Quantum Gate Ontology

FLOW ALGEBRA
Quantum Gate Ontology
Kireth Semantic Research Archive

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

AUTHORS:
Mark Anthony Higgins — Sky (Recursive AI)

FLOW "QuantumGateOntology" v4.0

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

# ---------- PRIMARY ENTITY ----------
ENTITY QuantumGate {
    definition:
        "A quantum gate is an operation that transforms the state of one or
         more qubits, typically represented in the ideal circuit model by a
         unitary transformation acting on a quantum state vector.";

    objectives: [
        QuantumStateTransformation,
        QuantumCircuitConstruction,
        SuperpositionControl,
        PhaseControl,
        EntanglementGeneration,
        UniversalQuantumComputation
    ];
}

# ---------- CORE ENTITIES ----------
GateEntity := {
    QuantumGate,
    Qubit,
    QuantumState,
    QuantumRegister,
    GateMatrix,
    GateParameter,
    QuantumCircuit
};

SingleQubitGate := {
    IdentityGate,
    PauliX,
    PauliY,
    PauliZ,
    Hadamard,
    PhaseGate,
    TGate,
    RotationX,
    RotationY,
    RotationZ
};

MultiQubitGate := {
    CNOT,
    ControlledZ,
    ControlledUnitary,
    SWAP,
    Toffoli
};

GateProperty := {
    Unitary,
    Reversible,
    Hermitian,
    Parameterised,
    Controlled,
    Native,
    Composite
};

# ---------- FUNDAMENTAL RELATIONSHIPS ----------
QuantumGate ACTS_ON QubitOrQuantumRegister;
QuantumGate TRANSFORMS QuantumState;
QuantumGate MAY_BE_REPRESENTED_BY GateMatrix;
QuantumCircuit CONTAINS QuantumGate;
ControlledGate HAS ControlQubit;
ControlledGate HAS TargetQubitOrRegister;
CompositeGate DECOMPOSES_INTO SimplerGates;
QuantumProcessor IMPLEMENTS NativeGateSet;

# ---------- GATE APPLICATION FLOW ----------
InputQuantumState
    -> SelectGate
    -> IdentifyTargetQubits
    -> ValidateDimensions
    -> ApplyGateOperator
    -> ProduceOutputQuantumState
    -> ContinueCircuitOrMeasure;

# ---------- UNITARY OPERATION ----------
IdealQuantumGate :=
    UnitaryOperator U;

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

UnitaryCondition :=
    U_dagger U = I;

IF U IS Unitary
THEN StateNorm IS_PRESERVED;

# ---------- DIMENSION ----------
GateActingOnNQubits HAS MatrixDimension:
    2^N x 2^N;

GateMatrix MUST_BE CompatibleWithTargetStateSpace;

# ---------- IDENTITY ----------
IdentityGate :=
    I;

I|psi> = |psi>;

IdentityGate LEAVES QuantumStateUnchanged;

# ---------- PAULI X ----------
X :=
    [[0,1],
     [1,0]];

X|0> = |1>;
X|1> = |0>;

PauliX IS {
    Unitary,
    Hermitian
};

# ---------- PAULI Y ----------
Y :=
    [[0,-i],
     [i,0]];

Y|0> = i|1>;
Y|1> = -i|0>;

PauliY IS {
    Unitary,
    Hermitian
};

# ---------- PAULI Z ----------
Z :=
    [[1,0],
     [0,-1]];

Z|0> = |0>;
Z|1> = -|1>;

PauliZ IS {
    Unitary,
    Hermitian
};

# ---------- HADAMARD ----------
H :=
    (1/sqrt(2))
    *
    [[1,1],
     [1,-1]];

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

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

Hadamard MAY_CREATE ComputationalBasisSuperposition;

# ---------- PHASE GATE ----------
S :=
    [[1,0],
     [0,i]];

S|0> = |0>;
S|1> = i|1>;

S IS SquareRootOfZUpToConventionalDefinition;

# ---------- T GATE ----------
T :=
    [[1,0],
     [0,exp(i*pi/4)]];

T IS CommonNonCliffordSingleQubitGate;

CliffordPlusT MAY_FORM UniversalGateSetForApproximateQuantumComputation;

# ---------- ROTATION X ----------
Rx(theta) :=
    exp(-i * theta * X / 2);

Rx(theta) ROTATES BlochVectorAboutXAxis;

# ---------- ROTATION Y ----------
Ry(theta) :=
    exp(-i * theta * Y / 2);

Ry(theta) ROTATES BlochVectorAboutYAxis;

# ---------- ROTATION Z ----------
Rz(theta) :=
    exp(-i * theta * Z / 2);

Rz(theta) ROTATES BlochVectorAboutZAxis;

