Algorithms

Variational, Trotterised, open-system, and error-mitigated quantum algorithms for coupled oscillator dynamics.

Variational Algorithms

VQE Production
Variational Quantum Eigensolver with Knm-informed ansätze. 0.05% ground state error on 4-qubit IBM hardware. COBYLA, SPSA, L-BFGS-B optimisers.
ADAPT-VQE Production
Adaptive circuit growth. Operator pool from XY generators. Gradient-driven gate selection minimises circuit depth while maintaining accuracy.
VarQITE Stable
Variational Quantum Imaginary Time Evolution. McLachlan variational principle. Converges to ground state without eigenvalue knowledge.
AVQDS Stable
Adaptive Variational Quantum Dynamics Solver. Real-time evolution with adaptive circuit depth. Tracks synchronisation order parameter R(t).
QSVT Experimental
Quantum Singular Value Transformation. Block-encoding of XY Hamiltonian. Optimal query complexity for time evolution.

Time Evolution

Trotter Decomposition Production
First and second-order Suzuki-Trotter. Configurable step count and ordering. Hardware-validated on IBM Heron r2 up to depth 770.
Floquet DTC Stable
Discrete Time Crystal simulation. Periodic driving of XY Hamiltonian. Subharmonic response detection. Many-body localisation analysis.

Open Quantum Systems

Lindblad Master Equation Production
Full Lindbladian evolution with configurable decay, dephasing, and jump operators. Models real hardware decoherence. Rust-accelerated operator construction.
MCWF Stable
Monte Carlo Wave Function method. Stochastic quantum trajectories with jump detection. Memory-efficient for large systems. Rust engine for sampling.
Trapped-Ion Noise Models Stable
Heating rate, motional dephasing, off-resonant coupling. Arbitrary trap topologies. Cross-platform comparison with superconducting results.

Error Mitigation

ZNE Production
Zero-Noise Extrapolation. Unitary folding with 4 extrapolation methods (linear, polynomial, exponential, Richardson). Hardware-validated.
PEC Production
Probabilistic Error Cancellation. Quasi-probability decomposition of noisy channels. Overhead scaling analysis included.
Dynamical Decoupling Stable
XY4, X2, CPMG pulse sequences. Inserted between Trotter steps. Suppresses low-frequency noise on idle qubits.
Z2 Parity Stable
Symmetry-based error detection. XY Hamiltonian conserves total Z parity. Post-selection discards parity-violating shots.
Symmetry Verification Stable
Generalised symmetry-based post-selection. Applicable to any conserved quantity of the XY model.
GUESS New April 2026
Guiding Extrapolations from Symmetry Decays (arXiv:2603.13060). The preferred default for the XY Hamiltonian: the symmetry observable ∑Zi is measured for free in the same Z-basis run as the target observable, so there is no shot-budget overhead. Calibrated on the ibm_kingston Phase 1 noise profile (8% leakage at depth 2 to 28% at depth 30).

Pulse Shaping & Quantum Optimal Control

ICI Three-Level Pulses New April 2026
Intuitive-Counterintuitive-Intuitive pulse sequences (Liu 2023). Pontryagin-Maximum-Principle-optimal control for three-level Λ-systems (STIRAP regime). Rust-native integrator runs 1,665× faster than the Python forward-Euler reference, parity-checked to 4.97×10−14.
(α,β)-Hypergeometric Pulses New April 2026
Analytically-solvable pulse family via Gauss 2F1 (arXiv:2504.08031, 2025). Closed-form envelope and detuning across a large family of shaped pulses. Custom Rust series expansion is 44× faster than a scipy 2F1 loop.
DynQ Qubit Placement New April 2026
Topology-agnostic qubit placement (arXiv:2601.19635). Scores candidate physical qubits by single-qubit gate, readout, and T1/T2 profile — independent of the connectivity graph. Used by the Phase 1 ibm_kingston campaign to select the four lowest-error qubits on Heron r2.

Analysis & Diagnostics

3 Novel Synchronisation Witnesses
Correlation witness, Fiedler witness, topological witness. NISQ-ready formalism for detecting quantum synchronisation without full state tomography.
Persistent Homology
Topological data analysis of quantum states. Betti curves, persistence diagrams. Detects phase transitions in synchronisation landscapes.
OTOC Scrambling
Out-of-Time-Order Correlators. Information scrambling rate. Butterfly velocity. Chaos detection in coupled oscillator dynamics. 4.4× Rust acceleration.
Krylov Complexity
Operator growth in Krylov basis. Lanczos coefficients. Complexity saturation analysis. Distinguishes integrable from chaotic dynamics.
DLA Parity Theorem Hardware-confirmed
Novel exact result: DLA(HXY) = su(2N−1) ⊕ su(2N−1) under the parity operator P = ∏iZi. First hardware confirmation on ibm_kingston (Heron r2, April 2026): 342 circuits across 8 Trotter depths show the odd (“feedback”) sector is systematically more decoherence-resistant than the even sector. Fisher's combined p ≪ 10−16, 7 of 8 depths significant at p < 0.05.
BKT Scaling
Berezinskii-Kosterlitz-Thouless finite-size scaling analysis. Critical coupling detection. Universal scaling collapse.

Quantum Error Correction

Toric Code Stable
2D toroidal stabiliser code. Anyonic excitations. Topological protection for quantum information stored in synchronised states.
Surface Code Stable
Planar code on arbitrary lattice topologies. Syndrome extraction. Minimum-weight perfect matching decoder.
UPDE Repetition Code Stable
16-layer SCPN protection scheme. Biological surface code analogy. Maps consciousness layers to QEC stabilisers.
Multiscale QEC New v0.9.5
Concatenated codes across 5 SCPN domains. Knill 2005 threshold. Syndrome flow analysis. Rust-accelerated decoding.

Code Example

# Kuramoto-to-quantum pipeline in 10 lines
from scpn_quantum_control.bridge import KnmCompiler
from scpn_quantum_control.phase import TrotterEvolution
from scpn_quantum_control.analysis import order_parameter

# Define coupling matrix (4 oscillators)
K = [[0, 1.2, 0.5, 0], [1.2, 0, 0.8, 0.3],
     [0.5, 0.8, 0, 1.0], [0, 0.3, 1.0, 0]]
omega = [1.0, 1.5, 0.8, 1.2] # natural frequencies

# Compile to quantum circuit
compiler = KnmCompiler(K, omega)
H = compiler.hamiltonian() # XY Hamiltonian
circuit = TrotterEvolution(H, dt=0.1, steps=20).circuit()

# Measure synchronisation
R = order_parameter(circuit, shots=8192)
print(f"Order parameter R = {R:.4f}")