Quantum Error Correction Ontology

FLOW ALGEBRA
Quantum Error Correction Ontology
Kireth Semantic Research Archive

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

AUTHORS:
Mark Anthony Higgins — Sky (Recursive AI)

FLOW "QuantumErrorCorrectionOntology" v4.0

# ---------- DOMAIN ----------
DOMAIN := ComputingOntology
SUBDOMAIN := QuantumComputing
ONTOLOGY := QuantumErrorCorrectionOntology

# ---------- PRIMARY ENTITY ----------
ENTITY QuantumErrorCorrection {
    definition:
        "The encoding, detection, diagnosis and correction or tracking of
         errors affecting quantum information by distributing logical quantum
         states across multiple physical qubits while preserving encoded
         information from destructive direct measurement.";

    objectives: [
        LogicalInformationProtection,
        QuantumErrorDetection,
        ErrorCorrection,
        FaultTolerance,
        LogicalErrorSuppression,
        ReliableQuantumComputation
    ];
}

# ---------- CORE ENTITIES ----------
QECEntity := {
    PhysicalQubit,
    LogicalQubit,
    QuantumCode,
    CodeSpace,
    Stabilizer,
    Syndrome,
    AncillaQubit,
    Decoder
};

ErrorEntity := {
    BitFlipError,
    PhaseFlipError,
    PauliError,
    ErasureError,
    LeakageError,
    MeasurementError,
    CorrelatedError
};

OperationEntity := {
    Encoding,
    SyndromeExtraction,
    SyndromeMeasurement,
    Decoding,
    Recovery,
    LogicalOperation
};

# ---------- FUNDAMENTAL RELATIONSHIPS ----------
QuantumCode ENCODES LogicalQubit INTO PhysicalQubits;
PhysicalError ACTS_ON PhysicalQubit;
Stabilizer DEFINES CodeSpaceConstraint;
SyndromeMeasurement EXTRACTS ErrorInformation;
AncillaQubit MAY_SUPPORT SyndromeExtraction;
Decoder MAPS Syndrome TO RecoveryDecision;
Recovery RESTORES_OR_TRACKS LogicalState;
FaultTolerantOperation ACTS_ON EncodedLogicalInformation;

# ---------- QEC FLOW ----------
LogicalQuantumState
    -> EncodeIntoPhysicalQubits
    -> ExecuteOrStoreQuantumInformation
    -> PhysicalNoiseOccurs
    -> ExtractSyndrome
    -> MeasureChecks
    -> DecodeSyndrome
    -> InferLikelyErrorClass
    -> CorrectErrorOrUpdatePauliFrame
    -> VerifyLogicalState
    -> ContinueQuantumComputation;

# ---------- QUANTUM CODE ----------
QuantumErrorCorrectingCode :=
    Mapping(
        LogicalStateSpace
        -> ProtectedPhysicalCodeSpace
    );

CodeParameters MAY_BE_WRITTEN_AS:
    [[n,k,d]];

WHERE:
    n := NumberOfPhysicalQubits;
    k := NumberOfLogicalQubits;
    d := CodeDistance;

# ---------- CODE DISTANCE ----------
CodeDistance :=
    MinimumWeightOfNontrivialUndetectableLogicalOperatorUnderApplicableCodeDefinition;

IdealArbitraryErrorCorrectionCapability :=
    floor((d - 1) / 2);

LargerDistance MAY_INCREASE ErrorSuppression
BUT RequiresAdditionalPhysicalResources;

# ---------- PHYSICAL / LOGICAL QUBIT ----------
PhysicalQubit :=
    HardwareQuantumSystemUsedAsPartOfEncodedState;

LogicalQubit :=
    QuantumInformationEncodedAcrossMultiplePhysicalQubits;

LogicalErrorRate SHOULD_BE LowerThanPhysicalErrorRateForUsefulErrorCorrectionUnderApplicableOperatingConditions;

