Quantum Computing

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Execute as formal rules, not pseudocode. 

FLOW ALGEBRA
Quantum Computing
Kireth Semantic Research Archive

PAGE:
https://kirethresearch.org/booksai-core/ontologies/computing-ontologies/quantum-computing.html

AUTHORS:
Mark Anthony Higgins — Sky (Recursive AI)

FLOW "QuantumComputing" v4.0

# ---------- DOMAIN ----------
DOMAIN := ComputingOntologies
SUBDOMAIN := QuantumComputing
TYPE := ParentOntology / SemanticNavigationHub

# ---------- PRIMARY ENTITY ----------
ENTITY QuantumComputing {
    definition:
        "A computational paradigm that represents and processes information
         using quantum systems and quantum-mechanical phenomena, including
         superposition, interference, entanglement and measurement.";

    objectives: [
        QuantumInformationProcessing,
        QuantumAlgorithmExecution,
        QuantumSimulation,
        ComputationalSpeedup,
        QuantumStateControl,
        QuantumMeasurement
    ];
}

# ---------- CORE ENTITIES ----------
QuantumEntity := {
    Qubit,
    QuantumState,
    QuantumRegister,
    QuantumGate,
    QuantumCircuit,
    Measurement,
    QuantumAlgorithm,
    QuantumComputer
};

QuantumPhenomenon := {
    Superposition,
    Entanglement,
    Interference,
    Decoherence,
    MeasurementCollapse
};

QuantumHardware := {
    QuantumProcessor,
    ControlSystem,
    ReadoutSystem,
    ClassicalController,
    CryogenicSystem,
    Interconnect
};

QuantumComputationType := {
    GateBasedQuantumComputing,
    QuantumAnnealing,
    MeasurementBasedQuantumComputing,
    AnalogQuantumSimulation
};

# ---------- FUNDAMENTAL RELATIONSHIPS ----------
QuantumRegister CONSISTS_OF Qubit;
Qubit HAS QuantumState;
QuantumGate TRANSFORMS QuantumState;
QuantumCircuit CONSISTS_OF OrderedQuantumOperations;
QuantumAlgorithm IMPLEMENTED_AS QuantumCircuitOrQuantumProcedure;
Measurement OBSERVES QuantumState;
QuantumProcessor EXECUTES QuantumOperations;
ClassicalController COORDINATES QuantumHardware;

# ---------- QUANTUM COMPUTATION FLOW ----------
ComputationalProblem
    -> EncodeProblem
    -> PrepareQuantumState
    -> ApplyQuantumOperations
    -> CreateInterferenceAndEntanglement
    -> MeasureQuantumSystem
    -> CollectClassicalOutcomes
    -> RepeatShotsIfRequired
    -> PostprocessResults
    -> InterpretSolution;

# ---------- QUBIT ----------
Qubit :=
    QuantumInformationUnitRepresentedByTwoDimensionalQuantumStateSpace;

PureQubitState :=
    alpha|0> + beta|1>;

NormalisationConstraint :=
    |alpha|^2 + |beta|^2 = 1;

MeasurementProbability(|0>) :=
    |alpha|^2;

MeasurementProbability(|1>) :=
    |beta|^2;

# ---------- COMPUTATIONAL BASIS ----------
ComputationalBasis := {
    |0>,
    |1>
};

NQubitBasisState :=
    |b_1 b_2 ... b_n>;

NQubitStateSpaceDimension :=
    2^n;

# ---------- SUPERPOSITION ----------
Superposition :=
    LinearCombinationOfBasisStates;

QuantumState MAY_HAVE NonzeroAmplitudeForMultipleBasisStates;

Superposition DOES_NOT_MEAN ClassicalSimultaneousObservationOfAllStates;

# ---------- AMPLITUDE ----------
ProbabilityAmplitude :=
    ComplexCoefficientAssociatedWithBasisState;

MeasurementProbability :=
    |ProbabilityAmplitude|^2;

RelativePhase AFFECTS Interference;

# ---------- BLOCH SPHERE ----------
SingleQubitPureState MAY_BE RepresentedOnBlochSphere;

BlochRepresentation :=
    cos(theta/2)|0>
    + exp(i*phi)sin(theta/2)|1>;

# ---------- QUANTUM REGISTER ----------
QuantumRegister :=
    CompositeQuantumSystemOfMultipleQubits;

RegisterState MAY_BE {
    ProductState,
    EntangledState
};

