The vocabulary of coupled-oscillator quantum control, in one place — from the Kuramoto model and the order parameter through the XY Hamiltonian, its dynamical Lie algebra and parity sectors, to Trotterisation, GUESS mitigation and the BKT transition. Start typing to filter every term at once.
| Kuramoto model | A network of phase oscillators, each with its own natural frequency, coupled through a matrix $K_{ij}$; the canonical model of spontaneous synchronisation. See phase synchronisation. |
| Order parameter $R$ | $R=\lvert\frac1N\sum_j e^{i\theta_j}\rvert$, the magnitude of the mean phasor. Runs from 0 (incoherent) to 1 (perfect lockstep); the scalar measure of how synchronised a population is. |
| Natural frequency | The rate $\omega_i$ at which oscillator $i$ would advance in isolation. A spread of natural frequencies is the disordering force that coupling must overcome. |
| Coupling matrix $K_{ij}$ | The strength with which oscillator $i$ feels oscillator $j$. Its structure (the topology) decides how easily a network locks. See network topology. |
| Phase locking | Two or more oscillators holding a constant phase difference. For two oscillators it occurs once the coupling exceeds the frequency mismatch, $K\ge\lvert\Delta\omega\rvert$ (the Adler condition). |
| Partial synchronisation | A regime between disorder and full lock where a cluster of oscillators entrains while others drift; $R$ sits between 0 and 1. |
| Mean field | The all-to-all coupling limit, where every oscillator is pulled toward the population average rather than named neighbours. |
| Fiedler eigenvalue | The smallest non-zero eigenvalue $\lambda_2$ of a network's Laplacian; a measure of how well-connected the graph is, and the input to the library's $T_{\mathrm{BKT}}$ estimate. |
| XY Hamiltonian | $H_{XY}=\sum_{i<j}K_{ij}(\sigma^x_i\sigma^x_j+\sigma^y_i\sigma^y_j)+\sum_i\omega_i Z_i$; the linear, quantum-native model the Kuramoto dynamics reduce to in the small-oscillation limit. |
| Trotterisation | Approximating a continuous evolution $e^{-i(A+B)t}$ by alternating short evolutions $(e^{-iAt/n}e^{-iBt/n})^n$; the error from the non-commuting terms shrinks with more steps. See Trotterisation. |
| VQE | Variational Quantum Eigensolver — optimise the parameters of a shallow circuit to minimise an energy expectation, rather than evolving in real time. See algorithms. |
| Parameter-shift rule | An exact circuit gradient from two evaluations shifted by $\pm\tfrac\pi2$: $\partial_\theta\langle H\rangle=\tfrac12[\langle H\rangle(\theta+\tfrac\pi2)-\langle H\rangle(\theta-\tfrac\pi2)]$. See algorithms. |
| Ansatz | The parametrised circuit template a variational method optimises over; its structure sets what states are reachable. |
| Expectation value | $\langle H\rangle=\langle\psi\lvert H\rvert\psi\rangle$, the average of an observable in a state — the quantity read off a quantum run and the target of variational optimisation. |
| Dynamical Lie algebra (DLA) | The operator space generated by taking every nested commutator of a Hamiltonian's terms; it captures everything the dynamics can ever do. See DLA parity. |
| Parity operator $P$ | $P=\prod_i Z_i$, with eigenvalue $(-1)^{\text{popcount}}$ on a basis state; conserved by the XY Hamiltonian, so $[H_{XY},P]=0$. |
| Parity sector | One of the two equal blocks the parity symmetry splits the Hilbert space into — even ($P=+1$) and odd ($P=-1$), each of dimension $2^{n-1}$. |
| DLA parity decomposition | $\mathrm{DLA}(H_{XY})=\mathfrak{su}(2^{n-1})\oplus\mathfrak{su}(2^{n-1})$, dimension $2^{2n-1}-2$; the algebra splits with the parity sectors. See the theorem. |
| Parity leakage | Probability that has escaped a chosen parity sector; zero for the ideal parity-conserving Hamiltonian, and a direct fingerprint of decoherence on hardware. |
