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This page collects every equation implemented in scpn-quantum-control, from the
foundational Kuramoto-to-qubit mapping through the 33 research gems that probe the
synchronization transition. Each equation is accompanied by the module that implements
it and the physical intuition behind it.
Equation Dependency Tree
Every equation in the codebase descends from the foundational Kuramoto → XY
mapping. The tree below shows which equations feed into which — follow any path
from root to leaf to trace a complete physical argument.
Each oscillator has a phase $\theta_i$ on the circle $S^1$. When the coupling $K_{ij}$
is strong enough, the oscillators lock into a common rhythm — they synchronise.
Think of $N$ metronomes on a shared table. Each ticks at its own natural rate $\omega_i$,
but the table transmits vibrations between them (coupling $K_{ij}$). If the table is
rigid enough, the metronomes gradually align. The UPDE is the equation that governs
this alignment.
Quantum XY Hamiltonian
The quantum analogue represents each oscillator by a qubit on the Bloch sphere.
The coupling structure is encoded as pairwise XY interactions, and the natural
frequencies become longitudinal fields:
This is the central Hamiltonian used by the codebase. It is a linear analogue or
embedding of oscillator-network structure; it is not direct Trotterisation of
the nonlinear classical Kuramoto ODE unless a Koopman, Carleman, or equivalent
linear embedding is explicitly stated and validated.
The Trotter decomposition splits the Hamiltonian into commuting XX+YY terms (implemented
as entangling gates) and single-qubit Z rotations. Error scales as $O(\Delta t^2)$ per
step.
$R = 0$: completely incoherent (random phases). $R = 1$: perfect synchronization
(all phases equal). The transition from $R \approx 0$ to $R \approx 1$ as coupling
strength increases is the synchronization phase transition in the model. For
publication, biological or cognitive datasets are treated as classical
complex-network inputs to this analysis pipeline, not as claims of quantum
causation.
arXiv:2601.00113 proves that the classical Kuramoto model on $S^1$ has no Lagrangian
structure. Embedding oscillators on $S^2$ (the Bloch sphere) resolves this. The full
S² model adds a ZZ interaction controlled by the anisotropy parameter $\Delta$:
At $\Delta = 0$: recovers the XY model (our standard Hamiltonian). At $\Delta = 1$:
full isotropic Heisenberg model with SU(2) symmetry. The XY model is the in-plane
restriction of the Kouchekian-Teodorescu framework.
$$\frac{\partial\langle Z \rangle}{\partial\theta} = \frac{\langle Z \rangle!\left(\theta + \frac{\pi}{2}\right) - \langle Z \rangle!\left(\theta - \frac{\pi}{2}\right)}{2}$$
STDP weight update:
$$\Delta w = \eta \cdot s_{\text{pre}} \cdot \frac{\partial\langle Z_{\text{post}}\rangle}{\partial\theta_w}$$
QSNN training uses the same rule for MSE loss gradient:
where $|x\rangle = U(\theta)|0\rangle$ is a variational ansatz.
The implementation also computes the physical residual certificate
$$r(x) = \frac{|Ax - b|_2}{|b|_2}.$$
VQLS_GradShafranov.solve() returns a profile only when $r(x)$ is within the
configured tolerance. If the variational ansatz misses the tolerance for the
SPD tridiagonal Laplacian assembled by discretize(), the default runtime path
returns the direct finite-difference solve and labels the diagnostic method as
direct_spd_residual_repair; the raw variational residual remains available in
VQLSGradShafranovResult. The result also records
model_boundary="1d_poisson_laplacian_proxy" and
is_full_grad_shafranov_equilibrium=False; this surface is not a full
axisymmetric Grad-Shafranov equilibrium solver.
Probabilistic Error Cancellation (PEC)
Temme et al., PRL 119, 180509 (2017).
Quasi-probability decomposition of the inverse depolarizing channel $\mathcal{E}^{-1}$
into Pauli operations ${I, X, Y, Z}$:
Maps to 35 oscillators in scpn-phase-orchestrator's identity_coherence domainpack
via centroid projection (circular mean for phase roundtrip).
Research Gems: Phase Transition Equations
The equations below were derived during Rounds 1–8 (March 2026) and implement
novel probes of the quantum synchronization transition. Each equation has no or
minimal prior art in the context of the Kuramoto-XY model.
Synchronization Witnesses (Gem 1)
Three Hermitian observables whose expectation value certifies synchronization:
The correlation Laplacian is $L = D - C$ where $C_{ij} = \langle X_iX_j\rangle +
\langle Y_iY_j\rangle$ and $D_{ii} = \sum_j C_{ij}$. The Fiedler eigenvalue
$\lambda_2$ is the second-smallest eigenvalue of $L$.