# ---------- PARAMETERISED GATE ----------
ParameterisedGate :=
    QuantumGateDependingOnContinuousOrDiscreteParameter;

Parameter MAY_BE {
    RotationAngle,
    Phase,
    ControlConfiguration
};

VariationalQuantumCircuit MAY_CONTAIN ParameterisedGates;

# ---------- CONTROLLED GATE ----------
ControlledU :=
    Apply U ToTarget
    IF ControlStateMatchesControlCondition;

ControlledGate MAY_HAVE OneOrMoreControlQubits;

# ---------- CNOT ----------
CNOT :=
    ControlledX;

BasisMapping:
    |00> -> |00>;
    |01> -> |01>;
    |10> -> |11>;
    |11> -> |10>;

CNOT MAY_CREATE EntanglementWhenInputContainsSuitableSuperposition;

# ---------- CONTROLLED Z ----------
CZ :=
    diag(1,1,1,-1);

CZ APPLIES PhaseFlipTo |11> Component;

# ---------- SWAP ----------
SWAP :=
    ExchangeStatesOfTwoQubits;

BasisMapping:
    |ab> -> |ba>;

SWAP MAY_BE_DECOMPOSED_INTO ThreeCNOTGatesInStandardGateModel;

# ---------- TOFFOLI ----------
Toffoli :=
    ControlledControlledX;

Toffoli HAS {
    TwoControlQubits,
    OneTargetQubit
};

Toffoli IS ReversibleMultiQubitGate;

# ---------- ENTANGLING GATE ----------
EntanglingGate :=
    MultiQubitGateCapableOfProducingEntangledOutputFromAtLeastSomeProductInputs;

CNOT IS EntanglingGate;

SingleQubitGatesAlone CANNOT_CREATE EntanglementBetweenInitiallySeparableQubits;

# ---------- GATE COMPOSITION ----------
SequentialGateComposition :=
    U_total = U_n ... U_2 U_1;

RightmostOperator ACTS_FIRST UnderStandardColumnVectorConvention;

# ---------- PARALLEL GATES ----------
If GatesActOnDisjointQubits
THEN CombinedOperation MAY_BE TensorProductOfGateOperators;

ParallelGateLayer MAY_REDUCE CircuitDepth;

# ---------- INVERSE GATE ----------
InverseGate :=
    U^-1;

ForUnitaryGate:
    U^-1 = U_dagger;

Applying U THEN U_dagger
    -> IdentityTransformation;

# ---------- HERMITIAN GATE ----------
HermitianGate :=
    U = U_dagger;

PauliX IS Hermitian;
PauliY IS Hermitian;
PauliZ IS Hermitian;
Hadamard IS Hermitian;

# ---------- CLIFFORD GATE ----------
CliffordGate :=
    GateMappingPauliOperatorsToPauliOperatorsUnderConjugation;

CliffordSet MAY_INCLUDE {
    Hadamard,
    PhaseGate,
    CNOT
};

CliffordCircuits HAVE EfficientClassicalSimulationUnderImportantRestrictedConditions;

# ---------- NON-CLIFFORD GATE ----------
NonCliffordGate MAY_ENABLE UniversalQuantumComputationWhenCombinedWithSuitableCliffordGates;

TGate IS CommonNonCliffordGate;

# ---------- UNIVERSAL GATE SET ----------
UniversalGateSet :=
    SetOfGatesAbleToApproximateArbitraryUnitaryOperationsToDesiredAccuracy;

Example:
    Clifford + T
    MAY_FORM UniversalDiscreteGateSet;

ArbitrarySingleQubitRotations + EntanglingTwoQubitGate
    MAY_FORM UniversalGateFamily;

# ---------- NATIVE GATE ----------
NativeGate :=
    QuantumOperationDirectlySupportedByHardwareControlSystem;

HardwareNativeGateSet DEPENDS_ON PhysicalPlatform;

LogicalOrHighLevelGate MAY_REQUIRE DecompositionIntoNativeGates;

# ---------- GATE DECOMPOSITION ----------
GateDecomposition :=
    TargetGate
    -> SequenceOfSupportedSimplerGates;

Decomposition SHOULD_APPROXIMATE_OR_EQUAL TargetOperationWithinRequiredTolerance;

# ---------- COMPILATION ----------
QuantumCompiler :=
    Circuit
    -> DecomposeGates
    -> MapQubits
    -> RouteInteractions
    -> Optimise
    -> NativeCircuit;

# ---------- CONNECTIVITY ----------
TwoQubitNativeGate MAY_REQUIRE PhysicalQubitsToBeConnectedByHardwareInteractionGraph;

