The Generalized Superfast Encoding (GSE) is a fermion-to-qubit mapping: a way to represent electrons, which are antisymmetric under exchange, using qubits, which are not. In the standard choice, the Jordan-Wigner transformation, a hop between two modes becomes a Pauli operator whose weight grows with the distance between them. GSE spends extra qubits to remove both costs. Operators act only on the qubits of the modes they connect, so their weight is set by the molecule's interaction graph. The loops of that graph supply stabilizers for error detection and correction making the GSE a stabilizer code.
This tutorial builds the encoding from first principles and constructs two small codes — a ring and a three-mode, six-qubit code — reading off their stabilizers, code distance, and how the checks detect errors. It is a single, self-contained notebook.
- Fermions and mappings. The companion tutorial Introduction to Chemistry and Mappings covers the molecular Hamiltonian, second quantization, and the Jordan-Wigner transformation. This notebook picks up where that one ends and does not re-teach the chemistry.
- The stabilizer formalism. Familiarity with Pauli operators, stabilizers, and syndromes at the level of the Fault-Tolerant Syndrome Extraction tutorial. GSE is a stabilizer code, and the notebook uses that language throughout.
Use the Launch on qBraid button on this tutorial's page in the qBraid Explore hub. It clones the repository into qBraid Lab. The notebook runs with the pre-configured qBraid-SDK (v0.12.0) environment; once it's ready, select it as the notebook kernel and run the notebook from top to bottom. For selecting environments and kernels in qBraid Lab, see the environments guide.
The default qBraid-SDK environment ships with Qiskit and NumPy; the notebook's setup cell installs Qiskit Nature automatically if it isn't already present. The notebook also begins with a commented optional %pip install cell listing what it needs for other setups.
- Qiskit and Qiskit Nature — Pauli algebra and the Jordan-Wigner mapping of fermionic operators.
- NumPy — the codespace and code-distance calculations.
- Run the notebook from top to bottom, in order. If the kernel state becomes inconsistent, use Kernel > Restart Kernel and Run All Cells.
- The whole notebook runs in a few seconds on a laptop-class machine; there is no hardware submission and no cost.
[1] K. Setia, S. Bravyi, A. Mezzacapo, J. D. Whitfield, "Superfast encodings for fermionic quantum simulation," Phys. Rev. Research 1, 033033 (2019). arXiv:1810.05274.
[2] J. Brown, T. S. Hardikar, K. Heitritter, K. Setia, "Efficient and Noise-Resilient Molecular Quantum Simulation with the Generalized Superfast Encoding," arXiv:2511.09322 (2025).
[3] J. Brown, J. Iaconis, Y. Alexeev, L. Joseph, S. Churchill, K. Heitritter, W. Aguilar-Calvo, M. Roetteler, M. Suchara, "Mid-Circuit Measurements for Clifford Noise Reduction in Hamiltonian Simulations," arXiv:2605.06792 (2026).