Pauli Algebra

PauliString(nqubits::Int, pauli::Symbol, qind::Integer, coeff=1.0)
PauliString(nqubits::Int, paulis::Vector{Symbol}, qinds::Vector{Integer}, coeff=1.0)

Constructor for a PauliString on nqubits qubits from a Symbol (:I, :X, :Y, :Z) or Vector{Symbol}. Provide the index or indices for those symbols as qind or qinds. The coefficient of the Pauli string in the Pauli sum defaults to 1.0.

PauliString(nqubits::Int, term::TermType, coeff::CoeffType)

PauliString is a struct that represents a Pauli string on nqubits qubits. Commonly term is an unsigned Integer. See the other PauliString constructors for higher-level usage.

PauliSum(nqubits::Int, terms::Dict)

PauliSum is a struct that represents a sum of Pauli strings acting on nqubits qubits. It is a wrapper around a dictionary Dict(Pauli string => coefficient}, where the Pauli strings are typically unsigned Integers for efficiency reasons.

PauliSum(nqubits::Integer)

Contructor for an empty PauliSum on nqubits qubits. Element type defaults for Float64.

PauliSum(pstr::PauliString)

Constructor for a PauliSum on nqubits qubits from a PauliString.

PauliSum(pstrs::Vector{PauliString})

Constructor for a PauliSum on nqubits qubits from a PauliString.

PauliSum(nq::Integer, pstr::PauliString)

Constructor for a PauliSum on nqubits qubits from a PauliString.

PauliSum(CoeffType, nq::Int)

Contructor for an empty PauliSum on nqubits qubits. The type of the coefficients can be provided.

*(pstr::PauliString, c::Number)

Multiply a PauliString by a scalar c. Returns a new PauliString.

*(pstr1::PauliString, pstr2::PauliString)

Perform a Pauli product of two PauliStrings.

*(psum::PauliSum, c::Number)

Multiply a PauliSum by a scalar c. This copies the PauliSum.

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

Perform a Pauli product of a PauliString with a PauliSum. Returns a PauliSum with complex coefficients.

*(psum1::PauliSum, psum2::PauliSum)

Perform a Pauli product of two PauliSums. Returns a PauliSum with complex coefficients.

+(psum1::PauliSum, psum2::PauliSum)

Addition of two PauliSums. Returns a PauliSum.

+(pstr1::PauliString, pstr2::PauliString)

Addition of two PauliStrings. Returns a PauliSum.

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

Addition of a PauliString to a PauliSum. Returns a PauliSum.

+(psum::PauliSum, c::Number)
+(c::Number, psum::PauliSum)

Addition of c * Identity to a PauliSum. This copies the PauliSum.

-(pstr1::PauliString, pstr2::PauliString)

Subtract two PauliStrings. Returns a PauliSum.

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

Subtract a PauliString from a PauliSum or vice versa. Returns a PauliSum.

-(psum1::PauliSum, psum2::PauliSum)

Subtract two PauliSums. Returns a PauliSum.

/(pstr::PauliString, c::Number)

Divide a PauliString by a scalar c. Returns a new PauliString.

/(psum::PauliSum, c::Number)

Divide a PauliSum by a scalar c. This copies the PauliSum.

==(psum1::PauliSum, psum2::PauliSum)

Equality check for PauliSum.

≈(psum1::PauliSum, psum2::PauliSum)

Approximate equality check for PauliSum. Simply calls isapprox() on the coefficients of the contained Pauli strings.

Copy a PauliSum by copying its terms field.

empty!(psum::PauliSum)

Empty the PauliSum by emptying the dictionary on the terms fields.

Iterator for PauliSum returns an iterator over (pstr, coeff) pairs`.

length(psum::PauliSum)

Number of terms in the PauliSum.

similar(psum::PauliSum)

Create a new PauliSum with the same number of qubits and coefficient type as psum. Calls sizehint!() with length(psum) on the dictionary of the new PauliSum.

sizehint!(psum::PauliSum, n)

Hint to the PauliSum to reserve space for n terms.

