The Kitaev toric code remains a benchmark for fault-tolerant quantum computation, yet standard techniques for increasing its logical dimension—lattice surgery, punctures, or concatenation—incur substantial qubit overhead. I will present a unified construction and analysis framework that alleviates this cost by combining ring-theoretic methods with insights from topological order. Working directly in the polynomial ring theory, we reveal a code’s anyon properties under twisted boundary conditions and its logical dimension without assembling large parity-check matrices.
Applied to the torus geometries, this algebraic approach yields optimal weight-6 LDPC codes such as [[120, 8, 12]], [[186, 10, 14]], [[210, 10, 16]], and [[360, 12, 24]]. Each code stores markedly more logical qubits per physical qubit than the conventional toric code while retaining local, easily measurable stabilizers. The same stabilizers can be transferred to planar layouts with open boundaries by condensing suitable boundary anyons and performing a lattice-grafting optimization that removes redundant boundary qubits. The resulting planar bivariate-bicycle codes—examples include [[78, 6, 6]], [[268, 8, 12]], [[405, 9, 15]], and [[450, 11, 15]]—maintain weight-6 checks and achieve efficiency figures (kd²/n) nearly an order of magnitude higher than the surface code. Their minimal logical operators display truncated Sierpiński-triangle patterns, so the distance scales with the fractal area rather than with the system size in small lattices.
These results demonstrate that a topological and ring-theoretic viewpoint provides a systematic pathway to compact, hardware-friendly quantum LDPC codes, thereby advancing the prospect of near-term, fault-tolerant quantum processors.
References:
1. arXiv:2312.11170 (PRX Quantum 5, 030328 (2024))
2. arXiv:2410.11942
3. arXiv:2503.03827 (PRX Quantum 6, 020357 (2025))
4. arXiv:2503.04699 (PRL 135 (7), 076603 (2025))
5. arXiv:2504.08887 (PRX Quantum Accepted)