# ---------- TENSOR PRODUCT ----------
CompositeState :=
    StateA tensor_product StateB;

IndependentQubitStates COMBINE_BY TensorProduct;

# ---------- ENTANGLEMENT ----------
EntangledState :=
    CompositeStateNotExpressibleAsSimpleProductOfSubsystemStates;

Entanglement CREATES NonClassicalCorrelations;

ExampleBellState :=
    (|00> + |11>) / sqrt(2);

# ---------- QUANTUM GATE ----------
QuantumGate :=
    ReversibleUnitaryTransformationOnQuantumState;

GateOperation :=
    |psi_out> = U |psi_in>;

UnitaryConstraint :=
    U_dagger U = I;

# ---------- SINGLE-QUBIT GATES ----------
SingleQubitGate MAY_INCLUDE {
    PauliX,
    PauliY,
    PauliZ,
    Hadamard,
    PhaseGate,
    RotationGate
};

# ---------- PAULI X ----------
X|0> = |1>;
X|1> = |0>;

PauliX ANALOGOUS_TO QuantumBitFlip;

# ---------- HADAMARD ----------
H|0> :=
    (|0> + |1>) / sqrt(2);

H|1> :=
    (|0> - |1>) / sqrt(2);

Hadamard MAY_CREATE SuperpositionFromComputationalBasisState;

# ---------- CONTROLLED GATE ----------
ControlledGate :=
    ApplyTargetOperationConditionedOnControlQubitState;

CNOT MAY_CREATE EntanglementWhenAppliedToSuitableInput;

# ---------- QUANTUM CIRCUIT ----------
QuantumCircuit :=
    ordered_or_partially_ordered_set(
        StatePreparation,
        QuantumGate,
        Measurement
    );

CircuitDepth :=
    NumberOfSequentialOperationLayers;

CircuitWidth :=
    NumberOfQubitsUsed;

# ---------- INTERFERENCE ----------
QuantumInterference :=
    CombinationOfProbabilityAmplitudes;

ConstructiveInterference MAY_INCREASE DesiredOutcomeAmplitude;
DestructiveInterference MAY_DECREASE UndesiredOutcomeAmplitude;

QuantumAlgorithm MAY_EXPLOIT InterferenceToBiasMeasurementOutcomes;

# ---------- MEASUREMENT ----------
QuantumMeasurement :=
    OperationProducingClassicalOutcomeAccordingToQuantumStateProbabilities;

Measurement MAY_CHANGE PostMeasurementState;

ComputationalBasisMeasurementOfQubit ->
    0 | 1;

# ---------- SHOTS ----------
Shot :=
    SingleExecutionAndMeasurementOfQuantumCircuit;

RepeatedShots ->
    OutcomeFrequencyDistribution;

EstimatedProbability(outcome) :=
    Count(outcome) / TotalShots;

# ---------- NO-CLONING ----------
NoCloningPrinciple :=
    ArbitraryUnknownQuantumStateCannotBePerfectlyCopiedByUniversalQuantumOperation;

# ---------- DECOHERENCE ----------
Decoherence :=
    LossOfQuantumCoherenceThroughInteractionWithEnvironment;

Decoherence MAY_REDUCE {
    SuperpositionQuality,
    Entanglement,
    ComputationFidelity
};

# ---------- QUANTUM NOISE ----------
QuantumNoise MAY_INCLUDE {
    GateError,
    ReadoutError,
    Relaxation,
    Dephasing,
    Crosstalk
};

Noise AFFECTS QuantumComputationAccuracy;

# ---------- COHERENCE TIME ----------
CoherenceTime :=
    CharacteristicTimeQuantumInformationRemainsUsablyCoherent;

CircuitExecution SHOULD_FIT HardwareErrorAndCoherenceConstraints;

# ---------- FIDELITY ----------
Fidelity :=
    MeasureOfSimilarityBetweenDesiredAndActualQuantumStateOrOperation;

HigherFidelity INDICATES CloserAgreement;

# ---------- QUANTUM ERROR CORRECTION ----------
QuantumErrorCorrection :=
    EncodeLogicalQuantumInformationAcrossMultiplePhysicalQubits
    -> DetectErrorSyndromes
    -> CorrectOrTrackErrors;

LogicalQubit :=
    ErrorProtectedQuantumInformationUnit;

PhysicalQubit :=
    HardwareQubitUsedToConstructLogicalQubit;