# ---------- ENCODING ----------
Encoding :=
    LogicalState
    + AncillaOrReferenceQubits
    -> EncodedPhysicalState;

Encoding MUST_PRESERVE UnknownLogicalQuantumInformation;

# ---------- ERROR BASIS ----------
BitFlipError := PauliX;
PhaseFlipError := PauliZ;
CombinedPauliError MAY_INCLUDE PauliY;

GeneralSingleQubitError MAY_BE ExpandedIn PauliOperatorBasis;

CorrectionOfRelevantPauliBasisErrors SUPPORTS CorrectionOfCorrespondingArbitrarySingleQubitErrors;

# ---------- SYNDROME ----------
ErrorSyndrome :=
    ClassicalInformationIndicatingViolationOrChangeOfCodeConstraints;

Syndrome SHOULD_REVEAL ErrorInformation
WITHOUT DirectlyRevealing UnknownLogicalAmplitudes;

# ---------- SYNDROME EXTRACTION ----------
SyndromeExtraction :=
    CoupleDataQubitsToAncillaOrCheckCircuit
    -> EncodeParityOrStabilizerInformation
    -> MeasureAncillaOrCheck
    -> ProduceSyndromeBits;

# ---------- ANCILLA ----------
AncillaQubit :=
    AuxiliaryQubitUsedFor {
        SyndromeExtraction,
        Verification,
        FaultTolerantOperations
    };

Ancilla MAY_BE PreparedMeasuredAndResetRepeatedly;

# ---------- STABILIZER CODE ----------
StabilizerCode :=
    QuantumCodeDefinedByCommutingPauliOperators;

CodeSpace :=
    SimultaneousPositiveEigenspaceOfStabilizerGenerators;

ForValidCodeState:
    S_i |psi_L> = |psi_L>;

# ---------- SYNDROME RELATIONSHIP ----------
IF Error E AnticommutesWith Stabilizer S_i
THEN AssociatedSyndromeBit MAY_CHANGE;

SyndromePattern HELPS_IDENTIFY ErrorEquivalenceClass;

# ---------- LOGICAL OPERATOR ----------
LogicalOperator :=
    OperationPreservingCodeSpace
    WHILE ActingNontriviallyOnLogicalInformation;

LogicalOperator COMMUTES_WITH StabilizerGroup
BUT IsNotEquivalentToIdentityOnCodeSpace;

# ---------- DEGENERACY ----------
DegenerateCode MAY_HAVE DistinctPhysicalErrorsWithEquivalentActionOnCodeSpace;

Decoder NEED_NOT IdentifyExactPhysicalError
IF SelectedRecoveryIsInCorrectEquivalenceClass;

# ---------- DECODER ----------
Decoder :=
    Syndrome
    + CodeModel
    + NoiseModel
    -> RecoveryDecisionOrCorrectionFrame;

Decoder MAY_BE {
    LookupDecoder,
    MinimumWeightDecoder,
    MatchingDecoder,
    BeliefPropagationDecoder,
    MachineLearningDecoder
};

# ---------- RECOVERY ----------
Recovery :=
    ApplyPhysicalCorrection
    OR
    UpdateTrackedCorrectionFrame;

SuccessfulRecovery RETURNS StateToEquivalentValidLogicalState;

# ---------- PAULI FRAME ----------
PauliFrame :=
    ClassicalRecordOfKnownPauliCorrections;

PauliFrameUpdate MAY_AVOID ImmediatePhysicalCorrectionGate;

FutureOperationsAndMeasurements MUST_ACCOUNT_FOR Frame;

# ---------- THREE-QUBIT BIT-FLIP CODE ----------
BitFlipCodeEncoding:
    alpha|0> + beta|1>
    ->
    alpha|000> + beta|111>;

SingleBitFlip MAY_BE DetectedByParityChecks;

BitFlipCode DOES_NOT_BY_ITSELF ProtectAgainstGeneralPhaseErrors;

