PauliStrings.jl

PauliStrings.jl is a Julia package for many-body quantum mechanics with Pauli string represented as binary integers (as in https://journals.aps.org/pra/abstract/10.1103/PhysRevA.68.042318). It is particularly adapted for running Lanczos, time evolving noisy systems and simulating spin systems on arbitrary graphs.

Documentation

Installation

using Pkg; Pkg.add(PauliStrings) or ] add PauliStrings

Initializing an operator

Import the library and initialize a operator of 4 qubits

using PauliStrings
import PauliStrings as ps
H = ps.Operator(4)

Add a Pauli strings to the operator

H += "XYZ1"
H += "1YZY"

julia> H
(1.0 - 0.0im) XYZ1
(1.0 - 0.0im) 1YZY

Add a Pauli string with a coeficient

H += -1.2,"XXXZ" #coeficient can be complex

Add a 2-qubit string coupling qubits i and j with X and Y:

H += 2, "X", i, "Y", j # with a coeficient=2
H += "X", i, "Y", j # with a coeficient=1

Add a 1-qubit string:

H += 2, "Z", i # with a coeficient=2
H += "Z", i # with a coeficient=1
H += "S+", i

Supported sites operators are X, Y, Z, Sx$=X/2$, Sy$=Y/2$, Sz$=Z/2$, S+$=(X+iY)/2$, S-$=(X-iY)/2$.

Basic Algebra

The Operator type supports the +,-,* operators with other Operators and Numbers:

H3 = H1*H2
H3 = H1+H2
H3 = H1-H2
H3 = H1+2 # adding a scalar is equivalent to adding the unit times the scalar
H = 5*H # multiply operator by a scalar

Trace : ps.trace(H)

Frobenius norm : ps.opnorm(H)

Conjugate transpose : ps.dagger(H)

Number of terms: length(H)

Commutator: ps.com(H1, H2). This is much faster than H1*H2-H2*H1

Print and export

print shows a list of terms with coeficients e.g :

julia> println(H)
(10.0 - 0.0im) 1ZZ
(5.0 - 0.0im) 1Z1
(15.0 + 0.0im) XYZ
(5.0 + 0.0im) 1YY

Export a list of strings with coeficients:

coefs, strings = ps.op_to_strings(H)

Truncate, Cutoff, Trim, Noise

ps.truncate(H,M) removes Pauli strings longer than M (returns a new Operator) ps.cutoff(H,c) removes Pauli strings with coeficient smaller than c in absolute value (returns a new Operator) ps.trim(H,N) keeps the first N trings with higest weight (returns a new Operator) ps.prune(H,alpha) keeps terms with probability 1-exp(-alpha*abs(c)) (returns a new Operator)

ps.add_noise(H,g) adds depolarizing noise that make each strings decay like $e^{gw}$ where $w$ is the lenght of the string. This is usefull when used with trim to keep the number of strings manageable during time evolution.

Time evolution

ps.rk4(H, O, dt; hbar=1, heisenberg=false) performs a step of Runge Kutta and returns the new updated O(t+dt)

H can be an Operator, or a function that takes a time and return an Operator. In case H is a function, a time also needs to be passed to rk4(H, O, dt, t). O is an Observable or a density matrix to time evolve. If evolving an observable in the heisenberg picture, set heisenberg=true.

An example is in time_evolve_example.jl. The following will time evolve O in the Heisenberg picture. At each step, we add depolarizing noise and trim the operator to keep the number of strings manageable

function evolve(H, O, M, times, noise)
    dt = times[2]-times[1]
    for t in times
        O = ps.rk4(H, O, dt; heisenberg=true, M=M) #preform one step of rk4, keep only M strings
        O = ps.add_noise(O, noise*dt) #add depolarizingn noise
        O = ps.trim(O, M) # keep the M strings with the largest weight
    end
    return O
end

Time evolution of the spin correlation function $\textup{Tr}(Z_1(0)Z_1(t))$ in the chaotic spin chain. Check time_evolve_example.jl to reproduce the plot.

Lanczos

Compute lanczos coeficients

bs = ps.lanczos(H, O, steps, nterms)

H : Hamiltonian

O : starting operator

nterms : maximum number of terms in the operator. Used by trim at every step

Results for X in XX from https://journals.aps.org/prx/pdf/10.1103/PhysRevX.9.041017 :