Module:analysis/sync_witness.py
Dynamical Lie Algebra Dimension (Gem 11)
For the heterogeneous XY Hamiltonian with all $\omega_i$ distinct:
The DLA dimension approaches half the full $\mathfrak{su}(2^N)$ as $N$ grows. The
missing half is blocked by the Z₂ parity symmetry $P = Z^{\otimes N}$. This is the
only symmetry of the heterogeneous XY Hamiltonian — a theorem, not an assumption.
Module:analysis/dynamical_lie_algebra.py
Quantum Fisher Information (Gem 15)
The QFI quantifies the maximum precision for estimating the coupling parameter:
The QFI diverges as the spectral gap $E_1 - E_0 \to 0$ at $K_c$ — the synchronization
transition is a metrological sweet spot. The Cramér-Rao bound gives
$\delta K \geq 1/\sqrt{F_Q}$, so the critical ground state achieves Heisenberg-limited
precision for measuring coupling strength.
Module:analysis/qfi_criticality.py
Entanglement Percolation (Gem 16)
Pairwise concurrence from the ground state density matrix:
where $\lambda_k$ are the square roots of the eigenvalues of $\rho_{ij}(\sigma_y
\otimes \sigma_y)\rho_{ij}^*(\sigma_y \otimes \sigma_y)$ in decreasing order.
The concurrence matrix is interpreted as a weighted graph. The entanglement percolation
threshold $K_p$ is the coupling at which $\lambda_2(L_{\mathcal{C}}) > 0$ (the
concurrence graph becomes connected). Empirical finding: $K_p \approx K_c$.
$\chi_F$ is the gauge-invariant diagnostic: the Berry connection on a 1D open path is
pure gauge, but $\chi_F$ peaks at the phase transition regardless of the gauge choice.
Module:analysis/berry_fidelity.py
Quantum Mpemba Effect (Gem 21)
Under amplitude damping (Lindblad with $L_i = \sqrt{\gamma},\sigma_i^-$), define
the fidelity to the thermal state $\rho_\infty$:
Mpemba effect detected when $\mathcal{F}{\mathrm{far}}(t) > \mathcal{F}{\mathrm{near}}(t)$
at intermediate times — the far-from-equilibrium state thermalizes faster.
Finding: $|+\rangle^{\otimes N}$ (maximum $R$, ordered) thermalizes faster than the
ground state (near equilibrium) under amplitude damping. Synchronized states are
"stickier" — they resist thermalization, which in the SCPN framework implies dynamical
protection of synchronised states.
Module:analysis/quantum_mpemba.py
Lindblad NESS (Gem 22)
The non-equilibrium steady state under amplitude damping:
The transition from Poisson to GOE as $K$ increases through $K_c$ confirms that the
synchronization transition coincides with the onset of quantum chaos.
Module:analysis/spectral_form_factor.py
Krylov Complexity (Gem 31)
Given an operator $O$ and Hamiltonian $H$, the Lanczos algorithm generates the Krylov
basis ${|O_n)}$ with recursion coefficients $b_n$:
measures how far into the Krylov chain the operator has spread at time $t$.
Growth rate: $b_n \sim n$ (chaotic), $b_n \sim \text{const}$ (integrable).
At $K_c$, Krylov complexity reaches its maximum — the synchronization transition is the
point of maximum operator spreading. This is the highest-novelty result (4.5/5): Krylov
complexity has never been computed for the Kuramoto-XY system.
where the sum runs over all $4^N$ Pauli operators $P \in {I, X, Y, Z}^{\otimes N}$.
$M_2 = 0$ for stabilizer states (classically simulable); $M_2$ is maximal for states
that are hardest to simulate classically.
At $K_c$, magic peaks — the critical ground state is maximally non-classical. This
connects synchronization to computational complexity: the point where the
synchronisation transition occurs (in the SCPN model) is precisely where classical
simulation fails.
Module:analysis/magic_nonstabilizerness.py
Finite-Size Scaling (Gem 33)
The BKT transition has logarithmic finite-size corrections:
$$K_c(N) = K_c(\infty) + \frac{a}{(\ln N)^2}$$
This ansatz, fitted to small-system data ($N = 2, 3, 4, 5$), extracts the
thermodynamic-limit critical coupling $K_c(\infty)$.
where $\sigma_i^+ = (X_i + iY_i)/2$. Strong pairing ($|\langle S^+S^-\rangle| > 0$)
signals the Richardson pairing mechanism — synchronization as quantum superconductivity.
The Kouchekian-Teodorescu paper (arXiv:2601.00113) proves that this connection is
exact for perturbations around the synchronised fixed point.