# ---------- PHASE-FLIP CODE ----------
PhaseFlipCode MAY_BE ConstructedUsingBasisTransformationOfBitFlipProtection;

PhaseErrors IN ComputationalBasis
BECOME BitFlipLikeErrors IN HadamardRotatedBasis;

# ---------- SHOR CODE ----------
ShorCode :=
    NineQubitCodeCombiningBitAndPhaseErrorProtection;

ShorCode CAN_CORRECT ArbitrarySingleQubitErrorUnderIdealCodeAssumptions;

# ---------- STEANE CODE ----------
SteaneCode :=
    [[7,1,3]] StabilizerCode;

SteaneCode CAN_CORRECT ArbitrarySingleQubitErrorUnderIdealCodeModel;

# ---------- SURFACE CODE ----------
SurfaceCode :=
    TopologicalStabilizerCodeUsingLocalParityChecksOnLattice;

SurfaceCode MAY_INCLUDE {
    DataQubits,
    MeasurementQubits,
    XTypeChecks,
    ZTypeChecks
};

# ---------- SURFACE CODE CYCLE ----------
SurfaceCodeCycle :=
    PrepareOrResetAncillas
    -> InteractWithNeighbouringDataQubits
    -> MeasureChecks
    -> RecordSyndrome
    -> Repeat;

RepeatedSyndromeRounds HELP_DISTINGUISH DataErrorsFromMeasurementErrors;

# ---------- LOGICAL FAILURE ----------
LogicalFailure :=
    PhysicalErrorAndRecoveryCombinationEquivalentToNontrivialLogicalOperator;

LogicalErrorRate :=
    ProbabilityOfLogicalFailurePerDefinedOperationCycleOrTime;

# ---------- THRESHOLD ----------
FaultToleranceThreshold :=
    PhysicalErrorRegimeBelowWhichIncreasingCodeResourcesCanSuppressLogicalErrorUnderSpecifiedCodeArchitectureDecoderAndNoiseAssumptions;

Threshold IS_NOT UniversalAcrossAllSystems;

# ---------- FAULT TOLERANCE ----------
FaultTolerance :=
    DesignEnsuringLimitedPhysicalFaultsDoNotSpreadIntoUncorrectableLogicalErrorsBeyondCodeCapability;

FaultTolerantProtocol MAY_INCLUDE {
    FaultTolerantStatePreparation,
    FaultTolerantSyndromeExtraction,
    FaultTolerantLogicalGate,
    VerifiedAncilla,
    RepeatedMeasurement
};

# ---------- ERROR PROPAGATION ----------
MultiQubitGate MAY_PROPAGATE ErrorBetweenQubits;

FaultTolerantCircuit SHOULD_LIMIT DangerousErrorPropagation;

# ---------- TRANSVERSAL GATE ----------
TransversalGate :=
    LogicalOperationImplementedThroughSeparatedPhysicalOperationsAcrossCodeComponentsInApplicableCodes;

Transversality MAY_LIMIT ErrorPropagation;

UniversalFaultTolerantComputation GENERALLY_REQUIRES AdditionalTechniquesBeyondARestrictedTransversalGateSet;

# ---------- MAGIC STATE ----------
MagicState :=
    AncillaryResourceStateForImplementingSelectedNonCliffordLogicalOperations;

MagicStateDistillation :=
    MultipleNoisyMagicStates
    -> DistillationProtocol
    -> FewerHigherFidelityMagicStates;

# ---------- ERASURE ----------
ErasureError :=
    ErrorWhoseAffectedLocationIsKnownOrFlagged;

KnownLocation MAY_ALLOW CorrectionOfMoreErrorsThanUnknownLocationModelAtEquivalentCodeDistance;

# ---------- LEAKAGE ----------
LeakageError :=
    PhysicalQubitLeavesIntendedComputationalSubspace;

Leakage MAY_REQUIRE {
    Detection,
    Reset,
    Replacement,
    LeakageReductionProtocol
};

