Canonicalization

Canonicalization is used to remove redundancy in the representation.

Consider the following two Hamiltonians:

\[ H_{1} = X \otimes I + I \otimes X \]

\[ H_{2} = I \otimes X + X \otimes I \]

\(H_{1}\) is equivalent to \(H_{2}\). Hence we convert the operators to a canonical form and the canonical form of the above operator is:

\[ H_{c} = 1\cdot(I \otimes X) + 1\cdot(X \otimes I) \]

These canonicalization steps (e.g. distribution) are done by implementing a RewriteRule with the corresponding logic.

Canonicalization Rules

Distribution

Distribution distributes the multiplication, scalar multiplication and tensor product of operators over the addition of operators.

Example

\[X \otimes (Y + Z) \longrightarrow X \otimes Y + X \otimes Z\]

Original GraphTransformed Graph

    graph TD
    element0("PauliX"):::Pauli
    element1("PauliY"):::Pauli
    element2("PauliZ"):::Pauli
    element3("OperatorAdd"):::OperatorAdd
    element3 --> element1 & element2
    element4("OperatorKron"):::OperatorKron
    element4 --> element0 & element3
    classDef Pauli stroke:#800000,stroke-width:3px
    classDef Ladder stroke:#800000,stroke-width:3px
    classDef OperatorAdd stroke:#800000,stroke-width:3px
    classDef OperatorScalarMul stroke:#800000,stroke-width:3px
    classDef OperatorKron stroke:#800000,stroke-width:3px
    classDef OperatorMul stroke:#800000,stroke-width:3px
    classDef MathExpr stroke:#800000,stroke-width:3px

    graph TD
    element0("PauliX"):::Pauli
    element1("PauliY"):::Pauli
    element2("OperatorKron"):::OperatorKron
    element2 --> element0 & element1
    element3("PauliX"):::Pauli
    element4("PauliZ"):::Pauli
    element5("OperatorKron"):::OperatorKron
    element5 --> element3 & element4
    element6("OperatorAdd"):::OperatorAdd
    element6 --> element2 & element5
    classDef Pauli stroke:#800000,stroke-width:3px
    classDef Ladder stroke:#800000,stroke-width:3px
    classDef OperatorAdd stroke:#800000,stroke-width:3px
    classDef OperatorScalarMul stroke:#800000,stroke-width:3px
    classDef OperatorKron stroke:#800000,stroke-width:3px
    classDef OperatorMul stroke:#800000,stroke-width:3px
    classDef MathExpr stroke:#800000,stroke-width:3px

Gather Math Expression

GatherMath centralizes the coefficients of the operators by gathering them.

Example

\[ X \times 3 \times I \longrightarrow 3 \times (X \times I)\]

Original GraphTransformed Graph

    graph TD
    element0("MathExpr<br/>--------<br/>expr = #quot;3#quot;"):::MathExpr
    element1("PauliX"):::Pauli
    element2("OperatorScalarMul"):::OperatorScalarMul
    element2 --> element0 & element1
    element3("PauliI"):::Pauli
    element4("OperatorMul"):::OperatorMul
    element4 --> element2 & element3
    classDef Pauli stroke:#800000,stroke-width:3px
    classDef Ladder stroke:#800000,stroke-width:3px
    classDef OperatorAdd stroke:#800000,stroke-width:3px
    classDef OperatorScalarMul stroke:#800000,stroke-width:3px
    classDef OperatorKron stroke:#800000,stroke-width:3px
    classDef OperatorMul stroke:#800000,stroke-width:3px
    classDef MathExpr stroke:#800000,stroke-width:3px

    graph TD
    element0("MathExpr<br/>--------<br/>expr = #quot;3#quot;"):::MathExpr
    element1("PauliX"):::Pauli
    element2("PauliI"):::Pauli
    element3("OperatorMul"):::OperatorMul
    element3 --> element1 & element2
    element4("OperatorScalarMul"):::OperatorScalarMul
    element4 --> element0 & element3
    classDef Pauli stroke:#800000,stroke-width:3px
    classDef Ladder stroke:#800000,stroke-width:3px
    classDef OperatorAdd stroke:#800000,stroke-width:3px
    classDef OperatorScalarMul stroke:#800000,stroke-width:3px
    classDef OperatorKron stroke:#800000,stroke-width:3px
    classDef OperatorMul stroke:#800000,stroke-width:3px
    classDef MathExpr stroke:#800000,stroke-width:3px

Proper Order

ProperOrder uses the associative property to convert a chain of OperatorAdd or a chain of OperatorMul and reorder the operation order from left to right.

