Documentation

Basics

PauliStrings.Operator — Method

Operator(N::Int)

Initialize an empty operator on N qubits

source

Base.length — Method

Base.length(o::Operator)

Number of pauli strings in an operator

Example

julia> A = Operator(4)
julia> A += "X111"
julia> A += "XYZ1"
julia> A += 2, "Y", 4
julia> length(A)
3

source

PauliStrings.eye — Method

eye(N::Int)

Identity operator on N qubits

source

Base.:+ — Method

Base.:+(o::Operator, args::Tuple{Number, Vararg{Any}})
Base.:+(o::Operator, args::Tuple{Vararg{Any}})
Base.:+(o::Operator, term::Tuple{Number, String})
Base.:+(o::Operator, term::String)

Main functions to contruct spin operators. Identical signatures are available for -.

Examples

k-local terms can be added by adding a tuple to the operator. The first element of the tuple is an optional coeficient. The other element are couples (symbol,site) where symbol can be "X", "Y", "Z", "Sx", "Sy", "Sz", "S+", "S-" and site is an integer specifying the site on wich the symbol is acting.

A = Operator(4)
A += 2, "X",1,"X",2
A += 3, "Y",1,"X",2
A += "X",3,"X",4
A += 4,"Z",3
A += 5.2,"X",1,"Y",2,"Z",3

julia> A
(4.0 + 0.0im) 11Z1
(3.0 - 0.0im) YX11
(1.0 + 0.0im) 11XX
(2.0 + 0.0im) XX11
(5.2 - 0.0im) XYZ1

Full strings can also be added:

A = Operator(4)
A += 2, "1XXY"
A += 2im, "11Z1"

julia> A
(0.0 + 2.0im) 11Z1
(2.0 - 0.0im) 1XXY

source

Base.:+ — Method

+(x, y...)

Addition operator. x+y+z+... calls this function with all arguments, i.e. +(x, y, z, ...).

Examples

julia> 1 + 20 + 4
25

julia> +(1, 20, 4)
25

source

Base.:+ — Method

+(x, y...)

Addition operator. x+y+z+... calls this function with all arguments, i.e. +(x, y, z, ...).

Examples

julia> 1 + 20 + 4
25

julia> +(1, 20, 4)
25

source

Base.:+(o::Operator, args::Tuple{Number, Vararg{Any}})
Base.:+(o::Operator, args::Tuple{Vararg{Any}})
Base.:+(o::Operator, term::Tuple{Number, String})
Base.:+(o::Operator, term::String)

Main functions to contruct spin operators. Identical signatures are available for -.

Examples

A = Operator(4)
A += 2, "X",1,"X",2
A += 3, "Y",1,"X",2
A += "X",3,"X",4
A += 4,"Z",3
A += 5.2,"X",1,"Y",2,"Z",3

julia> A
(4.0 + 0.0im) 11Z1
(3.0 - 0.0im) YX11
(1.0 + 0.0im) 11XX
(2.0 + 0.0im) XX11
(5.2 - 0.0im) XYZ1

Full strings can also be added:

A = Operator(4)
A += 2, "1XXY"
A += 2im, "11Z1"

julia> A
(0.0 + 2.0im) 11Z1
(2.0 - 0.0im) 1XXY

source

Base.:+ — Method

+(x, y...)

Addition operator. x+y+z+... calls this function with all arguments, i.e. +(x, y, z, ...).

Examples

julia> 1 + 20 + 4
25

julia> +(1, 20, 4)
25

source

Operations

PauliStrings.add — Method

add(o1::Operator, o2::Operator)
Base.:+(o1::Operator, o2::Operator)
Base.:+(o::Operator, a::Number)
Base.:+(a::Number, o::Operator)

Add two operators together or add a number to an operator

Example

A = Operator(4)
A += "XYZ1"
A += 1, "Y", 4
B = Operator(4)
B += 2, "Y", 2, "Y", 4
B += 1, "Z", 3

julia> A
(1.0 - 0.0im) 111Y
(1.0 - 0.0im) XYZ1

julia> B
(1.0 + 0.0im) 11Z1
(2.0 - 0.0im) 1Y1Y

julia> A+B
(1.0 + 0.0im) 11Z1
(2.0 - 0.0im) 1Y1Y
(1.0 - 0.0im) 111Y
(1.0 - 0.0im) XYZ1

julia> A+5
(1.0 - 0.0im) 111Y
(1.0 - 0.0im) XYZ1
(5.0 + 0.0im) 1111

source

Base.:+ — Method

+(x, y...)