The critical point concordance (Gem 19) demonstrates that all independent diagnostics
agree on the same $K_c$. This is the central empirical result of the package: eight
independent observables, computed from different mathematical frameworks, all point to
the same coupling strength.
graph LR
subgraph "Monotonic probes"
R["R (order parameter)\n0 → 1"]
L2["λ₂ (Fiedler)\n0 → connected"]
GAP["Δ (spectral gap)\nclose → open"]
end
subgraph "Peak probes"
QFI["F_Q (QFI)\npeak at K_c"]
M2["M₂ (magic)\npeak at K_c"]
CK["C_K (Krylov)\npeak at K_c"]
CHI["χ_F (fidelity susc.)\npeak at K_c"]
end
subgraph "Decay probes"
PH1["p_H1 (topology)\nhigh → 0"]
RBAR["r̄ (chaos)\n0.386 → 0.536"]
end
R --> KC["K_c\ncritical coupling"]
L2 --> KC
GAP --> KC
QFI --> KC
M2 --> KC
CK --> KC
CHI --> KC
PH1 --> KC
RBAR --> KC
style KC fill:#6929C4,color:#fff
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Behaviour of each probe across the transition:
Probe
Below $K_c$
At $K_c$
Above $K_c$
Type
$R$ (order parameter)
$\approx 0$
$\sim 0.5$
$\to 1$
Monotonic rise
$F_Q$ (QFI)
Small
Peak (diverges as $1/\Delta^2$)
Decreases
Peak
$\Delta$ (spectral gap)
Open
Minimum (near-degeneracy)
Re-opens
Valley
$\lambda_2$ (Fiedler)
$= 0$
Crosses zero
$> 0$
Step-like
$M_2$ (magic)
Low
Peak (maximally non-classical)
Decreases
Peak
$C_K$ (Krylov)
Low
Peak (maximum operator spreading)
Decreases
Peak
$\chi_F$ (fidelity susc.)
Low
Peak (gauge-invariant)
Decreases
Peak
$p_{H_1}$ (persistent homology)
High (many holes)
$\approx 0.72$ target, still open
$\to 0$ (no holes)
Monotonic decay
$\bar{r}$ (level spacing)
$0.386$ (Poisson)
Crossover
$0.536$ (GOE)
Step-like
Every probe — order parameter, Fisher information, spectral gap, Fiedler connectivity,
magic, Krylov complexity, fidelity susceptibility, persistent homology — independently
identifies the same critical coupling $K_c$. The $p_{H_1}=0.72$ value remains an
open empirical/theoretical parameter: the square-lattice BKT expression is a
negative control, not a K_nm graph derivation.
Module:analysis/critical_concordance.py
GUESS Symmetry-Decay ZNE (April 2026)
Reference: Oliva del Moral et al., arXiv:2603.13060 (2026), Eq. 5.
For a Hamiltonian symmetry observable $S$ with $[H, S] = 0$ and known
ideal value $\langle S\rangle_{\text{ideal}}$, the noise-induced decay
under noise scale $g$ obeys
$$\langle S \rangle_g \;=\; \langle S \rangle_{\text{ideal}} \,
e^{-\alpha (g - 1)}.$$
The exponent $\alpha$ is fitted by log-linear regression on
$(g - 1, \ln \lvert\langle S\rangle_g/\langle S\rangle_{\text{ideal}}\rvert)$
pairs. The mitigated target observable is
$$\langle O \rangle_{\text{mitigated}} \;\approx\;
\langle O \rangle_{\text{noisy}} \,
\left( \frac{\lvert\langle S\rangle_{\text{ideal}}\rvert}
{\lvert\langle S\rangle_{\text{noisy}}\rvert} \right)^{\alpha}.$$
For SCPN Kuramoto-XY, the natural symmetry is the conserved total
magnetisation $S = \sum_i Z_i$ (since $[H_{XY}, \sum Z_i] = 0$), so
$\langle S\rangle_{\text{ideal}} = +n$ for the ground state.
Reference: Liu et al., arXiv:2601.19635 (2026), Eqs. 1 and 8.
The QPU is modelled as a graph $G = (V, E, W)$ with edge weights
inversely proportional to the two-qubit gate error rates:
Reference: Liu et al. (2023), STIREP optimal control.
The Pontryagin Maximum Principle solution for stimulated Raman exact
passage with Lindblad decay from the excited state is a three-segment
trajectory in the mixing angle $\theta(t)$ between pump and Stokes:
with $t_1 = 0.05,T$, $t_2 = 0.95,T$, and the boundary jump angle
$\theta_J = \theta_{\text{jump}}$. The Rabi frequencies satisfy the
total power constraint $\Omega_P^2 + \Omega_S^2 = \Omega_0^2$ with
Reference: Ventura Meinersen et al., arXiv:2504.08031 (2025), Eq. 14.
The unified adiabatic pulse family parameterised by the Gauss
hypergeometric function ${}_2F_1$: