Error Modelling on Qiskit

Updated 4 December 2025

Error modelling on Qiskit is a systematic framework that simulates and mitigates quantum noise using Kraus maps and Lindblad equations.
It integrates device-calibrated parameters for both circuit-level and pulse-level simulations to support robust quantum algorithm validation.
Practical workflows include error mitigation methods such as zero-noise extrapolation, measurement calibration, and dynamical decoupling.

Error modeling on Qiskit refers to the systematic representation, simulation, and mitigation of physical noise processes that degrade quantum computations in both circuit-level and pulse-level contexts. Qiskit’s open architecture enables implementation of Kraus-operator channels, Lindblad master equations, device-calibrated noise models, and error-mitigation protocols, supporting research in quantum information, device calibration, quantum simulation, and hybrid qubit-bosonic systems.

1. Fundamental Noise Channels and Representations

Qiskit models errors using Kraus maps and Lindblad equations for qubits and bosonic modes. Common error channels for qubits include:

Depolarizing channel with error rate $p$ : $\rho \to (1-p)\rho + \frac{p}{3}(\sigma_x\rho\sigma_x + \sigma_y\rho\sigma_y + \sigma_z\rho\sigma_z)$ .
Amplitude damping (T₁): $E_0 = \begin{pmatrix}1&0\0&\sqrt{1-\gamma}\end{pmatrix}$, $E_1 = \begin{pmatrix}0&\sqrt{\gamma}\0&0\end{pmatrix}$, $\gamma=1-e^{-\Delta t/T_1}$ .
Phase damping (T₂): $E_0 = \sqrt{1-\gamma_\phi} I$ , $E_1 = \sqrt{\gamma_\phi} \sigma_z$ , $\gamma_\phi=1-e^{-\Delta t/T_\phi}$ .
Readout errors: modeled by classical confusion matrices.

For bosonic channels, Qiskit supports photon-loss under Lindblad dynamics:

$\dot{\rho} = \kappa(a\,\rho\,a^\dagger - \frac{1}{2}\{a^\dagger a, \rho\}),$

with Kraus expansion using Fock cutoff $N_{\rm cut}$ (Stavenger et al., 2022).

2. Device-Informed Error Models and Calibration

Device-specific error models are constructed using experimental parameters (gate fidelities, relaxation times, readout matrices) obtained from IBM Qiskit backends. This includes:

Depolarizing error rates: $p_1$ (single-qubit), $p_2$ (two-qubit) extracted from hardware properties, e.g., $p_1=2.2\times10^{-4}$ , $p_2=8.0\times10^{-3}$ (Mirzakhani et al., 5 Jun 2025).
Thermal relaxation: $T_1$ , $T_2$ , and gate durations.
Crosstalk and leakage: off-diagonal Hamiltonian terms and explicit modeling of multi-level systems (Roy et al., 2022).
Control-pulse distortions: transfer functions of AWG, mixers, and filters parameterized in the classical model (Roy et al., 2022).
Bosonic channels: cavity decay rate $\kappa$ , photon-loss coupling.

Parameters are mapped into the simulation’s global vector $\theta$ , which is then refined via closed-loop calibration using measured data and optimization of mismatch objectives such as log-likelihood losses $E(\theta)$ (Roy et al., 2022).

3. Implementation in Qiskit: Circuit and Pulse Level

Qiskit provides tools for error modeling at both the abstract circuit and concrete pulse levels:

Circuit-level simulation: Qiskit Aer’s NoiseModel allows attachment of quantum errors to gate names, durations, or idle times; also supports readout error and custom noise channels (Stavenger et al., 2022, Mirzakhani et al., 5 Jun 2025, Roy et al., 12 Jan 2024).
Pulse-level control: Qiskit Pulse enables scheduling explicit waveforms, implementing advanced mitigation (echoing, dynamical decoupling), performing Hamiltonian characterization, and resolving measurement errors at IQ level (Alexander et al., 2020).
Hybrid systems: Bosonic-Qiskit extends the AMM for qubit/bosonic interactions, supporting error channels for both discrete qubits and continuous-variable modes, requiring careful management of Fock space truncation and photon-loss boundaries (Stavenger et al., 2022).