Addition operator. x+y+z+... calls this function with all arguments, i.e. +(x, y, z, ...).

Examples

julia> 1 + 20 + 4
25

julia> +(1, 20, 4)
25

source

Base.:+ — Method

+(x, y...)

Addition operator. x+y+z+... calls this function with all arguments, i.e. +(x, y, z, ...).

Examples

julia> 1 + 20 + 4
25

julia> +(1, 20, 4)
25

source

Base.:+ — Method

+(x, y...)

Addition operator. x+y+z+... calls this function with all arguments, i.e. +(x, y, z, ...).

Examples

julia> 1 + 20 + 4
25

julia> +(1, 20, 4)
25

source

Base.:* — Method

Base.:*(o1::Operator, o2::Operator)
Base.:*(o::Operator, a::Number)
Base.:*(a::Number, o::Operator)

Multiply two operators together or an operator with a number

Example

A = Operator(4)
A += "XYZ1"
A += 1, "Y", 4
B = Operator(4)
B += 2, "Y", 2, "Y", 4
B += 1, "Z", 3

julia> A
(1.0 - 0.0im) 111Y
(1.0 - 0.0im) XYZ1


julia> B
(1.0 + 0.0im) 11Z1
(2.0 - 0.0im) 1Y1Y

julia> A*B
(2.0 - 0.0im) X1ZY
(1.0 - 0.0im) 11ZY
(2.0 - 0.0im) 1Y11
(1.0 - 0.0im) XY11

julia> A*5
(5.0 - 0.0im) 111Y
(5.0 - 0.0im) XYZ1

source

Base.:* — Method

*(x, y...)

Multiplication operator. x*y*z*... calls this function with all arguments, i.e. *(x, y, z, ...).

Examples

julia> 2 * 7 * 8
112

julia> *(2, 7, 8)
112

source

Base.:* — Method

*(x, y...)

Multiplication operator. x*y*z*... calls this function with all arguments, i.e. *(x, y, z, ...).

Examples

julia> 2 * 7 * 8
112

julia> *(2, 7, 8)
112

source

Base.:- — Method

Base.:-(o::Operator)
Base.:-(o1::Operator, o2::Operator)
Base.:-(o::Operator, a::Real)
Base.:-(a::Real, o::Operator)

Subtraction between operators and numbers

source

Base.:- — Method

-(x, y)

Subtraction operator.

Examples

julia> 2 - 3
-1

julia> -(2, 4.5)
-2.5

source

Base.:- — Method

-(x, y)

Subtraction operator.

Examples

julia> 2 - 3
-1

julia> -(2, 4.5)
-2.5

source

Base.:- — Method

-(x, y)

Subtraction operator.

Examples

julia> 2 - 3
-1

julia> -(2, 4.5)
-2.5

source

PauliStrings.com — Method

com(o1::Operator, o2::Operator; epsilon::Real=0, maxlength::Int=1000)

Commutator of two operators

Example

julia> A = Operator(4)
julia> A += "X111"
julia> B = Operator(4)
julia> B += "Z111"
julia> B += "XYZ1"
julia> com(A,B)
(0.0 - 2.0im) Y111

source

PauliStrings.trace — Method

trace(o::Operator)

Trace of an operator

Example

julia> A = Operator(4)
julia> A += 2,"1111"
julia> A += 3,"XYZ1"
julia> trace(A)
32.0 + 0.0im

source

PauliStrings.opnorm — Method

opnorm(o::Operator)

Frobenius norm

Example

julia> A = Operator(4)
julia> A += 2,"X",2
julia> A += 1,"Z",1,"Z",3
julia> opnorm(A)
8.94427190999916

source

PauliStrings.dagger — Method

dagger(o::Operator)

Conjugate transpose

Example

A = Operator(3)
A += 1im,"X",2
A += 1,"Z",1,"Z",3

julia> A

(1.0 + 0.0im) Z1Z
(0.0 + 1.0im) 1X1


julia> dagger(A)
(1.0 - 0.0im) Z1Z
(0.0 - 1.0im) 1X1

source

PauliStrings.ptrace — Method

ptrace(o::Operator, keep::Vector{Int})

Partial trace.

keep is list of qubits indices to keep starting at 1 note that this still returns an operator of size N and doesnt permute the qubits this only gets rid of Pauli strings that have no support on keep and add their coeficient*2^N to the identity string

Example

A = Operator(5)
A += "XY1XZ"
A += "XY11Z"

julia> ptrace(A, [3,4])
(1.0 - 0.0im) XY1XZ
(8.0 - 0.0im) 11111

julia> ptrace(A, [1,5])
(1.0 - 0.0im) XY1XZ
(1.0 - 0.0im) XY11Z

source

Random operators

PauliStrings.rand_local1 — Method

rand_local2(N::Int)

