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//! Simulated bifurcation — the Toshiba Ising-machine algorithm line (Goto et al.; ballistic bSB
//! and discrete dSB per the 2021 Science Advances formulation), as portable deterministic Rust.
//!
//! Classical mechanics rather than sampling: each spin is a particle x_i in [-1, 1] with momentum
//! y_i under a bifurcating potential ramped by a(t): 0 -> a0, coupled through the Ising J. The
//! symplectic (momentum-first) update with perfectly inelastic walls at |x| = 1:
//! y <- y + { -(a0 - a(t)) x + c0 * force } dt ; x <- x + a0 y dt ;
//! wall: |x| > 1 => x = sgn(x), y = 0.
//! bSB uses force_i = sum_j J_ij x_j; dSB uses force_i = sum_j J_ij sgn(x_j) (the discretisation
//! is what suppresses analog error). Read out sgn(x) EVERY step and keep the best-so-far — the
//! trajectory is ergodic and the final state is not the best visited. c0 = 0.5 / (J_rms sqrt(N)).
use crate::graph::Graph;
use crate::rng::Pcg;
#[derive(Clone, Copy, PartialEq, Eq)]
pub enum Variant {
Ballistic,
Discrete,
}
fn sgn(x: f64) -> f64 {
if x < 0.0 {
-1.0
} else {
1.0 // sgn(0) = +1, fixed for determinism
}
}
fn spins_of(x: &[f64]) -> Vec<i8> {
x.iter().map(|&v| if v < 0.0 { -1 } else { 1 }).collect()
}
/// One simulated-bifurcation run. Returns (best spins, best energy).
pub fn run(g: &Graph, variant: Variant, n_steps: usize, dt: f64, seed: u64) -> (Vec<i8>, f64) {
let n = g.n;
let a0 = 1.0f64;
// c0 = 0.5 / (J_rms * sqrt(N)), J_rms over ordered node pairs (CSR stores both directions)
let sum_j2: f64 = g.w.iter().map(|&w| w * w).sum();
let j_rms = (sum_j2 / (n as f64 * (n as f64 - 1.0))).sqrt();
let c0 = if j_rms > 0.0 { 0.5 / (j_rms * (n as f64).sqrt()) } else { 0.5 };
let mut rng = Pcg::new(seed, 0x5B);
// small random x-init in addition to y: with x = 0 exactly, symmetric graphs drive all
// particles identically, every site hits the wall on the same step, and the wall's y = 0
// reset erases the y-asymmetry — a synchronized trap (measured on K8 antiferromagnetic:
// dSB converged to the WORST state). Breaking spatial symmetry at init removes it.
let mut x: Vec<f64> = (0..n).map(|_| (rng.f64() - 0.5) * 0.02).collect();
let mut y: Vec<f64> = (0..n).map(|_| (rng.f64() - 0.5) * 0.2).collect();
let mut best = spins_of(&x);
let mut best_e = g.energy(&best);
let mut force = vec![0.0f64; n];
for k in 0..n_steps {
let a_t = a0 * k as f64 / n_steps as f64;
// force from the PRE-step x (separate pass = full-copy semantics)
for i in 0..n {
let mut s = 0.0;
for e in g.offset[i]..g.offset[i + 1] {
let xj = x[g.nbr[e] as usize];
s += g.w[e] * if variant == Variant::Discrete { sgn(xj) } else { xj };
}
force[i] = s + g.h[i];
}
for i in 0..n {
y[i] += (-(a0 - a_t) * x[i] + c0 * force[i]) * dt;
x[i] += a0 * y[i] * dt;
if x[i].abs() > 1.0 {
x[i] = sgn(x[i]);
y[i] = 0.0;
}
}
let s = spins_of(&x);
let e = g.energy(&s);
if e < best_e {
best_e = e;
best = s;
}
}
(best, best_e)
}
/// Multi-restart wrapper: `restarts` seeded runs, best result kept.