Checks whether the number of qubits nqubits is the same between in some collection.

add!(psum::PauliSum, pauli::Symbol, qind::Integer, coeff=1.0)
add!(psum::PauliSum, paulis::Vector{Symbol}, qinds::Vector{Integer}, coeff=1.0)

Add a Pauli string to a PauliSum psum. Changes psum in-place. Provide the Pauli string as a Symbol (:I, :X, :Y, :Z) or Vector{Symbol}. Provide the index or indices for those symbols as qind or qinds. The coefficient of the Pauli string in the Pauli sum defaults to 1.0.

add!(psum::PauliSum, pstr::PauliString)

Add a PauliString pstr to a PauliSum psum. Changes psum in-place. psum and pstr need to be defined on the same number of qubits and have the same coefficient type.

add!(psum1::PauliSum, psum2::PauliSum)

Add two PauliSums psum1 and psum2. Changes psum1 in-place. psum1 and psum2 need to be defined on the same number of qubits and have the same coefficient type.

add!(psum::PauliSum{Integer, CoeffType}, pstr::Integer, coeff::CoeffType)

Add a Pauli string pstr with coefficient coeff to a PauliSum psum. This changes psum in-place. pstr needs to have the same type as paulitype(psum), and coeff needs to have the same type as coefftype(psum).

coefficients(psum::PauliSum)

Returns an iterator over the coefficients of a PauliSum. Call topaulistrings to receive entries as PauliStrings.

coefftype(pstr::PauliString)

Get the coefficient type of a PauliString.

coefftype(psum::PauliSum)

Get the coefficient type of a PauliSum.

getcoeff(psum::PauliSum, pstr::Vector{Symbol}, qinds::Vector{Int})

Get the coefficient of a Pauli string in a PauliSum by providing the Pauli string pstr as a vector of Symbols acting on qubits qinds. This is consistent with how Pauli strings can be added to a PauliSum via add!(). Defaults to 0 if the Pauli string is not in the PauliSum.

getcoeff(psum::PauliSum, pauli::Symbol, qind::Integer)

Get the coefficient of a Pauli string in a PauliSum by providing the Pauli string as a Symbol acting on qubit qind. This is consistent with how Pauli strings can be added to a PauliSum via add!(). Defaults to 0 if the Pauli string is not in the PauliSum.

getcoeff(psum::PauliSum, pstr::Vector{Symbol})

Get the coefficient of a Pauli string in a PauliSum by providing the Pauli string pstr as a vector of Symbols acting on all qubits. This is consistent with how Pauli strings can be added to a PauliSum via add!(). Defaults to 0 if the Pauli string is not in the PauliSum.

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

Get the coefficient of a PauliString in a PauliSum. Defaults to 0 if the Pauli string is not in the PauliSum. Requires that the integer Pauli string in pstr is the same type as the integer Pauli strings in psum.

getcoeff(psum::PauliSum, pstr::Integer)

Get the coefficient of an integer Pauli string in a PauliSum. Defaults to 0 if the Pauli string is not in the PauliSum. Requires that the integer Pauli string pstr is the same type as the integer Pauli strings in psum.

mult!(psum::PauliSum, c::Number)

Multiply a PauliSum by a scalar c in-place.

norm(psum::PauliSum, L=2)

Calculate the norm of a PauliSum with respect to the L-norm. Calls LinearAlgebra.norm on the coefficients of the PauliSum.

numcoefftype(pstr::PauliString)

Get the type of the numerical coefficient of a PauliString. Will return the type of the output of tonumber(pstr.coeff).

numcoefftype(psum::PauliSum)

Get the type of the numerical coefficient of a PauliSum by calling numcoefftype() on the coefficients. If the PauliSum is empty, an error is thrown because the type cannot be inferred.

paulis(psum::PauliSum)

Returns an iterator over the integer pauli strings of a PauliSum. Call topaulistrings to receive entries as PauliStrings.

paulitype(pstr::PauliString)

Get the Pauli integer type of a PauliString.

paulitype(psum::PauliSum)

Get the Pauli integer type of a PauliSum.

set!(psum::PauliSum{TermType, CoeffType}, pstr::TermType, coeff::CoeffType)

In-place setting the coefficient of a Pauli string in a PauliSum dictionary. The type of the Pauli string needs to be the keytype=TermType of the dictionary, and the coefficient coeff needs to be the valuetype=CoeffType.

tonumber(val::Number)

Trivial function returning a numerical value of a number. Will be overloaded for custom wrapper types like PathProperties.

topaulistrings(psum::PauliSum)

Returns the Pauli strings in a PauliSum and their coefficients as a list of PauliString.

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{TT, CT}: The Pauli string to trace.

Returns

• CT: The trace value of the PauliString.

LinearAlgebra.tr(psum::PauliSum)

Compute the trace of a PauliSum 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^psum.nqubits (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.

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.

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(pstr::PauliString)

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

Arguments

• pstr::PauliString{TT, CT}: The Pauli string to trace.

Returns

• CT: The trace value of the PauliString.

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.

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.