Accurate simulations of planar topological codes cannot use cyclic boundaries

Abstract

Cyclic boundaries are used in many branches of physics and mathematics, typically to assist the approximation of a large space. We show that when determining the performance of planar, fault-tolerant, topological quantum error correction, using cyclic boundaries leads to a significant underestimate of the logical error rate. We present cyclic and non-cyclic surface code simulations exhibiting this discrepancy, and analytic formulae precisely reproducing the observed behavior in the limit of low physical error. These asymptotic formulae are then used to prove that the underestimate is exponentially large in the code distance d at any fixed physical error rate p below the threshold error rate p_th.

12 pages, 12 figures, initial referee comments incorporated

References in corpus (6)

• Surface codes: Towards practical large-scale quantum computation
• Fault-tolerant quantum computation with high threshold in two dimensions
• Topological fault-tolerance in cluster state quantum computation
• Quantum computing with nearest neighbor interactions and error rates over 1%
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• Logical Error Rate Scaling of the Toric Code
• A simple decoder for topological codes
• One-dimensional quantum computing with a 'segmented chain' is feasible with today's gate fidelities
• Logic Synthesis for Fault-Tolerant Quantum Computers
• Noise threshold and resource cost of fault-tolerant quantum computing with Majorana fermions in hybrid systems