Example: Building and attaching a thermal relaxation channel in Qiskit Aer:

from qiskit.providers.aer.noise import NoiseModel, thermal_relaxation_error
T1, T2, dt = 50e-6, 70e-6, 50e-9
noise_model = NoiseModel()
amp_err = thermal_relaxation_error(T1, T2, dt)
noise_model.add_quantum_error(amp_err, ['u3'], [0])

Example: Bosonic channel attachment:

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from qiskit.quantum_info import Kraus
bos_loss = Kraus(E_ops)
noise_model.add_quantum_error(bos_loss, ['cv_d'], [0])

(Stavenger et al., 2022)

4. Error Mitigation and Calibration Procedures

Qiskit integrates a suite of error-mitigation strategies at both simulation and hardware levels:

Zero-noise extrapolation (ZNE): Circuit-folding using Mitiq scales gates (e.g. $G \to G G^\dagger G$ ) across multiple factors; extrapolate expectation values to zero noise using Richardson or linear factories (Roy et al., 12 Jan 2024).
Measurement error mitigation: Calibration matrices constructed from dedicated calibration circuits are inverted to recover unbiased probabilities, using complete_meas_cal and MeasurementFilter (Mirzakhani et al., 5 Jun 2025).
Dynamical decoupling: Insertion of XY4 or similar sequences to suppress T₂ dephasing, via apply_dynamic_decoupling (Biswas et al., 14 Apr 2025).
Gate twirling/randomized compiling: Converts coherent errors to stochastic errors via Pauli randomization, implemented in Qiskit Ignis (Biswas et al., 14 Apr 2025).
Pulse-level extrapolation: Varying pulse amplitude/duration and extrapolating outcomes to zero-error regime (Alexander et al., 2020).

Mitigation performance is quantified by improvement in fidelity, expectation values, singlet yields, and reduction of deviation from theoretical models (e.g., error reduction by $3-5 \times$ in open quantum system simulation using ZNE) (Roy et al., 12 Jan 2024).

5. Analysis of Error Impact in Quantum Algorithms

Comprehensive studies using custom noise models reveal algorithm-specific error scaling:

IQPE and adiabatic simulations: Two-qubit depolarizing errors ( $p_2$ ) dominate infidelity, circuit depth leads to linear error accumulation, and energy estimates degrade with increasing $p_2$ and gate time (Mirzakhani et al., 5 Jun 2025).
Hybrid qubit-bosonic algorithms: Bosonic errors (photon loss, dephasing) require large Fock cutoffs in simulations; physical cavity lifetimes are orders of magnitude longer than pulse durations (Stavenger et al., 2022).
Radical pair magnetoreception models: Thermal relaxation channel with $T_1=50$ ms, $T_2=30$ ms and error mitigation (DD, twirling) preserves singlet yields and system purity (Biswas et al., 14 Apr 2025).
Benchmark algorithms (QFT, adder): Memory errors (T₁, T₂) dominate fidelity loss at scale, gate miscalibration parameters have moderate impact, initialization and measurement errors contribute at endpoints (Chaudhary et al., 2019).

Summary of hardware parameters and their typical effect: | Parameter | Typical Value (IBM) | Dominant Impact | |----------------|-------------------------|-------------------------------| | $p_1$ | $2.2 \times 10^{-4}$ | Minor (single-qubit errors) | | $p_2$ | $8.0 \times 10^{-3}$ | Major (two-qubit errors) | | $T_1$ | $300~\mu$ s | Energy relaxation | | $T_2$ | $160~\mu$ s | Dephasing memory | | Readout Error | $2\%$ | Generally negligible in GSE | (Mirzakhani et al., 5 Jun 2025)

6. Integration, Computational Scaling, and Limitations

Qiskit’s simulators (AerSimulator, custom provider backends such as qiskit-aakash) operate efficiently up to $\sim$ 12–15 qubits using density-matrix or Pauli-basis expansions (storage $4^n$ , classical effort per operation $O(4^n)$ ). Hybrid bosonic-qubit simulations become costly due to Fock space scaling $N_{\rm cut}^2$ . The digital twin approach in C³ enables differentiable high-fidelity device modeling and closed-loop calibration via Qiskit, augmented by TensorFlow-based gradient optimization and automatic parameter mapping (Roy et al., 2022).

Mitigation routines and error-model construction are hardware-aware, permitting validation against experimental runs. Practical guidelines recommend regular device calibration, small benchmark circuits for validation, and careful tuning of cutoff and noise parameters (Mirzakhani et al., 5 Jun 2025, Stavenger et al., 2022).

7. Practical Workflow and Reproducibility

Reproducible error modeling workflow in Qiskit involves:

Deriving device parameters from backend properties.
Building a Kraus/Lindblad-informed NoiseModel.
Attaching channels to gates, durations, or pulse instructions explicitly.
Transpiling circuits to match hardware connectivity and native gate sets.
Running noisy simulations and hardware jobs, applying mitigation (ZNE, DD, twirling).
Quantitatively comparing mitigated and raw outputs against theoretical predictions.
Refining parameters through closed-loop optimization, e.g., digital twin calibration.

This integrated workflow enables robust assessment and mitigation of quantum errors, supporting algorithm development, quantum device characterization, and benchmarking on NISQ hardware (Roy et al., 12 Jan 2024, Roy et al., 2022).