# Replication protocol: quantum tests on the 108-bead (4, 27) network

This is a blind replication package. It defines a model and four tests exactly, and asks for
specific numbers. It deliberately contains **no results and no predictions**. Run it, report
the numbers, and compare against the answer key held separately.

You may run the attached script `replicate_mala_tests.py` (Python 3, NumPy only), or
implement the model independently from this description. Independent implementation is the
stronger check.

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## 1. The model

**Sites.** N = 108 sites, indexed k = 0 … 107, arranged in a closed ring (site 107 is
adjacent to site 0). Sites are referred to as "beads"; bead number = k + 1.

**Hamiltonian.** A single particle hops between sites. The Hamiltonian is a real symmetric
108 × 108 matrix H built from up to three "hopping rules", each with a strength t ≥ 0:

- **thread** (step 1): connects k to k + 1 and k − 1 (mod 108)
- **squares** (step 27): connects k to k + 27 and k − 27 (mod 108)
- **rings** (step 4): connects k to k + 4 and k − 4 (mod 108)

For each active rule with strength t and step s: `H[k, (k+s) mod 108] -= t` and
`H[k, (k−s) mod 108] -= t` for every k. All diagonal entries are 0. (Note: if two rules map
to the same neighbour, contributions add.)

**Dynamics.** The state ψ is a complex vector of length 108, normalised to 1. It evolves by
the Schrödinger equation with ħ = 1:

    i dψ/dt = H ψ        ⇒        ψ(t) = exp(−i H t) ψ(0)

Use exact diagonalisation (H is real symmetric) or any integrator accurate to better than
1e-6 over the times below.

**Probability** at site k is |ψ_k|².

**Initial states used.** A "delta" at site k means ψ_k = 1, all others 0, then normalise.

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## 2. The four tests

Report every quantity marked ▶ to at least 4 significant figures.

### Test 1 — Relative phase between two places

- Hopping: squares only, t = 1 (thread 0, rings 0).
- Initial state: ψ_0 = 1, ψ_54 = e^{iφ}, all others 0, then normalise.
- For each φ ∈ {0°, 45°, 90°, 135°, 180°}, evolve from t = 0 to t = 4.
- ▶ Report **max over t of P_empty(t)**, where P_empty(t) = |ψ_27(t)|² + |ψ_81(t)|².
- ▶ Also report, for φ = 0°, the time of the first maximum of P_empty and its value.

### Test 2 — Factorisation of the network

- Hopping: squares t = 1 AND rings t = 1 (thread 0).
- Initial state: delta at site 0.
- Compute the return probability F(t) = |⟨ψ(0)|ψ(t)⟩|² for t from 0 to 4π (use ≥ 1500
  evenly spaced samples).
- Separately, define two smaller systems:
  - a 4-site ring with nearest-neighbour hopping t = 1 (H4[k,(k±1) mod 4] = −1),
    delta start at site 0, return probability F4(t);
  - a 27-site ring with nearest-neighbour hopping t = 1, delta start at site 0,
    return probability F27(t).
- ▶ Report **max over t of |F(t) − F4(t)·F27(t)|**.
- ▶ Report F(t) at t = π, 2π, 3π, 4π.

### Test 3 — Quantum walk versus classical walk

- Hopping: thread only, t = 1.
- Quantum: delta at site 0; evolve to t = 25. Define the spread
  σ_q(t) = sqrt( Σ_k |ψ_k(t)|² · d_k² ), where d_k = min(k, 108 − k) is the ring distance
  from site 0.
- Classical: a continuous-time random walker on the same ring, starting at site 0, hopping
  to each neighbour at rate 1: dp/dt = L p with L[k,(k±1)] = +1, L[k,k] = −2. Define
  σ_c(t) the same way using p_k(t).
- ▶ Report σ_q and σ_c at t = 5, 10, 20.
- ▶ State whether σ_q grows linearly in t or as a square root, and the same for σ_c,
  and give the fitted constant (σ ≈ a·t or σ ≈ a·√t).

### Test 4 — Other bead counts

For each N ∈ {27, 54, 108, 1008}:

- List every way to write N = p × q with 2 ≤ p ≤ q and gcd(p, q) = 1 ("coprime splits").
- For each split, build an N-site ring with a single hopping rule of step q and strength 1
  (this joins the sites into p-cycles).
- Initial state: ψ_a = 1 at a = 0 and ψ_b = 1 at b = (⌊p/2⌋ · q) mod N, normalised. If
  b = a (which happens when p = 1 — not applicable here — or by coincidence), use the delta
  at a alone.
- Evolve from t = 0 to t = 12 with ≥ 1000 samples. Define P_start(t) = |ψ_a|² + |ψ_b|²
  (just |ψ_a|² if b = a).
- ▶ Report, per split: **min over t of P_start(t)**.
- ▶ Report, per split: **the number of distinct oscillation frequencies present in
  P_start(t)** that carry more than 1% of the total oscillating weight. (Any reasonable
  method is acceptable — e.g. from eigenvalue differences of H weighted by the initial-state
  overlaps, or a spectral analysis of P_start(t) with leakage handled. State your method.)
- If N has no coprime split, say so.

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## 3. What to hand back

A table or JSON with every ▶ quantity, plus one sentence per test stating what the numbers
show, written before looking at any answer key. Please also state: the software used, the
integration method, and the time step or sample count.

## 4. Things that must NOT be in the report

- No numbers taken from any prior discussion of this model.
- No geometry: the 3D shape of the "thread" (torus, lotus, etc.) does not appear in the
  model and must not be used. Only the hopping rules above define H.