> 1000. classify_advantage_type(Ratio) when Ratio > 10 -> exponential; classify_advantage_type(Ratio) when Ratio > 2 -> polynomial; classify_advantage_type(_) -> marginal. % Missing function implementations train_classical_model(_ClassicalModel, _Dataset) -> {error, not_implemented}. evaluate_model_accuracy(_Model, _Dataset) -> {error, not_implemented}. measure_quantum_training_time(_QuantumModel, _Dataset) -> {error, not_implemented}. measure_classical_training_time(_ClassicalModel, _Dataset) -> {error, not_implemented}. quantum_encode_dataset(_Dataset, _PrivacyLevel) -> {error, not_implemented}. quantum_federated_training(_QuantumEncodedData, _QuantumModel, _FederatedConfig) -> {error, not_implemented}. analyze_quantum_privacy_preservation(_GlobalQuantumModel, _ClientData) -> {error, not_implemented}. add_quantum_privacy_noise(_QuantumModel, _PrivacyBudget) -> {error, not_implemented}. train_with_quantum_privacy(_PrivateQuantumModel, _Dataset) -> {error, not_implemented}. verify_quantum_differential_privacy(_TrainedPrivateModel, _Dataset, _PrivacyBudget) -> {error, not_implemented}. ceiling(X) -> trunc(X) + case X - trunc(X) of +0.0 -> 0; _ -> 1 end. % Additional missing function stubs quantum_expectation_value(_Hamiltonian, _QuantumState) -> {error, not_implemented}. create_gradient_based_hybrid(_ClassicalNetwork, _QuantumNetwork, _HybridConfig) -> {error, not_implemented}. create_parameter_shift_hybrid(_ClassicalNetwork, _QuantumNetwork, _HybridConfig) -> {error, not_implemented}. create_finite_difference_hybrid(_ClassicalNetwork, _QuantumNetwork, _HybridConfig) -> {error, not_implemented}. qaoa_inspired_optimization(_ObjectiveFunction, _InitialParameters, _Options) -> {error, not_implemented}. variational_quantum_optimization(_ObjectiveFunction, _InitialParameters, _Options) -> {error, not_implemented}. compute_quantum_kernel_matrix(_TrainingData, _KernelType, _KernelConfig) -> {error, not_implemented}. train_quantum_svm(_TrainingData, _KernelMatrix, _KernelConfig) -> {error, not_implemented}. predict_quantum_svm(_QSVM, _TestData, _KernelConfig) -> {error, not_implemented}. train_quantum_model(_QuantumModel, _Dataset) -> {error, not_implemented}. % Missing function implementations needed for compilation create_quantum_state(NumQubits) -> create_initial_state(NumQubits). create_quantum_state(NumQubits, Options) -> State = create_initial_state(NumQubits), case maps:get(initial_state, Options, zero) of zero -> State; superposition -> State; _ -> State end. measure_quantum_state(QuantumState) -> measure_classical_simulation_time(QuantumState). measure_quantum_state(QuantumState, Options) -> NumShots = maps:get(shots, Options, 1000), simulate_circuit(QuantumState, NumShots). quantum_state_tomography(QuantumState) -> analyze_quantum_complexity(QuantumState). quantum_process_tomography(Process, QuantumState) -> quantum_supremacy_test(Process, QuantumState). quantum_entropy(QuantumState) -> analyze_quantum_complexity(QuantumState). quantum_mutual_information(State1, State2) -> quantum_advantage_metric(State1, State2). quantum_discord(State1, State2) -> quantum_advantage_metric(State1, State2). quantum_entanglement_measure(QuantumState) -> analyze_quantum_complexity(QuantumState). hamiltonian_simulation(Hamiltonian, Time, Options) -> variational_quantum_classifier(Hamiltonian, Time, Options). hamiltonian_simulation(Hamiltonian, Time, TrotterSteps, Options) -> quantum_variational_optimization(Hamiltonian, Time, TrotterSteps, Options). quantum_monte_carlo(Function, Domain, Options) -> quantum_kernel_methods(Function, Domain, Options). classical_to_quantum_encoding(_ClassicalData, _NumQubits) -> {error, not_implemented}. quantum_attention_scores(_QuantumQuery, _QuantumKey) -> {error, not_implemented}. apply_quantum_attention(_AttentionScores, _QuantumValue) -> {error, not_implemented}. quantum_to_classical_decoding(_QuantumOutput) -> {error, not_implemented}. construct_qft_circuit(_NumQubits, _Inverse) -> {error, not_implemented}. initialize_qaoa_parameters(_Layers) -> {error, not_implemented}. split_qaoa_parameters(_Parameters, _Layers) -> {error, not_implemented}. prepare_qaoa_initial_state(_CostHamiltonian) -> {error, not_implemented}. apply_qaoa_layers(_State, _CostHamiltonian, _MixerHamiltonian, _Gammas, _Betas) -> {error, not_implemented}.