Example

\[ X \otimes (Y + Z) \longrightarrow (X + Y) + Z \]

Original GraphTransformed Graph

    graph TD
    element0("PauliX"):::Pauli
    element1("PauliY"):::Pauli
    element2("PauliZ"):::Pauli
    element3("OperatorAdd"):::OperatorAdd
    element3 --> element1 & element2
    element4("OperatorAdd"):::OperatorAdd
    element4 --> element0 & element3
    classDef Pauli stroke:#800000,stroke-width:3px
    classDef Ladder stroke:#800000,stroke-width:3px
    classDef OperatorAdd stroke:#800000,stroke-width:3px
    classDef OperatorScalarMul stroke:#800000,stroke-width:3px
    classDef OperatorKron stroke:#800000,stroke-width:3px
    classDef OperatorMul stroke:#800000,stroke-width:3px
    classDef MathExpr stroke:#800000,stroke-width:3px

    graph TD
    element0("PauliX"):::Pauli
    element1("PauliY"):::Pauli
    element2("OperatorAdd"):::OperatorAdd
    element2 --> element0 & element1
    element3("PauliZ"):::Pauli
    element4("OperatorAdd"):::OperatorAdd
    element4 --> element2 & element3
    classDef Pauli stroke:#800000,stroke-width:3px
    classDef Ladder stroke:#800000,stroke-width:3px
    classDef OperatorAdd stroke:#800000,stroke-width:3px
    classDef OperatorScalarMul stroke:#800000,stroke-width:3px
    classDef OperatorKron stroke:#800000,stroke-width:3px
    classDef OperatorMul stroke:#800000,stroke-width:3px
    classDef MathExpr stroke:#800000,stroke-width:3px

Pauli Algebra

PauliAlgebra applies the Pauli algebra to simplify the operator.

Example

\[ X \times Y + I \times I \longrightarrow iZ + I \]

Original GraphTransformed Graph

    graph TD
    element0("PauliX"):::Pauli
    element1("PauliY"):::Pauli
    element2("OperatorMul"):::OperatorMul
    element2 --> element0 & element1
    element3("PauliI"):::Pauli
    element4("PauliI"):::Pauli
    element5("OperatorMul"):::OperatorMul
    element5 --> element3 & element4
    element6("OperatorAdd"):::OperatorAdd
    element6 --> element2 & element5
    classDef Pauli stroke:#800000,stroke-width:3px
    classDef Ladder stroke:#800000,stroke-width:3px
    classDef OperatorAdd stroke:#800000,stroke-width:3px
    classDef OperatorScalarMul stroke:#800000,stroke-width:3px
    classDef OperatorKron stroke:#800000,stroke-width:3px
    classDef OperatorMul stroke:#800000,stroke-width:3px
    classDef MathExpr stroke:#800000,stroke-width:3px

    graph TD
    element0("MathExpr<br/>--------<br/>expr = #quot;1j#quot;"):::MathExpr
    element1("PauliZ"):::Pauli
    element2("OperatorScalarMul"):::OperatorScalarMul
    element2 --> element0 & element1
    element3("PauliI"):::Pauli
    element4("OperatorAdd"):::OperatorAdd
    element4 --> element2 & element3
    classDef Pauli stroke:#800000,stroke-width:3px
    classDef Ladder stroke:#800000,stroke-width:3px
    classDef OperatorAdd stroke:#800000,stroke-width:3px
    classDef OperatorScalarMul stroke:#800000,stroke-width:3px
    classDef OperatorKron stroke:#800000,stroke-width:3px
    classDef OperatorMul stroke:#800000,stroke-width:3px
    classDef MathExpr stroke:#800000,stroke-width:3px

Normal Order

NormalOrder puts the ladder operators into normal order.

