Pauli Algebra

LinearAlgebra.tr(psum::AbstractPauliSum)

Compute the trace of an AbstractPauliSum operator.

The trace is a linear operation: Tr(A + B) = Tr(A) + Tr(B). Since individual non-identity PauliString terms have a trace of zero (as per tr(::PauliString)), only the coefficient of the identity operator contributes to the total trace of a PauliSum.

The coefficient of the identity term (0x00) is retrieved from this mapping using get(psum.terms, 0x00, 0). This coefficient is then multiplied by 2^nqubits(psum) (the dimension of the Hilbert space). If the identity term (0x00) is not explicitly present in psum.terms, its coefficient is implicitly zero, resulting in a total trace of 0.0.

Arguments

psum::PauliSum: The Pauli sum to trace.

Returns

CT: The trace value of the PauliSum.

LinearAlgebra.tr(pstr::PauliString)

Compute the trace of a PauliString operator.

The trace of any non-identity Pauli operator (e.g., X, Y, Z, or their tensor products) is zero. The identity operator I has a trace equal to the dimension of the Hilbert space, 2^N, where N is the number of qubits.

A PauliString represents a single term, typically of the form coeff * P_1 P_2 ... P_N. If the pstr.term bitmask is 0x00, it signifies the identity operator across all pstr.nqubits. In this case, the trace is pstr.coeff * 2^pstr.nqubits. For any other pstr.term value (representing a non-identity Pauli operator), the trace is 0.0.

Arguments

• pstr::PauliString: The Pauli string to trace.

Returns

• CT: The trace value of the PauliString.

commutator(pstr1::Integer, pstr2::Integer)

Calculate the commutator of two integer Pauli strings. Returns a tuple of the coefficient and the potentially integer Pauli string. The coefficient is zero if the Pauli strings commute.

commutator(pstr1::PauliString, pstr2::PauliString)

Calculate the commutator of two PauliStrings.

commutator(psum::PauliSum, pstr::PauliString)
commutator(pstr::PauliString, psum::PauliSum)

Calculate the commutator of a PauliSum and a PauliString.

commutator(psum1::PauliSum, psum2::PauliSum)

Calculate the commutator of two PauliSums.

commutator(vpsum1::VectorPauliSum, vpsum2::VectorPauliSum)

Calculate the commutator of two VectorPauliSums. Returns a VectorPauliSum with complex coefficients.

Example

vpsum = VectorPauliSum(3, [1, 2], [1.]) # 1.0 * X + 1.0 * Y
vpsum2 = VectorPauliSum(3, [7], [0.25]) # 0.25 * ZX
commutator(vpsum, vpsum2) # should return a VectorPauliSum with - 0.5 YX + 0.5 XX

commutes(pstr1::Integer, pstr2::Integer)

Check if two integer Pauli strings commute.

commutes(pstr1::PauliString, pstr2::PauliString)

Check if two Pauli strings of type PauliString commute.

commutes(psum1::PauliSum, psum2::PauliSum)

Check if two Pauli sums of type PauliSum commute.

containsXorY(pstr::Integer)

Check if an integer Pauli string contains an X or Y Pauli.

containsXorY(pstr::PauliString)

Check if a Pauli string contains an X or Y Pauli.

containsYorZ(pstr::Integer)

Check if an integer Pauli string contains a Y or Z Pauli.

containsXorY(pstr::PauliString)

Check if a Pauli string contains a Y or Z Pauli.

countweight(pstr::Integer)

Function to count the weight of an integer Pauli string.

countweight(pstr::PauliString)

Function to count the weight of a PauliString.

countweight(psum::PauliSum)

Function to count the weight Pauli strings in a PauliSum. Returns an array of weights.

countx(pstr::Integer)

Function to count the number of X Paulis in an integer Pauli string.

countx(pstr::PauliString)

Function to count the number of X Paulis in a PauliString.

countx(psum::PauliSum)

Function to count the number of X Paulis in a PauliSum. Returns an array of counts.

countxy(pstr::Integer)

Function to count the number of X and Y Paulis in an integer Pauli string.

countxy(pstr::PauliString)

Function to count the number of X and Y Paulis in a PauliString.

countxy(psum::PauliSum)

Function to count the number of X and Y Paulis in a PauliSum. Returns an array of counts.

county(pstr::Integer)

Function to count the number of Y Paulis in an integer Pauli string.

county(pstr::PauliString)

Function to count the number of Y Paulis in a PauliString.

county(psum::PauliSum)

Function to count the number of Y Paulis in a PauliSum. Returns an array of counts.

countyz(pstr::Integer)

Function to count the number of Y and Z Paulis in an integer Pauli string.

countyz(pstr::PauliString)

Function to count the number of Y and Z Paulis in a PauliString.

countyz(psum::PauliSum)

Function to count the number of Y and Z Paulis in a PauliSum. Returns an array of counts.

countz(pstr::Integer)

Function to count the number of Z Paulis in an integer Pauli string.

countz(pstr::PauliString)

Function to count the number of Z Paulis in a PauliString.

countz(psum::PauliSum)

Function to count the number of Z Paulis in a PauliSum. Returns an array of counts.

pauliprod(pstr1::Integer, pstr2::Integer)

Calculate the product of two integer Pauli strings.

pauliprod(pstr1::PauliString, pstr2::PauliString)

Calculate the product of two PauliStrings.

