Graphical algorithms and threshold error rates for the 2d color code

Recent work on fault-tolerant quantum computation making use of topological error correction shows great potential, with the 2d surface code possessing a threshold error rate approaching 1% [1, 2]. However, the 2d surface code requires the use of a complex state distillation procedure to achieve universal quantum computation. The colour code of [3] is a related scheme partially solving the problem, providing a means to perform all Clifford group gates transversally. We review the colour code and its error correcting methodology, discussing one approximate technique based on graph matching. We derive an analytic lower bound to the threshold error rate of 6.25% under error-free syndrome extraction, while numerical simulations indicate it may be as high as 13.3%. Inclusion of faulty syndrome extraction circuits drops the threshold to approximately 0.1%. 1 Introduction The development of quantum error correcting codes in 1995 [4–6] is a major milestone in the journey towards realising a quantum computer that is able to outperform classical computers for large problems. Error correction allows the suppression of decoherence rate during a quantum algorithm, allowing one to perform lengthy calculations such as Shor's algorithm for prime number factorisation [7] with high fidelity results. The threshold theorem [8] states that, provided all gates are constructed with a failure rate below some threshold error rate, arbitrary length quantum computation can be achieved by employing quantum error correction with polylogarithmic overhead. The act of concatenation, the recursive grouping of logical qubits into successively higher level logical qubits, is one method to form codes with a threshold. However, this concatenation procedure creates non-local stabilisers involving an ever increasing number of physical qubits. As such, threshold error rates for codes formed in this manner suffer when one is limited to local interactions in few dimensions. For example, the 7-qubit Steane code has a threshold of p th = 1.85 × 10 −5 [9] when restricted to a 2d lattice with only nearest-neighbour couplings, and the Bacon-Shor code performs similarly, p th = 2.02 × 10 −5 [10]. On the other hand, topological error correcting codes are designed with such locality constraints in mind and hence are particularly well adapted to these architectures. It has been shown that the 2d surface code [11] possesses a threshold error rate approaching 1% [1, 2]. Additionally, use of defect braiding permits for long-range, multi-qubit interactions [12, 13].