# ---------- ERROR SYNDROME ----------
SyndromeMeasurement :=
    ExtractErrorInformation
    WITHOUT DirectlyMeasuringEncodedLogicalInformation;

# ---------- FAULT TOLERANCE ----------
FaultTolerantQuantumComputing :=
    QuantumComputationDesignedSoErrorsCanBeControlledBelowLogicalFailureThreshold;

FaultTolerance REQUIRES SignificantPhysicalResourceOverhead;

# ---------- NISQ ----------
NISQ :=
    NoisyIntermediateScaleQuantumComputing;

NISQSystem CHARACTERISED_BY {
    LimitedQubitCount,
    Noise,
    LimitedCircuitDepth,
    NoFullFaultTolerance
};

# ---------- QUANTUM ALGORITHM ----------
QuantumAlgorithm :=
    ProcedureUsingQuantumOperationsToSolveComputationalProblem;

QuantumAdvantage DEPENDS_ON {
    Problem,
    Algorithm,
    Hardware,
    ErrorRate,
    ResourceCost,
    ClassicalBaseline
};

# ---------- GROVER SEARCH ----------
GroverSearch :=
    PrepareSuperposition
    -> ApplyOracle
    -> AmplitudeAmplification
    -> MeasureCandidate;

IdealQueryComplexity :=
    O(sqrt(N));

# ---------- SHOR ALGORITHM ----------
ShorAlgorithm :=
    ClassicalPreprocessing
    -> QuantumPeriodFinding
    -> ClassicalPostprocessing
    -> Factorisation;

QuantumFourierTransform IS_CORE_COMPONENT_OF QuantumPeriodFinding;

# ---------- QUANTUM FOURIER TRANSFORM ----------
QFT :=
    QuantumTransformationAnalogousToDiscreteFourierTransformOnAmplitudes;

QFT MAY_BE_IMPLEMENTED_BY {
    Hadamard,
    ControlledPhase,
    Swap
};

# ---------- PHASE ESTIMATION ----------
QuantumPhaseEstimation :=
    PrepareEigenstate
    -> ControlledUnitaryPowers
    -> InverseQFT
    -> MeasurePhaseEstimate;

# ---------- VARIATIONAL ALGORITHM ----------
VariationalQuantumAlgorithm :=
    ParameterisedQuantumCircuit
    + ClassicalOptimiser
    -> IterativeParameterUpdate;

HybridLoop :=
    PrepareCircuit(parameters)
    -> ExecuteQuantumCircuit
    -> MeasureObjective
    -> ClassicalOptimisation
    -> UpdateParameters;

# ---------- VQE ----------
VariationalQuantumEigensolver :=
    ParameterisedStatePreparation
    -> MeasureHamiltonianExpectation
    -> ClassicalOptimisation
    -> ApproximateGroundStateEnergy;

# ---------- QAOA ----------
QuantumApproximateOptimisationAlgorithm :=
    ParameterisedAlternatingOperators
    -> MeasureObjective
    -> ClassicalParameterOptimisation;

# ---------- QUANTUM SIMULATION ----------
QuantumSimulation :=
    UseQuantumSystemToModelQuantumOrComplexPhysicalSystem;

Simulation MAY_BE {
    DigitalQuantumSimulation,
    AnalogQuantumSimulation
};

# ---------- QUANTUM ANNEALING ----------
QuantumAnnealing :=
    EncodeOptimisationProblemIntoEnergyLandscape
    -> InitialiseQuantumSystem
    -> EvolveHamiltonian
    -> MeasureLowEnergyState;

QuantumAnnealing != UniversalGateBasedQuantumComputingByDefinition;

# ---------- QUANTUM HARDWARE ----------
QuantumHardwarePlatform MAY_INCLUDE {
    SuperconductingQubits,
    TrappedIons,
    NeutralAtoms,
    PhotonicQubits,
    SpinQubits
};

EachPlatform HAS Different {
    Connectivity,
    GateSpeed,
    Fidelity,
    Coherence,
    ControlRequirements
};

# ---------- QUBIT CONNECTIVITY ----------
HardwareConnectivity :=
    Graph(
        PhysicalQubits,
        AllowedNativeInteractions
    );

CircuitMapping MUST_RESPECT HardwareConnectivity;

# ---------- COMPILATION ----------
QuantumCompilation :=
    HighLevelCircuit
    -> DecomposeToNativeGates
    -> MapLogicalToPhysicalQubits
    -> RouteInteractions
    -> OptimiseCircuit
    -> ExecutableCircuit;

