Practical challenge of Ising interaction

[nzy1997]

Practical challenge: Simulate the 2D kicked Ising dynamics in IBM's quantum experiment this page by classical computation. Can you simulate the dynamics for $\pi/2-\epsilon$ pulse of Ising interaction i.e. away from the maximally entangling $CZ$ gate?

[nzy1997]

@zzh-cycling Can you figure out what does this challenge mean?

[zzh-cycling]

Benchmark for Ising model:

$$ H = -\sum_{i, j} Z_i Z_j - \sum_i X_i $$

where initial state $| 0\rangle ^{\otimes N}$, t = collect(0.0:0.05:1.5), magnetization $M:=(\sum Z_i)/N, N=8$

1D PBC ED:

2D OBC ED:

where initial state $| 0\rangle ^{\otimes N^2}$, t = collect(0.0:0.05:10.0), $H = -\sum_{i, j} Z_i Z_j - 0.5 \sum_i X_i $, magnetization $M:=(\sum Z_i)/N^2, N=3 \times 3$. In such parameters, matter tends to keep ferromagnetic phases. ($M \sim 1$). If we set $J=0.5, h=1.0$, thus it evolves to paramagnetic phases (where $M \sim 0$)

[nzy1997]

Parameter setting: N=8, t_min = 0, t_max = 1.5, t_step = 0.05

[nzy1997]

The results with ItensorNetworks.jl. The parameters are the same with above.

1D OBC ED:

2D OBC ED:

Test code:

using BPGauge
using Test
using ITensorNetworks
using ITensors
using BPGauge: generate_zero_itensor

@testset "1D obc Ising" begin
    unit = 3.0
    r = 3.5
    bd = 1
    t_max = 1.5
    dt = 0.01
    maxdim = 10
    lattice_size = (8,1)


    lattice = SquareLattice()
    atoms = map(x -> unit .* x, generate_sites(lattice, lattice_size...));
    tn,ind_vec,g = create_zero_state(atoms,r, bd)
    expc_vec = Float64[]
    for t in dt:dt:t_max
        tn = BPGauge.apply_ising_hamiltonian(dt,ind_vec,g,tn;maxdim)
        expc = ITensorNetworks.expect(tn, "Sz"; alg="bp")
        push!(expc_vec,real(sum(expc)/4))
    end
    expc = ITensorNetworks.expect(tn, "Sz"; alg="bp")
    @show real(expc).> 0.0
    @show expc

    # using CairoMakie
    # fig = Figure()
    # ax = Axis(fig[1,1])
    # lines!(ax, collect(dt:dt:t_max),expc_vec)
    # fig
end

@testset "2D obc Ising" begin
    unit = 3.0
    r = 3.5
    bd = 1
    t_max = 10.0
    dt = 0.05
    maxdim = 20
    lattice_size = (3,3)


    lattice = SquareLattice()
    atoms = map(x -> unit .* x, generate_sites(lattice, lattice_size...));
    tn,ind_vec,g = create_zero_state(atoms,r, bd)
    expc_vec = Float64[]
    for t in dt:dt:t_max
        tn = BPGauge.apply_ising_hamiltonian(dt,ind_vec,g,tn;maxdim,omega = 0.5)
        expc = ITensorNetworks.expect(tn, "Sz"; alg="bp")
        push!(expc_vec,real(sum(expc)/4.5))
    end
    expc = ITensorNetworks.expect(tn, "Sz"; alg="bp")
    @show real(expc).> 0.0
    @show expc

    # using CairoMakie
    # fig = Figure()
    # ax = Axis(fig[1,1])
    # lines!(ax, collect(dt:dt:t_max),expc_vec)
    # fig
end