Automatic Optimization

When to use automatic optimization

Automatic optimization should be used when the Hamiltonian has a simple structure, in particular when all offdiagonal elements have the same value (or only a few distinct values).

In that regime, we do not need to repeatedly look up matrix entries, read them, and multiply them in a generic sparse-matrix path. Instead, POLFED can decompose mapping into diagonal and offdiagonal parts and use a lower-memory execution path.

This idea was already introduced in the beginner section Optimized Mapping, but in Quantum Sun (QSun) the optimized mapping was beneficial because of the structure of the previously mentioned Hamiltonian.

Speedup of the XXZ Hamiltonian

For the XXZ model, you only need to construct the Hamiltonian with xxz_hamiltonian from the XXZ model page, then enable optimize_mapping=true; here one should observe a large speedup compared to the generic mul! mapping.

using Polfed
using Polfed.Models: xxz_hamiltonian
using LinearAlgebra

L = 18
Lup = L ÷ 2
Delta = 0.55

mat = xxz_hamiltonian(L, Lup, 1.0, Delta, 0.0; boundary=:periodic, field=0.0, use_sparse=true)
x0 = rand(size(mat, 1)); x0 ./= norm(x0)
howmany = 120
target = 0.0

mapping_auto = MappingConfig(optimize_mapping=true)
vals_auto, vecs_auto, report_auto = polfed(mat, x0, howmany, target; produce_report=true, mapping=mapping_auto)
display_report(report_auto)

Timing behavior should be similar to what you observed in Custom Mapping: for this structured XXZ case, automatic optimization typically reaches nearly the same speedup while keeping the interface simpler.

In other words, optimize_mapping essentially does what is done in the Custom Mapping section.

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