Add DenseLayout trial to SabreLayout (Qiskit#12453)

Building on the work done in Qiskit#10829, Qiskit#10721, and Qiskit#12104 this commit adds
a new trial to all runs of `SabreLayout` that runs the dense layout
pass. In general the sabre layout algorithm starts from a random layout
and then runs a routing algorithm to permute that layout virtually where
swaps would be inserted to select a layout that would result in fewer
swaps. As was discussed in Qiskit#10721 and Qiskit#10829 that random starting point
is often not ideal especially for larger targets where the distance
between qubits can be quite far. Especially when the circuit qubit count
is low relative to the target's qubit count this can result it poor
layouts as the distance between the qubits is too large. In qiskit we
have an existing pass, `DenseLayout`, which tries to find the most
densely connected n qubit subgraph of a connectivity graph. This
algorithm necessarily will select a starting layout where the qubits are
near each other and for those large backends where the random starting
layout doesn't work well this can improve the output quality.

As the overhead of `DenseLayout` is quite low and the core algorithm is
written in rust already this commit adds a default trial that uses
DenseLayout as a starting point on top of the random trials (and any
input starting points). For example if the user specifies to run
SabreLayout with 20 layout trials this will run 20 random trials and
one trial with `DenseLayout` as the starting point. This is all done
directly in the sabre layout rust code for efficiency. The other
difference between the standalone `DenseLayout` pass is that in the
standalone pass a sparse matrix is built and a reverse Cuthill-McKee
permutation is run on the densest subgraph qubits to pick the final
layout. This permutation is skipped because in the case of Sabre's
use of dense layout we're relying on the sabre algorithm to perform
the permutation.

Depends on: Qiskit#12104

crates/accelerate/src/sabre/layout.rs

@@ -15,6 +15,7 @@ use pyo3::prelude::*;
 use pyo3::Python;
 
 use hashbrown::HashSet;
+use ndarray::prelude::*;
 use numpy::{IntoPyArray, PyArray, PyReadonlyArray2};
 use rand::prelude::*;
 use rand_pcg::Pcg64Mcg;
@@ -29,6 +30,8 @@ use super::sabre_dag::SabreDAG;
 use super::swap_map::SwapMap;
 use super::{Heuristic, NodeBlockResults, SabreResult};
 
+use crate::dense_layout::best_subset_inner;
+
 #[pyfunction]
 #[pyo3(signature = (dag, neighbor_table, distance_matrix, heuristic, max_iterations, num_swap_trials, num_random_trials, seed=None, partial_layouts=vec![]))]
 pub fn sabre_layout_and_routing(
@@ -52,6 +55,12 @@ pub fn sabre_layout_and_routing(
     let mut starting_layouts: Vec<Vec<Option<u32>>> =
         (0..num_random_trials).map(|_| vec![]).collect();
     starting_layouts.append(&mut partial_layouts);
+    // Run a dense layout trial
+    starting_layouts.push(compute_dense_starting_layout(
+        dag.num_qubits,
+        &target,
+        run_in_parallel,
+    ));
     let outer_rng = match seed {
         Some(seed) => Pcg64Mcg::seed_from_u64(seed),
         None => Pcg64Mcg::from_entropy(),
@@ -208,3 +217,26 @@ fn layout_trial(
         .collect();
     (initial_layout, final_permutation, sabre_result)
 }
+
+fn compute_dense_starting_layout(
+    num_qubits: usize,
+    target: &RoutingTargetView,
+    run_in_parallel: bool,
+) -> Vec<Option<u32>> {
+    let mut adj_matrix = target.distance.to_owned();
+    if run_in_parallel {
+        adj_matrix.par_mapv_inplace(|x| if x == 1. { 1. } else { 0. });
+    } else {
+        adj_matrix.mapv_inplace(|x| if x == 1. { 1. } else { 0. });
+    }
+    let [_rows, _cols, map] = best_subset_inner(
+        num_qubits,
+        adj_matrix.view(),
+        0,
+        0,
+        false,
+        true,
+        aview2(&[[0.]]),
+    );
+    map.into_iter().map(|x| Some(x as u32)).collect()
+}