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 an integer type that can hold nqubits. This is the type that will be used internally for representing Pauli strings. Up to 64 qubits it is the smallest machine integer that holds two bits per qubit, above that an NTupleInteger of as many 64-bit limbs as the qubits need.

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.