Sherrignton-Kirkpatrick Model

The cost function of the SK model is defined as

\[\begin{align*} \hat H_P = \frac{1}{\sqrt{N}}\sum_{i<j\leq N} J_{ij} \hat{Z}_i \hat{Z}_j, \end{align*}\]

where the couplings $J_{ij}$ are i.i.d. standard Gaussian variables, i.e. with zero mean $\left\langle J_{ij} \right\rangle = 0$ and variance $\left\langle J_{ij}^2 \right\rangle = J^2$.

We can set this model up as follows:

using QAOA, LinearAlgebra
import Random, Distributions

N = 4
σ2 = 1.0

Random.seed!(1)
J = rand(Distributions.Normal(0, σ2), N, N) ./ sqrt(N) 
J[diagind(J)] .= 0.0
J = UpperTriangular(J)
J = J + transpose(J)

We have two options to get the corresponding Problem from QAOA.jl. We can pass the coupling matrix $J$ directly:

p = 2
SK_problem = QAOA.Problem(p, zeros(N), J)

or we can use a predefined wrapper function that constructs $J$ from the above parameters and directly returns a Problem:

SK_problem = QAOA.sherrington_kirkpatrick(N, σ2, num_layers=p, seed=1)

Given SK_problem, we can then call the gradient optimizer:

learning_rate = 0.02
cost, params, probs = QAOA.optimize_parameters(SK_problem, vcat([0.5 for _ in 1:p], [0.5 for _ in 1:p]); learning_rate=learning_rate)

Alternatively, we can use NLsolve.jl:

cost, params, probs = QAOA.optimize_parameters(SK_problem, vcat([0.5 for _ in 1:p], [0.5 for _ in 1:p]), :LN_COBYLA)