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: $r = \sqrt{N^{-1} \sum_i (y_i - \hat{y}_i)^2}$. A large residual ($r > 0.1$) flags non-Markovian noise or systematic calibration drift and triggers a fallback to raw values.
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.
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.