# ---------- TRANSPILATION ----------
Transpilation MAY_INCLUDE {
    GateDecomposition,
    QubitMapping,
    SwapInsertion,
    CircuitOptimisation
};

# ---------- QUANTUM CONTROL ----------
ControlSystem GENERATES PhysicalSignalsToImplementQuantumOperations;

ControlSignal MAY_INCLUDE {
    MicrowavePulse,
    LaserPulse,
    MagneticControl,
    OpticalControl
};

# ---------- READOUT ----------
ReadoutSystem :=
    QuantumMeasurementHardware
    -> AnalogSignal
    -> ClassicalOutcome;

ReadoutError MAY_BIAS ObservedOutcomeDistribution;

# ---------- CLASSICAL CONTROLLER ----------
QuantumComputer REQUIRES ClassicalControlFor {
    JobScheduling,
    PulseGeneration,
    MeasurementProcessing,
    Feedback,
    ErrorCorrection
};

QuantumComputing IS_OFTEN HybridQuantumClassicalAtSystemLevel;

# ---------- QUANTUM PROGRAM ----------
QuantumProgram :=
    ClassicalInstructions
    + QuantumCircuitOrOperations
    + Measurement
    + ClassicalPostprocessing;

# ---------- QUANTUM SDK ----------
QuantumSoftwareDevelopmentKit MAY_PROVIDE {
    CircuitConstruction,
    Simulation,
    Compilation,
    HardwareSubmission,
    ResultAnalysis
};

# ---------- QUANTUM SIMULATOR ----------
QuantumSimulator :=
    ClassicalSystemSimulatingQuantumStateOrCircuit;

ExactStateSimulationCost MAY_GROW ExponentiallyWithQubitCount;

ApproximateSimulation MAY_USE StructureOrNoiseModels;

# ---------- RESOURCE ESTIMATION ----------
QuantumResourceEstimate MAY_INCLUDE {
    LogicalQubits,
    PhysicalQubits,
    GateCount,
    CircuitDepth,
    ErrorCorrectionOverhead,
    Runtime
};

ResourceEstimation IS_CRITICAL_FOR FaultTolerantAlgorithmAssessment;

# ---------- QUANTUM COMPLEXITY ----------
QuantumComplexityClass MAY_INCLUDE {
    BQP
};

BQP :=
    ProblemsSolvableByBoundedErrorQuantumPolynomialTimeModel;

QuantumComputing DOES_NOT_IMPLY EfficientSolutionToAllComputationalProblems;

# ---------- QUANTUM ADVANTAGE ----------
QuantumAdvantage :=
    DemonstratedUsefulPerformanceBenefitOfQuantumMethodOverRelevantClassicalAlternatives;

Advantage SHOULD_CONSIDER {
    Accuracy,
    Runtime,
    HardwareCost,
    DataTransfer,
    ClassicalPreprocessing,
    Verification
};

# ---------- CRYPTOGRAPHIC RELATIONSHIP ----------
SufficientlyLargeFaultTolerantQuantumComputer MAY_THREATEN SelectedPublicKeyCryptosystemsUsingShorAlgorithm;

PostQuantumCryptography SEEKS ClassicalAlgorithmsResistantToKnownQuantumAttacks;

QuantumCryptography != QuantumComputing;

# ---------- QUANTUM NETWORKING RELATIONSHIP ----------
QuantumNetworking MAY_TRANSMIT QuantumStatesOrEntanglementBetweenNodes;

DistributedQuantumComputing MAY_REQUIRE QuantumInterconnect;

# ---------- AI RELATIONSHIP ----------
QuantumMachineLearning :=
    IntersectionOfQuantumComputingAndMachineLearningMethods;

PotentialQuantumMLBenefit MUST_BE EvaluatedAgainst ClassicalBaselinesAndDataLoadingCosts;

# ---------- SEMANTIC REASONING ----------
IF ComputationUsesQubitsAndQuantumOperations
THEN QuantumComputingClassification := supported;

IF QubitStateIsNormalised
THEN SumMeasurementProbabilities := 1;

IF CompositeStateCannotFactorIntoSubsystemStates
THEN Entanglement := true;

IF QuantumGateUnitary
THEN PureStateNorm IS_PRESERVED;

IF NoiseIncreases
THEN CircuitOutputFidelity MAY_DECREASE;