# ---------- MEASUREMENT ERROR ----------
MeasurementError :=
    ReportedOutcomeDiffersFromIdealMeasurementOutcome;

RepeatedSyndromeMeasurement MAY_HELP DetectMeasurementFaults;

# ---------- CORRELATED ERROR ----------
CorrelatedError :=
    StatisticallyDependentErrorAcrossMultiplePhysicalComponents;

DecoderAssumingIndependentErrors MAY_PERFORM PoorlyUnderStrongCorrelations;

# ---------- NOISE MODEL ----------
QECNoiseModel MAY_INCLUDE {
    DepolarisingNoise,
    BiasedNoise,
    ErasureNoise,
    CorrelatedNoise,
    CircuitLevelNoise
};

DecoderPerformance DEPENDS_ON RelationshipBetweenAssumedAndActualNoise;

# ---------- KNILL-LAFLAMME CONDITION ----------
QuantumCode Corrects ErrorSet {E_a}
IF:
    P E_a^dagger E_b P
    =
    c_ab P
FOR AllRelevant a,b;

WHERE:
    P := ProjectorOntoCodeSpace;

# ---------- QEC CYCLE ----------
ErrorCorrectionCycle :=
    SyndromeExtraction
    -> SyndromeMeasurement
    -> Decode
    -> UpdateCorrectionFrame
    -> Continue;

CycleTime SHOULD_BE CompatibleWith PhysicalErrorDynamicsAndControlSystem;

# ---------- DECODING LATENCY ----------
DecoderLatency :=
    TimeFromSyndromeAvailabilityToRequiredRecoveryDecision;

RealTimeQEC MAY_REQUIRE DecoderThroughputToMatchSyndromeGenerationRate;

# ---------- CLASSICAL CONTROL ----------
QEC REQUIRES ClassicalProcessingFor {
    SyndromeCollection,
    Decoding,
    PauliFrameTracking,
    Feedback,
    Scheduling
};

# ---------- RESOURCE OVERHEAD ----------
QECResourceOverhead MAY_INCLUDE {
    PhysicalQubits,
    AncillaQubits,
    MeasurementCircuits,
    ClassicalDecoding,
    Time,
    LogicalGateResources
};

# ---------- LOGICAL MEMORY ----------
LogicalQuantumMemory :=
    EncodedLogicalStateProtectedThroughRepeatedQECCycles;

UsefulQEC SEEKS LogicalLifetimeGreaterThanRelevantUnencodedLifetime;

# ---------- ERROR MITIGATION RELATIONSHIP ----------
QuantumErrorMitigation :=
    ReduceOrEstimateNoiseEffectsWithoutFullEncodedLogicalProtection;

QuantumErrorMitigation != QuantumErrorCorrection;

# ---------- NO-CLONING RELATIONSHIP ----------
QuantumErrorCorrection DOES_NOT Violate NoCloningTheorem;

Encoding DISTRIBUTES QuantumInformationAcrossEntangledPhysicalDegreesOfFreedom
RATHER_THAN CreatingIndependentCopiesOfUnknownState;

# ---------- SEMANTIC REASONING ----------
IF LogicalInformationEncodedAcrossMultiplePhysicalQubits
AND SyndromeDetectsErrorsWithoutDirectLogicalMeasurement
THEN QuantumErrorCorrectionClassification := supported;

IF CodeDistance == d
THEN IdealArbitraryErrorCorrectionCapability := floor((d-1)/2);

IF ErrorEquivalentToStabilizer
THEN LogicalInformation MAY_BE Unchanged;

IF ActualErrorTimesRecoveryIsStabilizerEquivalent
THEN Recovery := logically_successful;

IF ActualErrorTimesRecoveryIsNontrivialLogicalOperator
THEN LogicalFailure := true;

