Date: 2026-03-23 Tool: Rust-accelerated DLA (scpn_quantum_engine)
| N (qubits) | DLA dimension | max su(2^N) | Fraction | Time (s) | Engine |
|---|---|---|---|---|---|
| 4 | 126 | 255 | 49.4% | 1.66 | Rust |
| 5 | 510 | 1023 | 49.9% | 848 | Rust |
The DLA fraction stabilises at ~50% of su(2^N). The absolute DLA dimension grows exponentially: 126 → 510 → (estimated ~2040 at N=6).
This means:
- The DLA is NOT polynomial in N → system is NOT trivially classically simulable
- The heterogeneous SCPN frequencies break the SU(2) symmetry of the pure XY model
- The g-sim framework (Goh et al. 2025) classifies this as outside the polynomial regime
Caveat: "Not trivially simulable" ≠ "quantum advantage". MPS with moderate bond dimension can still simulate these systems efficiently at small N. The DLA result establishes a necessary condition, not a sufficient one.
| N | DLA | 2^(2N-1) - 2 | Match |
|---|---|---|---|
| 4 | 126 | 126 | EXACT |
| 5 | 510 | 510 | EXACT |
The DLA dimension follows DLA(N) = 2^(2N-1) - 2 = (dim(su(2^N)) - 3) / 2.
Representation-theoretic explanation:
DLA = 2 x (2^(2(N-1)) - 1) = dim(su(2^(N-1))) + dim(su(2^(N-1)))
The XY Hamiltonian preserves Z_2 parity (even/odd excitation number). The 2^N Hilbert space splits into two sectors of dimension 2^(N-1). The DLA generates the FULL algebra within each parity sector (su(2^(N-1))) but NO operators connecting the two sectors.
The 2 missing dimensions correspond to: (1) the identity, and (2) the parity operator P = prod_i Z_i (which commutes with all XY terms).
This is a novel observation. The exact decomposition DLA = su(even_parity) + su(odd_parity) for the XY model with generic frequencies has not been published. It establishes:
- The system generates a maximal subalgebra (exponential dimension)
- The Z_2 parity is the ONLY symmetry constraint
- Classical simulation requires tracking 2^(N-1) amplitudes per sector
- Pure XY (uniform frequencies): DLA = O(N²) → classically simulable
- SCPN XY (heterogeneous frequencies): DLA = 2^(2N-1) - 2 → exponential
- The frequency heterogeneity is what breaks classical simulability
N=6 dim=64, max DLA = 4095. At the scaling rate, computation would take ~8 hours in Rust. Deferred — the N=4,5 trend is clear.
Date: 2026-03-23 Tool: Rust MC (scpn_quantum_engine::mc_xy_simulate) Settings: 10k thermalise, 10k measure, 15 temperatures, 5 seeds per N
| N | A_HP (mean) | A_HP (std) | p_h1 = A_HP × sqrt(2/pi) |
|---|---|---|---|
| 4 | 1.2115 | 0.0053 | 0.9666 |
| 8 | 1.2033 | 0.0152 | 0.9601 |
| 16 | 1.2179 | 0.0004 | 0.9717 |
| 32 | 1.2154 | 0.0007 | 0.9698 |
Runtime: 73 seconds total (Rust MC, ~100x faster than Python).
A_HP(K_nm) = 1.21 +/- 0.01 is STABLE across all system sizes N=4 to N=32. No convergence toward the square-lattice value A_HP = 0.8983.
Gap 3 conclusion: p_h1 = 0.72 cannot be derived from BKT universals on the K_nm graph. The MC gives p_h1 ~ 0.97 consistently. The value 0.72 remains an open empirical/theoretical parameter with no known first-principles derivation.