SPDX-License-Identifier: AGPL-3.0-or-later¶

Commercial license available¶

© Concepts 1996–2026 Miroslav Šotek. All rights reserved.¶

© Code 2020–2026 Miroslav Šotek. All rights reserved.¶

ORCID: 0009-0009-3560-0851¶

Contact: www.anulum.li | protoscience@anulum.li¶

scpn-quantum-control — Quantum Cryptography Research Branch¶

Quantum Cryptography Research Branch¶

Thesis¶

The SCPN coupling matrix K_nm encodes oscillator topology as quantum entanglement structure under the Kuramoto-XY isomorphism. Parties who share K_nm can generate correlated measurement statistics from the Hamiltonian's ground state — an eavesdropper without K_nm cannot reconstruct these correlations. This gives a topology-authenticated QKD protocol where the shared secret is not a bit string but a physical coupling matrix.

Literature Grounding¶

Paper	Key Result	K_nm Connection
Frequency-bin QKD (npj Quantum Inf. 2025)	Coupled ring resonators implement BBM92 via coupling topology	K_nm maps to photonic coupling graph
Huygens quantum sync (Nature Comms. 2025)	Coupled oscillators under shared noise co-generate phase sync + entanglement	Kuramoto sync = entanglement generation
Quantum network science (AAAI 2025)	Fidelity-weighted graph structure determines key rate	K_nm spectral properties set QKD channel capacity
Entanglement percolation (arXiv 2026)	Mixed-state threshold lower than pure-state	Weak K_nm entries still contribute above threshold
Hierarchical group QKD (Sci. Reports 2025)	Multi-party protocol with sub-group key derivation	16-layer hierarchy = natural key tree

Module Architecture¶

scpn_quantum_control/
└── crypto/                    # Topology-authenticated quantum cryptography
    ├── __init__.py
    ├── knm_key.py             # K_nm → key material pipeline
    ├── topology_auth.py       # Spectral fingerprint authentication
    ├── entanglement_qkd.py    # Entanglement-based key distribution
    ├── percolation.py         # K_nm entanglement percolation analysis
    ├── hierarchical_keys.py   # Multi-layer key derivation from SCPN
    └── noise_analysis.py      # Devetak-Winter key rates, noise channels

tests/
├── test_knm_key.py
├── test_topology_auth.py
├── test_entanglement_qkd.py
├── test_percolation.py
├── test_hierarchical_keys.py
└── test_noise_analysis.py

Module Specifications¶

1. `knm_key.py` — Coupling Matrix to Key Material¶

Core primitive: Given shared K_nm and agreed Hamiltonian H(K_nm), prepare the ground state |ψ₀⟩, measure in agreed basis, extract correlated bit string.

Pipeline:
  K_nm → knm_to_hamiltonian(K) → VQE ground state → measurement → sift → key

Functions: - prepare_key_state(K, omega, ansatz_reps) → QuantumCircuit Builds VQE-optimized circuit encoding K_nm's ground state. - extract_raw_key(counts, basis, n_qubits) → BitArray Sifts measurement results into raw key bits. - estimate_qber(alice_bits, bob_bits) → float Quantum bit error rate from shared subset. - privacy_amplification(raw_key, qber) → SecureKey Universal₂ hash family compression.

Security argument: K_nm has 16×15/2 = 120 independent off-diagonal entries (symmetric, zero diagonal). Each entry is a continuous real value. An eavesdropper must reconstruct all 120 values to reproduce the ground state — the search space is R^120. Measurement statistics from a wrong K_nm' produce statistically distinguishable correlations (detectable via QBER > threshold).

2. `topology_auth.py` — Spectral Fingerprint Authentication¶

Core idea: The Laplacian spectrum of K_nm (already computed by SSGF's spectral.py) provides a public authentication token. The Fiedler value λ₁ and spectral gap λ₁/λ₂ uniquely characterize the coupling topology without revealing K_nm itself.

Functions: - spectral_fingerprint(K) → dict Returns {fiedler, gap_ratio, spectral_entropy, n_components}. - verify_fingerprint(K, fingerprint, tol) → bool Checks K against a claimed fingerprint. - topology_distance(fp1, fp2) → float Metric between two fingerprints for drift detection.

Why it works: Spectral properties are graph invariants — many different K_nm matrices share the same spectrum (co-spectral graphs). Publishing the spectrum doesn't reveal K_nm, but any party with the true K_nm can verify consistency.