Examples

julia> pauliprod(PauliString(1, [:X], [1]), PauliString(1, [:Y], [1])) # X*Y=iZ

pauliprod(psum1::PauliSum, psum2::PauliSum)

Calculate the product of two PauliSums. Default returns a PauliSum{TT, ComplexF64} where TT is the type of the new Pauli Strings.

Examples

psum = PauliSum(PauliString(3, [:Y], [2])) 
psum_identity = PauliSum(PauliString(3, [:I], [1]))
pauliprod(psum, psum_identity) # Psum * I = Psum

trace(psum::PauliSum)

Wrapper for LinearAlgebra.tr(psum::PauliSum).

Arguments

psum::PauliSum: The Pauli sum to trace.

Returns

CT: The trace value of the PauliSum.

trace(pstr::PauliString)

Wrapper for LinearAlgebra.tr(pstr::PauliString).

Arguments

• pstr::PauliString: The Pauli string to trace.

Returns

• CT: The trace value of the PauliString.

getinttype(nqubits::Integer)

Function to return the smallest integer type that can hold nqubits. This is the type that will be used internally for representing Pauli strings.

getpauli(pstr::PauliStringType, qinds::Vector{Integer})

Gets the Paulis on indices qinds of a pstr in the integer representation.

getpauli(pstr::PauliStringType, qind1::Int, qind2::Int)

Gets the Paulis from qind1 to qind2 of a pstr in the integer representation. This function is useful for extracting a continuous sub-PauliString.

inttostring(pstr::PauliType, nqubits::Integer)

Returns a string representation of an integer Pauli string pstr on nqubits qubits. The characters of the string from left to right are the Paulis on the qubits from 1 to nqubits.

inttosymbol(pstr::PauliStringType, nqubits::Integer)

Maps an integer Pauli string to a vector of symbols.

inttosymbol(pauli::PauliType)

Maps an integer Pauli to its corresponding symbol.

ispauli(pauli1::Union{Symbol, PauliType}, pauli2::Union{Symbol, PauliType})

ispauli(pauli1::Union{Vector{Symbol}, PauliStringType}, pauli2::Union{Vector{Symbol}, PauliStringType})

Check if two Paulis are equal, where one is given as a symbol and the other as an integer.

setpauli(pstr::PauliStringType, target_paulis::PauliStringType, index1::Integer, index2::Integer)

Sets the Paulis from index1 to index2 of an integer Pauli string to target_paulis.

setpauli(
    pstr::PauliStringType, 
    target_paulis::Vector{Symbol}, 
    qinds::Vector{Integer}
)

Set the Paulis qinds of an integer Pauli string pstr to target_paulis. target_paulis is a vector of symbols. Use tuples in performance critical functions because they are immutable.

setpauli(
    pstr::PauliStringType, 
    target_paulis::PauliStringType, 
    qinds::Vector{Integer}
)

Set the Paulis qinds of an integer Pauli string pstr to target_paulis. Use Tuples for qinds in performance critical functions because they are immutable.

setpauli(pstr::PauliStringType, target_pauli::PauliType, index::Integer)

Sets the Pauli on index index of an integer Pauli string to target_pauli. That Pauli should be provided as integer (0, 1, 2, 3).

setpauli(pstr::PauliStringType, target_pauli::Symbol, index::Integer)

Sets the Pauli on index of an integer Pauli string to target_pauli. That Pauli should be provided as a symbol (:I, :X, :Y, :Z).

symboltoint(pstr::Union{Vector{Symbol}, Symbol})

Maps a symbol or a vector of symbols pstr to an integer Pauli string.

Example:

symboltoint([:X, :I])
>>> 0x01

symboltoint(nqubits::Integer, paulis::Vector{Symbol}, qinds::Vector{Int})

Maps a vector of symbols pstr acting on the indices qinds to an integer Pauli string. Other sites are set to the identity. qinds can be any iterable.

symboltoint(nqubits::Integer, pauli::Symbol, qind::Integer)

Maps a single symbol pauli acting on the index qind to an integer Pauli string. Other sites are set to the identity.

symboltoint(pauli::Symbol)

Maps a single symbol to its corresponding integer representation.

symboltoint(::PauliStringType, paulis, qinds)

Maps a vector of symbols paulis acting on the indices qinds to an integer Pauli string with type PauliStringType. Other sites are set to the identity. qinds can be any iterable.

symboltoint(::PauliStringType, pauli::Symbol, qind::Integer)

Maps a single symbol pauli acting on the index qind to an integer Pauli string with type PauliStringType. Other sites are set to the identity.