Example

\[ C \times A + A \times C \longrightarrow C \times A + C \times A + J\]

Original GraphTransformed Graph

    graph TD
    element0("Creation"):::Ladder
    element1("Annihilation"):::Ladder
    element2("OperatorMul"):::OperatorMul
    element2 --> element0 & element1
    element3("Annihilation"):::Ladder
    element4("Creation"):::Ladder
    element5("OperatorMul"):::OperatorMul
    element5 --> element3 & element4
    element6("OperatorAdd"):::OperatorAdd
    element6 --> element2 & element5
    classDef Pauli stroke:#800000,stroke-width:3px
    classDef Ladder stroke:#800000,stroke-width:3px
    classDef OperatorAdd stroke:#800000,stroke-width:3px
    classDef OperatorScalarMul stroke:#800000,stroke-width:3px
    classDef OperatorKron stroke:#800000,stroke-width:3px
    classDef OperatorMul stroke:#800000,stroke-width:3px
    classDef MathExpr stroke:#800000,stroke-width:3px

    graph TD
    element0("Creation"):::Ladder
    element1("Annihilation"):::Ladder
    element2("OperatorMul"):::OperatorMul
    element2 --> element0 & element1
    element3("Creation"):::Ladder
    element4("Annihilation"):::Ladder
    element5("OperatorMul"):::OperatorMul
    element5 --> element3 & element4
    element6("Identity"):::Ladder
    element7("OperatorAdd"):::OperatorAdd
    element7 --> element5 & element6
    element8("OperatorAdd"):::OperatorAdd
    element8 --> element2 & element7
    classDef Pauli stroke:#800000,stroke-width:3px
    classDef Ladder stroke:#800000,stroke-width:3px
    classDef OperatorAdd stroke:#800000,stroke-width:3px
    classDef OperatorScalarMul stroke:#800000,stroke-width:3px
    classDef OperatorKron stroke:#800000,stroke-width:3px
    classDef OperatorMul stroke:#800000,stroke-width:3px
    classDef MathExpr stroke:#800000,stroke-width:3px

Prune Identity

PruneIdentity prunes the unnecessary ladder identities from the graph.

Example

\[ C\times A \times J\longrightarrow C \times A\]

Original GraphTransformed Graph

    graph TD
    element0("Creation"):::Ladder
    element1("Annihilation"):::Ladder
    element2("OperatorMul"):::OperatorMul
    element2 --> element0 & element1
    element3("Identity"):::Ladder
    element4("OperatorMul"):::OperatorMul
    element4 --> element2 & element3
    classDef Pauli stroke:#800000,stroke-width:3px
    classDef Ladder stroke:#800000,stroke-width:3px
    classDef OperatorAdd stroke:#800000,stroke-width:3px
    classDef OperatorScalarMul stroke:#800000,stroke-width:3px
    classDef OperatorKron stroke:#800000,stroke-width:3px
    classDef OperatorMul stroke:#800000,stroke-width:3px
    classDef MathExpr stroke:#800000,stroke-width:3px

    graph TD
    element0("Creation"):::Ladder
    element1("Annihilation"):::Ladder
    element2("OperatorMul"):::OperatorMul
    element2 --> element0 & element1
    classDef Pauli stroke:#800000,stroke-width:3px
    classDef Ladder stroke:#800000,stroke-width:3px
    classDef OperatorAdd stroke:#800000,stroke-width:3px
    classDef OperatorScalarMul stroke:#800000,stroke-width:3px
    classDef OperatorKron stroke:#800000,stroke-width:3px
    classDef OperatorMul stroke:#800000,stroke-width:3px
    classDef MathExpr stroke:#800000,stroke-width:3px

Sorted Order

SortedOrder sorts the addition terms in operators into a predefined order.

Example

\[ X \otimes I + I \otimes X \longrightarrow I \otimes X + X \otimes I\]