pub fn run_restarts(
g: &Graph,
variant: Variant,
n_steps: usize,
dt: f64,
seed: u64,
restarts: usize,
) -> (Vec<i8>, f64) {
let mut best: Option<(Vec<i8>, f64)> = None;
for r in 0..restarts {
let (s, e) = run(g, variant, n_steps, dt, seed ^ (r as u64).wrapping_mul(0x9E3779B97F4A7C15));
if best.as_ref().is_none_or(|(_, be)| e < *be) {
best = Some((s, e));
}
}
best.unwrap()
}
#[cfg(test)]
mod tests {
use super::*;
use crate::graph::GraphBuilder;
fn exact_ground(g: &Graph) -> f64 {
let n = g.n;
assert!(n <= 20);
let mut e0 = f64::MAX;
let mut s = vec![-1i8; n];
for m in 0..(1u32 << n) {
for b in 0..n {
s[b] = if m >> b & 1 == 1 { 1 } else { -1 };
}
let e = g.energy(&s);
if e < e0 {
e0 = e;
}
}
e0
}
/// Wall rule: overshoot lands exactly on the wall with zero momentum.
#[test]
fn wall_rule() {
let mut x = 1.7f64;
let mut y = 0.9f64;
if x.abs() > 1.0 {
x = sgn(x);
y = 0.0;
}
assert_eq!(x, 1.0);
assert_eq!(y, 0.0);
}
/// Closed-form graphs: K8 (all antiferromagnetic), C7 ring, Petersen — both variants must
/// return the exhaustively-enumerated ground-state energy.
#[test]
fn closed_form_graphs_reach_ground_state() {
let mut graphs: Vec<Graph> = Vec::new();
let mut gb = GraphBuilder::new(8);
for i in 0..8 {
for j in (i + 1)..8 {
gb.couple(i, j, -1.0);
}
}
graphs.push(gb.build());
let mut gb = GraphBuilder::new(7);
for i in 0..7 {
gb.couple(i, (i + 1) % 7, -1.0);
}
graphs.push(gb.build());
let mut gb = GraphBuilder::new(10);
let outer = [(0, 1), (1, 2), (2, 3), (3, 4), (4, 0)];
let inner = [(5, 7), (7, 9), (9, 6), (6, 8), (8, 5)];
let spokes = [(0, 5), (1, 6), (2, 7), (3, 8), (4, 9)];
for &(a, b) in outer.iter().chain(&inner).chain(&spokes) {
gb.couple(a, b, -1.0);
}
graphs.push(gb.build());
for (gi, g) in graphs.iter().enumerate() {
let e0 = exact_ground(g);
for (vi, v) in [Variant::Ballistic, Variant::Discrete].into_iter().enumerate() {
let (_, e) = run_restarts(g, v, 2000, 1.0, 0x5B00 + gi as u64, 10);
assert!(
(e - e0).abs() < 1e-9,
"graph {gi} variant {vi}: found {e} vs exact ground {e0}"
);
}
}
}
/// Random Gaussian-J instances at N = 16: near-perfect ground-state hit rates with frozen
/// seeds (bSB >= 18/20, dSB >= 19/20 per the published tuning).
#[test]
fn random_instances_hit_rate() {
let mut hits_b = 0;
let mut hits_d = 0;
for inst in 0..20u64 {
let mut rng = Pcg::new(0x6A55 ^ inst, 7);
let mut gb = GraphBuilder::new(16);
for i in 0..16 {
for j in (i + 1)..16 {
let a = rng.f64().max(1e-12);
let b = rng.f64();
let gauss = (-2.0 * a.ln()).sqrt() * (std::f64::consts::TAU * b).cos();
gb.couple(i, j, gauss);
}
}
let g = gb.build();
let e0 = exact_ground(&g);
let (_, eb) = run_restarts(&g, Variant::Ballistic, 2000, 1.0, 0xB0 ^ inst, 10);
let (_, ed) = run_restarts(&g, Variant::Discrete, 2000, 1.0, 0xD0 ^ inst, 10);
if (eb - e0).abs() < 1e-9 {
hits_b += 1;
}
if (ed - e0).abs() < 1e-9 {
hits_d += 1;
}
}
assert!(hits_b >= 18, "bSB ground-state hits {hits_b}/20");
assert!(hits_d >= 19, "dSB ground-state hits {hits_d}/20");
}
}