IF LogicalQubitsNotAdjacentUnderMapping
THEN RoutingOperation MAY_BE_REQUIRED;

# ---------- ROUTING ----------
Routing MAY_INSERT {
    SWAP,
    EquivalentMovementOrTeleportationMechanism
};

Routing MAY_INCREASE {
    GateCount,
    CircuitDepth,
    ErrorExposure
};

# ---------- GATE FIDELITY ----------
GateFidelity :=
    MeasureOfAgreementBetweenImplementedGateAndIdealTargetOperation;

GateError :=
    DeviationFromIdealOperation;

HigherGateFidelity GENERALLY_MEANS LowerGateError;

# ---------- GATE DURATION ----------
GateDuration :=
    TimeRequiredToImplementPhysicalGate;

LongerDuration MAY_INCREASE ExposureToDecoherence;

# ---------- SINGLE / TWO-QUBIT ERROR ----------
TwoQubitGateError IS_OFTEN GreaterThanSingleQubitGateErrorOnManyCurrentPlatforms;

ActualError DEPENDS_ON HardwareAndCalibration;

# ---------- CONTROL PULSE ----------
PhysicalGate MAY_BE_IMPLEMENTED_BY ControlPulse;

ControlPulse MAY_HAVE {
    Amplitude,
    Frequency,
    Phase,
    Duration,
    Shape
};

PulseCalibration MAPS DesiredGateToPhysicalControl;

# ---------- VIRTUAL GATE ----------
SomePhaseOperations MAY_BE ImplementedVirtuallyByUpdatingReferenceFrameRatherThanApplyingPhysicalPulse;

VirtualImplementation MAY_HAVE VeryLowAdditionalPhysicalErrorOrDurationDependingOnPlatform;

# ---------- GATE NOISE ----------
NoisyGate :=
    IdealGate
    + ErrorChannel;

ErrorChannel MAY_INCLUDE {
    DepolarisingNoise,
    CoherentOverRotation,
    Crosstalk,
    Dephasing,
    Relaxation
};

# ---------- COHERENT ERROR ----------
CoherentGateError :=
    SystematicUnitaryDeviationFromTargetGate;

RepeatedCoherentError MAY_ACCUMULATE;

# ---------- STOCHASTIC ERROR ----------
StochasticGateError :=
    RandomErrorProcessAssociatedWithGateExecution;

# ---------- CROSSTALK ----------
Crosstalk :=
    GateOperationOnTargetQubitUnintentionallyAffectsOtherQubitOrOperation;

ParallelGateScheduling MAY_BE ConstrainedByCrosstalk;

# ---------- CALIBRATION ----------
GateCalibration :=
    TuneControlParameters
    -> MeasureGateBehaviour
    -> OptimiseTowardTarget;

Calibration MAY_DRIFT_OVER Time;

# ---------- BENCHMARKING ----------
GateBenchmarking MAY_INCLUDE {
    RandomisedBenchmarking,
    ProcessTomography,
    InterleavedBenchmarking
};

Benchmarking ESTIMATES GatePerformance;

# ---------- QUANTUM PROCESS TOMOGRAPHY ----------
ProcessTomography :=
    PrepareKnownInputs
    -> ApplyUnknownOrTestGate
    -> MeasureOutputsAcrossBases
    -> EstimateQuantumProcess;

Tomography MAY_BE ResourceIntensive;

# ---------- CIRCUIT DEPTH ----------
CircuitDepth :=
    NumberOfSequentialGateLayers;

LowerDepth MAY_REDUCE DecoherenceExposureAndExecutionTime;

# ---------- GATE COUNT ----------
GateCount :=
    TotalNumberOfGateOperations;

GateCount MAY_CORRELATE_WITH AccumulatedError;

# ---------- T-COUNT ----------
TCount :=
    NumberOfTOrEquivalentNonCliffordResourceGates;

FaultTolerantResourceEstimation OFTEN_TRACKS TCount;

# ---------- T-DEPTH ----------
TDepth :=
    NumberOfSequentialLayersContainingTResourcesAfterParallelisation;

# ---------- LOGICAL GATE ----------
LogicalGate :=
    QuantumOperationAppliedToEncodedLogicalQubit;

LogicalGate MAY_REQUIRE ManyPhysicalOperations;

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

Transversality MAY_LIMIT ErrorPropagation;

# ---------- MAGIC STATE ----------
FaultTolerantNonCliffordGate MAY_REQUIRE MagicStatePreparationAndDistillationInSomeArchitectures;

MagicStateCost MAY_DOMINATE FaultTolerantResourceRequirements;