Original GraphTransformed Graph

    graph TD
    element0("PauliX"):::Pauli
    element1("PauliI"):::Pauli
    element2("OperatorKron"):::OperatorKron
    element2 --> element0 & element1
    element3("PauliI"):::Pauli
    element4("PauliX"):::Pauli
    element5("OperatorKron"):::OperatorKron
    element5 --> element3 & element4
    element6("OperatorAdd"):::OperatorAdd
    element6 --> element2 & element5
    classDef Pauli stroke:#800000,stroke-width:3px
    classDef Ladder stroke:#800000,stroke-width:3px
    classDef OperatorAdd stroke:#800000,stroke-width:3px
    classDef OperatorScalarMul stroke:#800000,stroke-width:3px
    classDef OperatorKron stroke:#800000,stroke-width:3px
    classDef OperatorMul stroke:#800000,stroke-width:3px
    classDef MathExpr stroke:#800000,stroke-width:3px

    graph TD
    element0("PauliI"):::Pauli
    element1("PauliX"):::Pauli
    element2("OperatorKron"):::OperatorKron
    element2 --> element0 & element1
    element3("PauliX"):::Pauli
    element4("PauliI"):::Pauli
    element5("OperatorKron"):::OperatorKron
    element5 --> element3 & element4
    element6("OperatorAdd"):::OperatorAdd
    element6 --> element2 & element5
    classDef Pauli stroke:#800000,stroke-width:3px
    classDef Ladder stroke:#800000,stroke-width:3px
    classDef OperatorAdd stroke:#800000,stroke-width:3px
    classDef OperatorScalarMul stroke:#800000,stroke-width:3px
    classDef OperatorKron stroke:#800000,stroke-width:3px
    classDef OperatorMul stroke:#800000,stroke-width:3px
    classDef MathExpr stroke:#800000,stroke-width:3px

Scale Terms

ScaleTerms introduces scalar multiplication to terms without a coefficient for a more consistent reprensentation.

Example

\[ I \otimes X + X \otimes I \longrightarrow 1*(I \otimes X) + 1*(X \otimes I)\]

Original GraphTransformed Graph

    graph TD
    element0("PauliI"):::Pauli
    element1("PauliX"):::Pauli
    element2("OperatorKron"):::OperatorKron
    element2 --> element0 & element1
    element3("PauliX"):::Pauli
    element4("PauliI"):::Pauli
    element5("OperatorKron"):::OperatorKron
    element5 --> element3 & element4
    element6("OperatorAdd"):::OperatorAdd
    element6 --> element2 & element5
    classDef Pauli stroke:#800000,stroke-width:3px
    classDef Ladder stroke:#800000,stroke-width:3px
    classDef OperatorAdd stroke:#800000,stroke-width:3px
    classDef OperatorScalarMul stroke:#800000,stroke-width:3px
    classDef OperatorKron stroke:#800000,stroke-width:3px
    classDef OperatorMul stroke:#800000,stroke-width:3px
    classDef MathExpr stroke:#800000,stroke-width:3px

graph TD
element0("MathExpr<br/>--------<br/>expr = #quot;1#quot;"):::MathExpr
element1("PauliI"):::Pauli
element2("PauliX"):::Pauli
element3("OperatorKron"):::OperatorKron
element3 --> element1 & element2
element4("OperatorScalarMul"):::OperatorScalarMul
element4 --> element0 & element3
element5("MathExpr<br/>--------<br/>expr = #quot;1#quot;"):::MathExpr
element6("PauliX"):::Pauli
element7("PauliI"):::Pauli
element8("OperatorKron"):::OperatorKron
element8 --> element6 & element7
element9("OperatorScalarMul"):::OperatorScalarMul
element9 --> element5 & element8
element10("OperatorAdd"):::OperatorAdd
element10 --> element4 & element9
classDef Pauli stroke:#800000,stroke-width:3px
classDef Ladder stroke:#800000,stroke-width:3px
classDef OperatorAdd stroke:#800000,stroke-width:3px
classDef OperatorScalarMul stroke:#800000,stroke-width:3px
classDef OperatorKron stroke:#800000,stroke-width:3px
classDef OperatorMul stroke:#800000,stroke-width:3px
classDef MathExpr stroke:#800000,stroke-width:3px

Canonicalization Pass

stateDiagram-v2

[*] --> distribution_pass: done
distribution_pass --> ProperOrder: done
ProperOrder --> pauli_algebra_pass: done
pauli_algebra_pass --> GatherPauli: done
GatherPauli --> VerifyHilbertSpaceDim: done
VerifyHilbertSpaceDim --> normal_order_pass: done
normal_order_pass --> PruneIdentity: pass
PruneIdentity --> scale_terms_pass: done
scale_terms_pass --> SortedOrder: done
SortedOrder --> math_canonicalization_pass: done
math_canonicalization_pass --> end: done