Abstract
We numerically study coherent errors in surface codes on planar graphs, focusing on noise of the form of $Z$- or $X$-rotations of individual qubits. We find that, similarly to the case of incoherent bit- and phase-flips, a trade-off between resilience against coherent $X$- and $Z$-rotations can be made via the connectivity of the graph. However, our results indicate that, unlike in the incoherent case, the error-correction thresholds for the various graphs do not approach a universal bound. We also study the distribution of final states after error correction. We show that graphs fall into three distinct classes, each resulting in qualitatively distinct final-state distributions. In particular, we show that a graph class exists where the logical-level noise exhibits a decoherence threshold slightly above the error-correction threshold. In these classes, therefore, the logical level noise above the error-correction threshold can retain significant amount of coherence even for large-distance codes. To perform our analysis, we develop a Majorana-fermion representation of planar-graph surface codes and describe the characterization of logical-state storage using fermion-linear-optics-based simulations. We thereby generalize the approach introduced for the square lattice by Bravyi \textit{et al}. [npj Quantum Inf. 4, 55 (2018)] to surface codes on general planar graphs.
Highlights
In recent years, significant progress has been made to improve the coherence times of qubits [1,2,3], including demonstrations of key ingredients for quantum error correction (QEC) [4,5,6,7,8]
We described how the C4 encoding of qubits can be used to obtain a Majorana-fermion representation of surface codes on arbitrary planar graphs, and we characterized logical-state storage under coherent Z rotations using fermion linear optics (FLO)-based simulations
Comparing ηth to the thresholds ηtPh for the Pauli twirl of the physical-qubit coherent error, we found that while ηth and ηtPh are similar, the inequality ηth ηtPh holds for all considered systems
Summary
Significant progress has been made to improve the coherence times of qubits [1,2,3], including demonstrations of key ingredients for quantum error correction (QEC) [4,5,6,7,8]. The size of the system was not sufficient to establish a threshold but it provided evidence that using the so-called Pauli twirl to approximate coherent errors as incoherent noise on the level of physical qubits underestimates the logical error rate. We describe a general approach for representing surface codes with Majorana fermions on arbitrary planar graphs, including planar lattices, and show how the FLO-based algorithm can be adapted to this case For incoherent errors, it was found [31,32] that by changing the lattice geometry, one can trade off resilience against phase flips for resilience against bit flips. En route to our analysis of the logical-level coherence, we describe a coherent decoder that takes advantage of the deterministic nature of coherent errors
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