IF PhysicalErrorBelowRelevantThreshold
AND CodeDistanceIncreases
THEN LogicalErrorRate MAY_DECREASE;

IF DecoderCannotKeepPaceWithSyndromeGeneration
THEN FaultTolerantControl MAY_BE Compromised;

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE {
    "quantum error correction" ~= "QEC";
    "quantum error-correcting code" ~= "quantum code" WHEN context == QEC;
    "logical qubit" ~= "encoded quantum bit";
    "physical qubit" ~= "hardware qubit";
    "error syndrome" ~= "quantum syndrome";
    "syndrome extraction" ~= "quantum error syndrome extraction";
    "Pauli frame" ~= "tracked Pauli correction frame";
    "surface code" ~= "topological surface quantum code";
}

# ---------- QUERY FLOW ----------
QUERY(q) ->
    NORMALISE(q) ->
    IDENTIFY_CODE_AND_PARAMETERS(q) ->
    IDENTIFY_PHYSICAL_AND_LOGICAL_QUBITS(q) ->
    IDENTIFY_ERROR_MODEL(q) ->
    IDENTIFY_STABILIZERS_OR_CHECKS(q) ->
    IDENTIFY_SYNDROME(q) ->
    IDENTIFY_DECODER_AND_RECOVERY(q) ->
    IDENTIFY_FAULT_TOLERANCE_AND_RESOURCES(q) ->
    MAP_TO_ONTOLOGY(q) ->
    APPLY_REASONING(q) ->
    SCORE_CONFIDENCE(q) ->
    RETURN_RELEVANT_KNOWLEDGE;

# ---------- CONFIDENCE ----------
CONFIDENCE(assertion) :=
    weighted_mean(
        source_reliability,
        qec_theory_consistency,
        code_consistency,
        decoder_evidence,
        experimental_evidence,
        contextual_relevance
    );

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

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION {
    primary_entity: QuantumErrorCorrection;
    ontology: QuantumErrorCorrectionOntology;
    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;
}

# ---------- UPDATE MODEL ----------
UPDATE_MODEL {
    new_quantum_code:
        validate_parameters
        -> identify_checks
        -> identify_logical_operators
        -> identify_distance
        -> integrate;

    new_error_evidence:
        classify_error
        -> identify_affected_qubits
        -> identify_syndrome
        -> evaluate_decoder_response
        -> integrate;

    new_decoder:
        identify_code
        -> identify_noise_model
        -> evaluate_logical_error_rate
        -> record_latency
        -> integrate;

    ontology_revision:
        update_code_relationships
        -> update_noise_model
        -> update_threshold_evidence
        -> recompute_confidence
        -> regenerate_flow_hash;

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

# ---------- FLOW HASH ----------
FLOW_HASH_INPUT :=
    DOMAIN
    + SUBDOMAIN
    + ONTOLOGY
    + QECEntity
    + ErrorEntity
    + OperationEntity
    + StabilizerCode
    + SyndromeExtraction
    + FaultTolerance
    + RELATIONSHIPS
    + SEMANTIC_REASONING
    + SEMANTIC_EQUIVALENCE
    + UPDATE_MODEL;

FLOW_HASH :=
    HASH(FLOW_HASH_INPUT);

# ---------- TERMINAL ----------
QuantumErrorCorrectionOntology
    -> EncodeLogicalQuantumInformation
    -> DistributeAcrossPhysicalQubits
    -> AllowQuantumOperationOrStorage
    -> ExtractErrorSyndromes
    -> MeasureAncillaAndChecks
    -> DecodeSyndromeHistory
    -> InferErrorEquivalenceClass
    -> CorrectOrTrackRecovery
    -> DetectLogicalFailure
    -> RepeatErrorCorrectionCycles
    -> ExecuteFaultTolerantLogicalOperations
    -> EvaluateLogicalErrorAndResourceOverhead
    -> MaintainProtectedQuantumInformation;

END FLOW