3. `entanglement_qkd.py` — Topology-Authenticated QKD¶

Protocol (SCPN-QKD): 1. Alice and Bob share K_nm (pre-distributed secret). 2. Both construct H(K_nm) and prepare ground state |ψ₀⟩. 3. Alice measures qubits {0,...,7} in random {X, Z} basis. 4. Bob measures qubits {8,...,15} in random {X, Z} basis. 5. Public channel: announce basis choices, keep matching. 6. Sift → estimate QBER → privacy amplify → secure key.

Functions: - scpn_qkd_protocol(K, omega, alice_qubits, bob_qubits, shots) → QKDResult Full protocol execution on simulator. - correlator_matrix(counts, alice_qubits, bob_qubits) → ndarray Cross-correlation matrix between Alice and Bob measurements. - bell_inequality_test(correlator) → dict CHSH violation test to certify entanglement.

Key rate bound: From the Devetak-Winter formula, r ≥ 1 - h(QBER) - h(QBER) where h is binary entropy. The topology-dependent entanglement structure means different qubit pairs have different key rates — strongly coupled pairs (K_nm > 0.2) yield higher rates.

4. `percolation.py` — Entanglement Percolation on K_nm¶

Core question: Which (n,m) pairs in K_nm are entangled above threshold and usable as QKD channels?

Functions: - concurrence_map(K, omega) → ndarray Compute pairwise concurrence from ground state reduced density matrices. - percolation_threshold(K) → float Minimum K_nm value for end-to-end entanglement. - active_channel_graph(K, threshold) → nx.Graph Graph of above-threshold entangled pairs. - key_rate_per_channel(concurrence_map) → ndarray Devetak-Winter key rate for each link.

Connection to SSGF: The Fiedler value λ₁ from SSGF's spectral bridge is the algebraic connectivity — when λ₁ > 0, the graph is connected and end-to-end entanglement percolates.

5. `hierarchical_keys.py` — SCPN Layer Key Derivation¶

Core idea: The 16-layer SCPN hierarchy maps to a key tree.

Master key: hash(K_nm_full ‖ R_global)
  ├── L1 subkey: hash(K_nm[0,:] ‖ θ₁(t))
  ├── L2 subkey: hash(K_nm[1,:] ‖ θ₂(t))
  ├── ...
  └── L16 subkey: hash(K_nm[15,:] ‖ θ₁₆(t))

Functions: - derive_master_key(K, R_global, nonce) → bytes Master key from full coupling matrix + order parameter. - derive_layer_key(K, layer_idx, phase_sequence, nonce) → bytes Layer-specific subkey. - key_hierarchy(K, phases, n_layers) → dict[int, bytes] Full hierarchy derivation. - verify_key_chain(master, layer_keys, K) → bool Verify layer keys are consistent with master.

Time-varying keys: The Kuramoto phase sequence θ_n(t) adds temporal entropy. Different time windows produce different keys from the same K_nm — natural key rotation without re-keying.

Hardware Experiments¶

Experiment	Qubits	Status	Description
`bell_test_4q`	4	Implemented (v0.6.4)	CHSH violation with K_nm ground state on hardware
`correlator_4q`	4	Implemented (v0.6.4)	ZZ cross-correlation validates K_ij topology
`qkd_qber_4q`	4	Implemented (v0.6.4)	QBER from hardware vs BB84 threshold (< 0.11)
`correlator_8q`	8	Planned	Cross-correlation matrix on ibm_fez
`percolation_16q`	16	Planned	Full K_nm entanglement map on hardware

Dependencies on Existing Modules¶

Existing Module	Used By	Purpose
`bridge.knm_hamiltonian`	knm_key, entanglement_qkd	K_nm → H conversion
`phase.phase_vqe`	knm_key	Ground state preparation
`qec.control_qec`	entanglement_qkd	Error correction on key circuits
`mitigation.zne`	entanglement_qkd	Error mitigation for key extraction
`hardware.runner`	all experiments	IBM hardware execution

Research Timeline¶

Phase 1 — Complete: All 6 crypto modules implemented with full test coverage. knm_key, topology_auth, entanglement_qkd, percolation, hierarchical_keys, noise_analysis.

Phase 2 — Complete: Full SCPN-QKD protocol on simulator. Bell inequality verification, QBER estimation, Devetak-Winter key rates under noise.

Phase 3 — Complete: Entanglement percolation on K_nm graph, hierarchical key derivation from 16-layer SCPN structure.

Phase 4 — Complete: 3 hardware experiments executed on ibm_fez, March 28, 2026. Results in results/ibm_hardware_2026-03-28/. - bell_test_4q: CHSH S-value from hardware counts - correlator_4q: 4x4 connected correlation matrix - qkd_qber_4q: Z-basis and X-basis QBER

Phase 5 — Planned: 8-qubit correlator, 16-qubit percolation on hardware.