Physics-informed zero-noise extrapolation using Hamiltonian symmetries. The preferred default for the SCPN Kuramoto-XY framework: shot-budget-free, calibrated on real ibm_kingston noise profiles.
Paper: Oliva del Moral et al., Guiding extrapolations from symmetry decays for efficient error mitigation.
arXiv:2603.13060 (2026).
Python module: scpn_quantum_control.mitigation.symmetry_decay
Rust acceleration: scpn_quantum_engine.fit_symmetry_decay, scpn_quantum_engine.guess_extrapolate_batch
Tests: 25 strong tests in tests/test_symmetry_decay.py
Richardson zero-noise extrapolation fits a polynomial through observables measured at amplified noise levels (via circuit folding) and extrapolates to the zero-noise limit. At large circuit depths (100+ qubits, thousands of CZ gates), the polynomial model fails to capture the true noise profile, leading to systematic over- or under-correction. Oliva del Moral et al. document divergence of Richardson ZNE beyond depth 2000 CZ gates on IBM hardware (their Figure 7).
Polynomial extrapolation assumes noise is additive and scale-invariant, which breaks down for correlated noise channels. No standard method leverages physical conservation laws specific to the Hamiltonian being simulated.
If the Hamiltonian $H$ conserves a symmetry observable $S$ (i.e. $[H, S] = 0$), then $\langle S \rangle$ is analytically known for any initial state. The deviation of $\langle S \rangle$ from its ideal value under hardware noise directly reveals the noise-induced decay profile.
GUESS transfers this learned decay to target observables whose ideal values are unknown. Instead of fitting a generic polynomial, GUESS uses physics to constrain the extrapolation.
The SCPN Kuramoto-XY Hamiltonian
naturally conserves total magnetisation $S = \sum_i Z_i$ because the $XX + YY$ interaction flips pairs of spins in opposite directions and leaves the total $Z$-component invariant. Formally $[H_{XY}, \sum_i Z_i] = 0$.
This is the crucial point: measuring $S = \sum_i Z_i$ requires only Z-basis measurements, which are already part of every experiment run. Whereas generic ZNE costs an additional 2–4× shot budget per noise scale factor, GUESS on the XY Hamiltonian costs zero extra shots. The symmetry observable is a free rider on the target observable measurement.
Under noise at scale factor $g$ (where $g = 1$ is base noise), the symmetry observable decays exponentially:
where $\alpha \ge 0$ is the noise scaling exponent. This model follows from the Lindblad master equation under depolarising noise: each gate contributes an independent decay factor, and circuit folding multiplies the total decay rate by $g$.
Taking the logarithm:
This is a linear model $y = -\alpha \cdot x$ where $y_i = \ln(\langle S \rangle_{g_i} / \langle S \rangle_\text{ideal})$ and $x_i = g_i - 1$. We fit $\alpha$ via ordinary least-squares on $(x_i, y_i)$ pairs from $N \ge 2$ noise scale measurements.
Fit residual. $r = \sqrt{N^{-1} \sum_i (y_i - \hat{y}_i)^2}$ is returned alongside $\alpha$ as a fit-quality diagnostic (the model's fit_residual field). A large residual means the single-exponential decay is a poor description of the data — non-Markovian noise or calibration drift — and signals that the mitigated value should be distrusted. The library does not threshold on the residual automatically; its one built-in fallback to the raw value is when the symmetry observable has fully decayed ($\lvert\langle S\rangle_\text{noisy}\rvert \to 0$), where the correction is undefined.
Given the learned $\alpha$, the mitigated value of any target observable $O$ is (Oliva del Moral et al., 2026, Eq. 5):
Properties. When noise is absent, $C = 1$ and no correction is applied. When $\alpha = 0$, $C = 1$ regardless of symmetry values. When the symmetry fully decays ($\langle S \rangle_\text{noisy} \to 0$), the correction diverges and we fall back to the raw value. $C \ge 1$ for physical noise.
An observable is measured at odd noise scales $g=1,3,5$ (unitary folding amplifies the noise). Standard zero-noise extrapolation fits those points and reads the value back at $g=0$; a quadratic fit through three points removes more of the curvature than a line. The symmetry observable, whose ideal value is known exactly, decays as $\langle S\rangle_g=\langle S\rangle_\text{ideal}\,e^{-\alpha(g-1)}$ — fitting it recovers the noise rate $\alpha$, which is the extra information GUESS uses to correct the target. Drive the noise and watch the three estimates separate.
Synthetic illustration of the mechanism, computed live from the library's exact formulas — Richardson polyfit to $g=0$, the log-linear symmetry-decay fit for $\alpha$, and the GUESS correction $\langle O\rangle_\text{noisy}(\langle S\rangle_\text{ideal}/\langle S\rangle_\text{noisy})^{\alpha}$ (Oliva del Moral et al. 2026, Eq. 5) — verified against a NumPy reference to machine precision. The real GUESS runs on hardware counts; measured Phase 1 numbers are on the results page.
The ibm_kingston Phase 1 campaign (342 circuits, 8 Trotter depths) provides exactly the noise profile GUESS needs. Parity leakage rises smoothly from ~8 % at depth 2 to ~28 % at depth 30 — a textbook exponential-plus-saturation curve.
For the Phase 2 campaign, a noise-scaled sub-sweep with circuit folding factors $g \in \{1, 3, 5\}$ will use the measured parity leakage itself as the symmetry observable. Because parity leakage is the complement of $\langle P \rangle = \langle \prod_i Z_i \rangle$, this is mathematically equivalent to running GUESS on the $\mathbb{Z}_2$ parity operator, with the ideal value $\langle P \rangle_\text{ideal} = \pm 1$ known in closed form.
→ See the Phase 1 results page for the full leakage-vs-depth curve.
The symmetry observable $\langle S \rangle$ and the target observable $\langle O \rangle$ are extracted from the same Z-basis counts. No extra shots are required to measure $S$; it is a post-processing step on the same bitstrings.
| Method | Physics-aware | Shot overhead | Works at deep circuits | Best for |
|---|---|---|---|---|
| GUESS (this page) | ✓ | 0 (for XY) | ✓ | Hamiltonians with known symmetry |
| Richardson ZNE | — | 2–4× | diverges > 2000 CZ | generic circuits, shallow depth |
| Exponential ZNE | — | 2–4× | ✓ | generic circuits, deep depth |
| PEC | noise-model | exp. in depth | ✓ | highest accuracy when noise model is known |
| DDD | — | 0 (idle slots only) | ✓ | idle-qubit low-frequency noise |
| $\mathbb{Z}_2$ parity post-selection | ✓ | partial (shots rejected) | ✓ | hard discard of bad shots |
GUESS and DDD are complementary and can be stacked. GUESS and post-selection target the same symmetry; GUESS extrapolates while post-selection discards. They can also be stacked.