Numerical Implementation of Just-In-Time Decoding in Novel Lattice Slices Through the Three-Dimensional Surface Code

We build on recent work by B. Brown (Sci. Adv. 6, eaay4929 (2020)) to develop and simulate an explicit recipe for a just-in-time decoding scheme in three 3D surface codes, which can be used to implement a transversal (non-Clifford) $\overline{CCZ}$ between three 2D surface codes in time linear in the code distance. We present a fully detailed set of bounded-height lattice slices through the 3D codes which retain the code distance and measurement-error detecting properties of the full 3D code and admit a dimension-jumping process which expands from/collapses to 2D surface codes supported on the boundaries of each slice. At each timestep of the procedure the slices agree on a common set of overlapping qubits on which $CCZ$ should be applied. We use these slices to simulate the performance of a simple JIT decoder against stochastic $X$ and measurement errors and find evidence for a threshold $p_c \sim 0.1\%$ in all three codes. We expect that this threshold could be improved by optimisation of the decoder.Comment: 19 pages, 11 figures. Additional supplementary materials at https://github.com/tRowans/JIT-supplementary-materials. v2; removed some claims regarding issues with staircase slices and changed one referenc

Integrated care for migraine and chronic tension-type headaches: A prospective observational study

© 2019 Background and purpose: This prospective observational study aimed to investigate the effects of an interdisciplinary multimodal integrated care program in patients with chronic migraine and/or tension-type headache. Materials and methods: Patients (n = 158) underwent inpatient, outpatient and/or semi-stationary treatment including conventional as well as complementary headache treatment. Headache frequency was defined as the primary outcome; secondary outcomes included pain (VAS, PPS), medication use, quality of life (SF-36), function (HDI, PSFS), depression and anxiety (HADS), and pain self-efficacy (PSEQ). Results: Headache frequency decreased from 17.0 ± 8.8 days/month at treatment start to 11.4 ± 9.2 at treatment end and to 10.6 ± 9.3 at 6-month follow-up (p < 0.001). All other outcome measures also improved across the course of the study (all p < 0.001). Conclusions: An integrated care approach based on integrative medicine improved headache symptoms and functioning in patients with chronic migraine and/or tension-type headache. Interdisciplinary multimodal treatment approaches seem to adequately address the specific treatment needs of headache patients