SPDX-License-Identifier: AGPL-3.0-or-later¶

Commercial license available¶

© Concepts 1996–2026 Miroslav Šotek. All rights reserved.¶

© Code 2020–2026 Miroslav Šotek. All rights reserved.¶

ORCID: 0009-0009-3560-0851¶

Contact: www.anulum.li | protoscience@anulum.li¶

scpn-quantum-control — Error Mitigation Documentation¶

Error Mitigation¶

scpn-quantum-control ships three complementary error-mitigation backends. Choose based on whether you have a known symmetry observable, how much shot budget you can spend, and whether you want a generic black-box wrapper.

Backend	Module	Best for	Overhead	Reference
GUESS symmetry-decay ZNE	`mitigation.symmetry_decay`	XY Hamiltonian (uses conserved \(\sum Z_i\))	None — symmetry observable comes for free	symmetry_decay_guess.md, arXiv:2603.13060
Mitiq ZNE / DDD	`mitigation.mitiq_integration`	Generic circuits with no known symmetry	2–4× shots per scale factor	this page
PEC	`mitigation.pec`	Highest accuracy when you have a noise model	Exponential in circuit depth	`mitigation_api.md`

For the SCPN Kuramoto-XY framework specifically, GUESS is the preferred default: the XY Hamiltonian commutes with the total magnetisation operator \(S = \sum_i Z_i\), so the symmetry observable is measured for free in the same Z-basis run as the target observable. The Phase 1 hardware campaign on ibm_kingston in April 2026 used the parity leakage rate measured this way as both the noise calibration and the headline scientific result. See symmetry_decay_guess.md for the full GUESS theory and tutorial.

Mitiq integration (generic ZNE / DDD)¶

scpn_quantum_control.mitigation.mitiq_integration

Wraps Mitiq (Unitary Fund) to provide production-quality error mitigation for Kuramoto-XY circuits. Two techniques are supported:

ZNE (Zero Noise Extrapolation) — amplify noise by circuit folding, then extrapolate to the zero-noise limit
DDD (Digital Dynamical Decoupling) — insert identity sequences in idle periods to suppress low-frequency noise

Requires: pip install mitiq (and qiskit-aer for the default executor).

Known issue: Mitiq 1.0 has a bug where from __future__ import annotations breaks executor introspection. This module avoids that import as a workaround.

Theory¶

Zero Noise Extrapolation (ZNE)¶

NISQ circuits suffer from gate errors. ZNE works by:

Running the circuit at its original noise level (scale factor 1)
Artificially increasing noise by circuit folding: replacing each gate \(G\) with \(G G^\dagger G\) (scale factor 3), \(G G^\dagger G G^\dagger G\) (factor 5), etc.
Extrapolating the expectation value to scale factor 0 (zero noise) using Richardson extrapolation or polynomial fit.

This gives a bias-corrected estimate without requiring knowledge of the noise model.

Digital Dynamical Decoupling (DDD)¶

Idle qubits accumulate low-frequency decoherence. DDD inserts pairs of Pauli gates (e.g., \(XX\) or \(XYXY\)) during idle periods. These average out quasi-static noise without changing the ideal circuit output.

DDD is complementary to ZNE: DDD reduces idle noise, ZNE corrects gate noise.

API Reference¶

from scpn_quantum_control.mitigation.mitiq_integration import (
    is_mitiq_available,
    zne_mitigated_expectation,
    ddd_mitigated_expectation,
)

`zne_mitigated_expectation`¶

result = zne_mitigated_expectation(
    circuit: QuantumCircuit,           # Qiskit circuit (with measurements)
    executor: callable | None = None,  # custom executor (default: AerSimulator)
    scale_factors: list[float] | None = None,  # noise scale factors (default: [1, 3, 5])
    shots: int = 8192,
) -> float

Returns: ZNE-mitigated expectation value (float).