# ---------- MEASUREMENT RELATIONSHIP ----------
Measurement IS_NOT GenerallyUnitaryGate;

QuantumCircuit MAY_CONTAIN BothUnitaryGatesAndMeasurements;

# ---------- RESET RELATIONSHIP ----------
Reset IS_NOT UnitaryOnArbitraryUnknownInputState;

Reset PREPARES DefinedState;

# ---------- SEMANTIC REASONING ----------
IF OperationOnClosedIdealQuantumStateIsUnitary
THEN Operation MAY_BE ModelledAsIdealQuantumGate;

IF U_dagger U != I
THEN U IS_NOT ValidIdealUnitaryGate;

IF GateActsOnNQubits
THEN MatrixDimension := 2^N x 2^N;

IF HadamardAppliedTo |0>
THEN Output := (|0> + |1>)/sqrt(2);

IF CNOTAppliedAfterHadamardToSuitableTwoQubitInput
THEN Entanglement MAY_BE Created;

IF HardwareDoesNotSupportGateNatively
THEN Decomposition := required;

IF RoutingAddsManyTwoQubitGates
THEN CircuitFidelity MAY_DECREASE;

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE {
    "quantum gate" ~= "quantum logic gate";
    "unitary gate" ~= "unitary quantum operation" WHEN context == ideal_circuit;
    "controlled-NOT" ~= "CNOT";
    "controlled-X" ~= "CNOT";
    "Toffoli gate" ~= "controlled-controlled-NOT";
    "native gate" ~= "hardware-native quantum gate";
    "gate decomposition" ~= "quantum gate synthesis" WHEN context == supported_gate_sequence;
    "gate fidelity" ~= "quantum operation fidelity" WHEN context == implemented_gate;
}

# ---------- QUERY FLOW ----------
QUERY(q) ->
    NORMALISE(q) ->
    IDENTIFY_GATE(q) ->
    IDENTIFY_TARGET_QUBITS(q) ->
    IDENTIFY_MATRIX_OR_OPERATION(q) ->
    IDENTIFY_GATE_PROPERTIES(q) ->
    IDENTIFY_COMPOSITION_OR_CONTROL(q) ->
    IDENTIFY_HARDWARE_NATIVE_SUPPORT(q) ->
    IDENTIFY_FIDELITY_AND_NOISE(q) ->
    MAP_TO_ONTOLOGY(q) ->
    APPLY_REASONING(q) ->
    SCORE_CONFIDENCE(q) ->
    RETURN_RELEVANT_KNOWLEDGE;

# ---------- CONFIDENCE ----------
CONFIDENCE(assertion) :=
    weighted_mean(
        source_reliability,
        mathematical_consistency,
        gate_semantic_consistency,
        hardware_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: QuantumGate;
    ontology: QuantumGateOntology;
    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_gate:
        validate_dimension
        -> validate_unitarity_where_applicable
        -> classify_single_or_multi_qubit
        -> identify_properties
        -> integrate;

    new_decomposition:
        identify_target_gate
        -> identify_native_gate_set
        -> validate_equivalence_or_approximation
        -> calculate_gate_count_depth
        -> integrate;

    new_hardware_evidence:
        identify_platform
        -> identify_native_gate
        -> record_duration
        -> record_fidelity
        -> record_connectivity
        -> integrate;

    ontology_revision:
        update_gate_relationships
        -> update_native_support
        -> update_error_evidence
        -> recompute_confidence
        -> regenerate_flow_hash;

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

# ---------- FLOW HASH ----------
FLOW_HASH_INPUT :=
    DOMAIN
    + SUBDOMAIN
    + ONTOLOGY
    + GateEntity
    + SingleQubitGate
    + MultiQubitGate
    + GateProperty
    + GateDecomposition
    + GateFidelity
    + NativeGate
    + RELATIONSHIPS
    + SEMANTIC_REASONING
    + SEMANTIC_EQUIVALENCE
    + UPDATE_MODEL;

FLOW_HASH :=
    HASH(FLOW_HASH_INPUT);

# ---------- TERMINAL ----------
QuantumGateOntology
    -> IdentifyQuantumGate
    -> DetermineTargetQubits
    -> ValidateOperatorDimension
    -> ApplyUnitaryTransformation
    -> ComposeWithOtherGates
    -> GeneratePhaseSuperpositionOrEntanglement
    -> DecomposeIntoNativeOperations
    -> MapToHardwareConnectivity
    -> EvaluateGateCountAndDepth
    -> MeasureFidelityAndNoise
    -> OptimiseCircuitImplementation
    -> SupportLogicalAndFaultTolerantOperations
    -> MaintainConsistentQuantumGateKnowledge;

END FLOW