| DLA parity asymmetry | The Phase 1 hardware observation that the odd parity sector is more decoherence-resistant than the even one. See Phase 1 results. |
| BKT transition | Berezinskii–Kosterlitz–Thouless: a transition with no symmetry breaking, driven by vortex–antivortex unbinding; correlations turn from algebraic to exponential. See BKT. |
| Correlation length $\xi$ | The distance over which phases stay aligned. Above $T_{\mathrm{BKT}}$ it diverges with an essential singularity, $\xi\sim e^{b/\sqrt{T-T_{\mathrm{BKT}}}}$. |
| Nelson–Kosterlitz jump | The universal drop in spin stiffness at the BKT point, $\rho_s(T_{\mathrm{BKT}}^-)=\tfrac2\pi T_{\mathrm{BKT}}$. |
| Berry (geometric) phase | A phase a state accrues around a closed loop in parameter space that depends only on the loop's geometry. For a spin-½ it is $-\tfrac12$ the enclosed solid angle. See Berry phase. |
| Solid angle $\Omega$ | The area a loop encloses on the sphere of states; for a latitude loop, $\Omega=2\pi(1-\cos\theta_0)$. Sets the Berry phase. |
| Wilson loop | The gauge-invariant product of consecutive-state overlaps $\prod_i\langle\psi_i\lvert\psi_{i+1}\rangle$ around a loop; the discrete way to read off a geometric phase. |
| STIRAP | Stimulated Raman Adiabatic Passage — population transfer through a three-level system via a dark state, using a counterintuitive Stokes-before-pump ordering. |
| ICI pulse | Intuitive–Counterintuitive–Intuitive: a three-segment, time-optimal mixing-angle schedule for a STIRAP-like transfer. See ICI pulse shaping. |
| Hypergeometric pulse | A two-parameter $(\alpha,\beta)$ envelope family, $\operatorname{sech}(\gamma t)\,{}_2F_1(\alpha,\beta;\tfrac{\alpha+\beta+1}2;\tfrac{1+\tanh\gamma t}2)$, containing Allen–Eberly, STIRAP and Demkov–Kunike. See hypergeometric pulses. |
| Allen–Eberly / Demkov–Kunike | Classic pulse shapes recovered as special points of the hypergeometric family — $(\alpha,\beta)=(0,0)$ (a pure $\operatorname{sech}$) and $(1,\tfrac12)$ respectively. |
| Mixing angle $\theta(t)$ | The single angle whose sine and cosine set the pump and Stokes Rabi drives, $\Omega_P=\Omega_0\sin\theta,\ \Omega_S=\Omega_0\cos\theta$, keeping total power fixed. |
| Zero-noise extrapolation (ZNE) | Measure an observable at several amplified noise levels, then fit and extrapolate back to zero noise. See algorithms. |
| Gate folding | Amplifying a circuit's noise on purpose by inserting $G\,(G^\dagger G)^{k}$, the input ZNE needs at odd noise scales $1,3,5,\dots$ |
| GUESS | Symmetry-guided mitigation: use the known-ideal decay of a conserved symmetry to calibrate the noise and correct the target observable. See GUESS. |
| Richardson extrapolation | Fitting a polynomial through the noise-scaled points and reading its value at zero; a linear fit is order 1, a quadratic order 2. |
| PEC / DDD | Probabilistic Error Cancellation and Digital Dynamical Decoupling — further mitigation methods in the library's toolbox alongside ZNE and GUESS. |
| IBM Heron r2 | The superconducting processor generation on which the Phase 1 campaign ran (device ibm_kingston). See hardware validation. |
| Susceptance | The imaginary part of a power line's admittance; for the power-grid mapping it sets the Kuramoto coupling $K_{ij}=V_iV_jB_{ij}/M_i$. See applications. |
| Welch's t-test | A statistical test for a difference in means with unequal variances; used to compare the even and odd parity sectors in the Phase 1 analysis. |
| Raw counts | The unprocessed measurement outcomes from a quantum run, kept so every downstream figure can be recomputed. See reproducibility. |
| Result pack | A bundle of raw counts, metadata and provenance for a run, so a claim can be checked from its evidence rather than taken on trust. |