Default executor: Runs the circuit on qiskit_aer.AerSimulator and returns the parity expectation value \(\langle Z^{\otimes n} \rangle\).

Custom executor: Must be a callable circuit → float. Mitiq handles the circuit folding internally.

`ddd_mitigated_expectation`¶

result = ddd_mitigated_expectation(
    circuit: QuantumCircuit,
    executor: callable | None = None,
    shots: int = 8192,
) -> float

Returns: DDD-mitigated expectation value (float).

Inserts \(XX\) sequences in idle periods.

`is_mitiq_available`¶

is_mitiq_available() -> bool

Tutorial¶

ZNE on a Bell State¶

from qiskit import QuantumCircuit
from scpn_quantum_control.mitigation.mitiq_integration import (
    zne_mitigated_expectation, is_mitiq_available
)

if not is_mitiq_available():
    raise ImportError("Install mitiq: pip install mitiq")

# Bell state circuit
qc = QuantumCircuit(2)
qc.h(0)
qc.cx(0, 1)
qc.measure_all()

# ZNE with default scale factors [1, 3, 5]
mitigated = zne_mitigated_expectation(qc)
print(f"ZNE-mitigated ⟨ZZ⟩ = {mitigated:.4f}")

# Custom scale factors (more aggressive extrapolation)
mitigated_5pt = zne_mitigated_expectation(
    qc, scale_factors=[1.0, 2.0, 3.0, 4.0, 5.0]
)
print(f"5-point ZNE ⟨ZZ⟩ = {mitigated_5pt:.4f}")

DDD for Idle Noise¶

from scpn_quantum_control.mitigation.mitiq_integration import (
    ddd_mitigated_expectation
)

# Circuit with idle periods (barrier creates explicit idle slots)
qc = QuantumCircuit(3)
qc.h(0)
qc.barrier()
qc.cx(0, 1)
qc.barrier()
qc.cx(1, 2)
qc.measure_all()

mitigated = ddd_mitigated_expectation(qc)
print(f"DDD-mitigated ⟨ZZZ⟩ = {mitigated:.4f}")

With Kuramoto-XY Circuits¶

import numpy as np
from scpn_quantum_control.hardware.circuit_export import build_trotter_circuit
from scpn_quantum_control.mitigation.mitiq_integration import (
    zne_mitigated_expectation
)

n = 4
K = 0.45 * np.exp(-0.3 * np.abs(np.subtract.outer(range(n), range(n))))
np.fill_diagonal(K, 0.0)
omega = np.linspace(0.8, 1.2, n)

qc = build_trotter_circuit(K, omega, t=0.1, reps=3)
qc.measure_all()

mitigated = zne_mitigated_expectation(qc, scale_factors=[1, 3, 5])
print(f"Mitigated Kuramoto circuit: {mitigated:.4f}")

Comparison¶

Feature	This module	Mitiq standalone	Qiskit Transpiler Passes
ZNE	Yes (wrapped)	Yes	No
DDD	Yes (wrapped)	Yes	Yes (`DynamicalDecoupling`)
PEC	Via `scpn_quantum_control.mitigation.pec`	Yes	No
CDR	No	Yes	No
Circuit type	Qiskit `QuantumCircuit`	Any (via converters)	Qiskit only
Executor	Default AerSimulator	User-supplied	N/A

Our wrapper provides a single-function interface with sensible defaults. For advanced use (custom noise models, CDR, benchmarking), use Mitiq directly.

References¶

LaRose, R. et al. "Mitiq: A software package for error mitigation on noisy quantum computers." Quantum 6, 774 (2022).
Temme, K., Bravyi, S. & Gambetta, J. M. "Error mitigation for short-depth quantum circuits." PRL 119, 180509 (2017). (ZNE)
Viola, L., Knill, E. & Lloyd, S. "Dynamical decoupling of open quantum systems." PRL 82, 2417 (1999). (DD theory)