Random 1-local operator

source

PauliStrings.rand_local2 — Method

rand_local2(N::Int)

Random 2-local operator

source

Truncation and noise

Base.truncate — Method

truncate(o::Operator, N::Int; keepnorm::Bool = false)

Remove all terms of length > N. Keep all terms of length <= N. i.e remove all M-local terms with M>N

Example

A = Operator(4)
A += "X",1,"X",2
A += "Z",1,"Z",2,"Z",4

julia> A
(1.0 + 0.0im) ZZ1Z
(1.0 + 0.0im) XX11

julia> ps.truncate(A,2)
(1.0 + 0.0im) XX11

source

PauliStrings.trim — Method

trim(o::Operator, N::Int; keepnorm::Bool = false, keep::Operator=Operator(N))

Keep the first N terms with largest coeficients.

keepnorm is set to true to keep the norm of o.

keep is an operator that specify a set of strings that cannot be removed

Example

A = Operator(4)
A += 1,"XXXX"
A += 2,"XX11"
A += 3,"XX1X"
A += 4,"ZZXX"
B = Operator(4)
B += 1,"XX11"
B += 1,"XX1X"

julia> trim(A,2)
(4.0 + 0.0im) ZZXX
(3.0 + 0.0im) XX1X

julia> trim(A,2;keep=B)
(4.0 + 0.0im) ZZXX
(3.0 + 0.0im) XX1X
(2.0 + 0.0im) XX11

source

PauliStrings.prune — Method

 prune(o::Operator, alpha::Real; keepnorm::Bool = false)

Keep terms with probability 1-exp(-alpha*abs(c)) where c is the weight of the term

source

PauliStrings.cutoff — Method

cutoff(o::Operator, epsilon::Real; keepnorm::Bool = false)

Remove all terms with weight < epsilon

Example

A = Operator(4)
A += 1,"XXXX"
A += 2,"XX11"
A += 3,"XX1X"
A += 4,"ZZXX"

julia> cutoff(A, 2.5)
(3.0 + 0.0im) XX1X
(4.0 + 0.0im) ZZXX

source

PauliStrings.add_noise — Method

add_noise(o::Operator, g::Real)

Add depolarizing noise that make the long string decays. g is the noise amplitude.

Example

A = add_noise(A, 0.1)

Reference

https://arxiv.org/pdf/2407.12768

source

Algorithms

PauliStrings.lanczos — Method

lanczos(H::Operator, O::Operator, steps::Int, nterms::Int; keepnorm=true, maxlength=1000, localop=false)

Computer the first steps lanczos coeficients for Hamiltonian H and initial operator O

At every step, the operator is trimed with PauliStrings.trim and only nterms are kept.

If H and O are 1D-translation-invariant, it is possible to provide a single local term of O and set localop=true

Using maxlength speeds up the commutator by only keeping terms of length <= maxlength

source

PauliStrings.rk4 — Method

rk4(H::Operator, O::Operator, dt::Real; hbar::Real=1, heisenberg=false, M=2^20, keep::Operator=Operator(N))

Single step of Runge–Kutta-4 with time independant Hamiltonian. Returns O(t+dt). Set heisenberg=true for evolving an observable in the heisenberg picture. If heisenberg=false then it is assumed that O is a density matrix.

source

PauliStrings.rk4 — Method

rk4(H::Function, O::Operator, dt::Real, t::Real; hbar::Real=1, heisenberg=false)

Single step of Runge–Kutta-4 with time dependant Hamiltonian. H is a function that takes a number (time) and returns an operator.

source

Tools

PauliStrings.compress — Method

compress(o::Operator)

Accumulate repeated terms and remove terms with a coeficient smaller than 1e-20

source

PauliStrings.op_to_strings — Method

op_to_strings(o::Operator)

takes an operator, return (coefs, strings) where coefs is a list of numbers and strings is a list of pauli string coefs[i] multiply strings[i]

source

PauliStrings.shift_left — Method

shift_left(O::Operator)

Shift evey string left so they start on site 1. This usefull for using translation symmetry in 1D systems

Example

A = Operator(4)
A += "XYZ1"
A += "11X1"
A += "1XZY"
A += "ZZ11"

julia> shift_left(A)
(1.0 - 0.0im) XZY1
(1.0 + 0.0im) X111
(1.0 - 0.0im) XYZ1
(1.0 + 0.0im) ZZ11

source