QuantumNaturalGradient.jl

This package facilitates the computation of the Quantum Natural Gradient, a critical component in the optimization of quantum variational algorithms, especially during the time evolution process. Below, we detail how to integrate and use this package effectively.

Getting Started

Prerequisites

Before you begin, ensure that you have a working implementation of the function Oks_and_Eks(θ::Vector, sample_nr::Integer). This function is pivotal for the operation of the Quantum Natural Gradient algorithm, as it computes several key quantities based on a given parameter vector θ and a specified number of samples sample_nr.

Function Output Format

The Oks_and_Eks function should now return a dictionary. Required fields:

• :Oks (Matrix): Gradient of <s|ψ> w.r.t. θ, normalized by <s|ψ>. Dimensions: (sample_nr, length(θ)).
• :Eks (Vector): Expectation value of <s|H|ψ>/<s|ψ>.
• :logψs (Vector): Logarithm of <s|ψ> (or zeros(sample_nr) if unused).
• :samples (Vector{Int}): Sampled configurations (or zeros(Int, sample_nr) if unused).
• :weights (Vector{Float}): Importance sampling weights, i.e., ||<s|ψ>||^2 / p(s) (optional).

Example Output

Dict(
    :Oks => Oks,
    :Eks => Eks,
    :logψs => logψs,
    :samples => samples
)

Imaginary Time Evolution

To perform imaginary time evolution, use the following setup:

dt = 0.1  # Time step
eigen_cut = 0.1  # Eigenvalue cutoff for solver
integrator = QNG.Euler(lr=dt)  # Define the integrator with learning rate
solver = QNG.EigenSolver(eigen_cut, verbose=true)  # Eigenvalue solver with verbosity
θ = init()  # Initialize your parameter vector

# Evolve the system
@time loss_value, trained_θ, misc = QNG.evolve(Oks_and_Eks, θ; 
                                                integrator, 
                                                verbosity=2,
                                                solver,
                                                sample_nr,  # Number of samples
                                                maxiter,  # Maximum iterations
                                                callback,  # Callback function for each iteration
                                                )

Obtaining the Natural Gradient Object

For direct access to the Natural Gradient object without running the full evolution, you can initialize it as follows:

ng = NaturalGradient(θ, Oks_and_Eks; sample_nr=100)

This allows for more granular control and inspection of the gradient for advanced use cases.

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Logger Functions

The evolve function accepts a keyword argument logger_funcs, which is a list of functions whose outputs are saved after every iteration. For example:

logger_funcs = []
history_params(; θ) = θ
push!(logger_funcs, history_params)

In this example, the history_params function saves the parameters θ after each optimization step.

Currently supported variables include:

• natural_gradient (The natural gradient object)
• θ (parameters)
• niter (number of current iteration)
• energy (current energy)
• norm_natgrad (norm of the natural gradient)
• norm_θ (norm of the parameters)

Additionally, you can log any parameters output by the Oks_and_Eks function.