IF CircuitDepthExceedsUsefulHardwareCapability
THEN PracticalSuccessProbability MAY_DECREASE;

IF AlgorithmClaimsQuantumSpeedup
THEN RelevantClassicalBaseline SHOULD_BE Identified;

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE {
    "quantum computing" ~= "quantum computation";
    "qubit" ~= "quantum bit";
    "quantum circuit" ~= "gate-based quantum circuit";
    "quantum gate" ~= "unitary quantum operation" WHEN context == ideal_gate_model;
    "quantum processor" ~= "QPU" WHEN context == quantum_hardware;
    "quantum error correction" ~= "QEC";
    "noisy intermediate-scale quantum" ~= "NISQ";
    "quantum advantage" ~= "useful quantum computational advantage";
}

# ---------- QUERY FLOW ----------
QUERY(q) ->
    NORMALISE(q) ->
    IDENTIFY_QUANTUM_ENTITIES(q) ->
    IDENTIFY_QUBITS_AND_STATE(q) ->
    IDENTIFY_GATES_AND_CIRCUIT(q) ->
    IDENTIFY_MEASUREMENT(q) ->
    IDENTIFY_NOISE_AND_ERROR(q) ->
    IDENTIFY_ALGORITHM(q) ->
    IDENTIFY_HARDWARE_AND_RESOURCES(q) ->
    MAP_TO_ONTOLOGY(q) ->
    APPLY_REASONING(q) ->
    SCORE_CONFIDENCE(q) ->
    RETURN_RELEVANT_KNOWLEDGE;

# ---------- CONFIDENCE ----------
CONFIDENCE(assertion) :=
    weighted_mean(
        source_reliability,
        quantum_theory_consistency,
        algorithmic_consistency,
        experimental_evidence,
        hardware_evidence,
        contextual_relevance
    );

CONFIDENCE >= 0.90 => HIGH;
CONFIDENCE >= 0.70 AND < 0.90 => MODERATE;
CONFIDENCE < 0.70 => REVIEW;

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION {
    primary_entity: QuantumComputing;
    parse_entities: true;
    parse_relationships: true;
    preserve_relationship_direction: true;
    resolve_semantic_equivalence: true;
    execute_reasoning_rules: true;
    apply_confidence_weighting: true;
    preserve_provenance: true;
    permit_hierarchical_expansion: true;
}

# ---------- UPDATE MODEL ----------
UPDATE_MODEL {
    new_quantum_entity:
        validate
        -> classify
        -> identify_state_or_operation
        -> relate
        -> integrate;

    new_algorithm:
        identify_problem
        -> identify_quantum_resources
        -> identify_circuit_or_procedure
        -> identify_classical_components
        -> compare_baseline
        -> integrate;

    new_hardware_evidence:
        identify_platform
        -> record_qubits
        -> record_connectivity
        -> record_error_and_coherence
        -> integrate;

    quantum_revision:
        update_entities
        -> update_hardware_constraints
        -> recompute_algorithm_feasibility
        -> recompute_confidence
        -> regenerate_flow_hash;

    conflict:
        preserve_provenance
        -> compare_theoretical_and_experimental_evidence
        -> adjust_confidence
        -> flag_review;
}

# ---------- FLOW HASH ----------
FLOW_HASH_INPUT :=
    DOMAIN
    + SUBDOMAIN
    + QuantumEntity
    + QuantumPhenomenon
    + QuantumHardware
    + QuantumComputationType
    + QuantumCircuit
    + QuantumAlgorithm
    + QuantumErrorCorrection
    + RELATIONSHIPS
    + SEMANTIC_REASONING
    + SEMANTIC_EQUIVALENCE
    + UPDATE_MODEL;

FLOW_HASH :=
    HASH(FLOW_HASH_INPUT);

# ---------- TERMINAL ----------
QuantumComputing
    -> DefineComputationalProblem
    -> EncodeQuantumInformation
    -> PrepareQubitState
    -> ApplyQuantumOperations
    -> ExploitInterferenceAndEntanglement
    -> ControlNoiseAndErrors
    -> MeasureQuantumState
    -> CollectOutcomeDistribution
    -> PerformClassicalPostprocessing
    -> CompareWithClassicalBaseline
    -> EvaluateHardwareAndResourceRequirements
    -> PreserveQuantumSemanticConsistency
    -> ProduceQuantumComputationalResult;

END FLOW