bibkey: hastings2017smallmajorana
authors: Mathew B. Hastings
year: 2017
title: "Small Majorana fermion codes"
doi: 10.26421/QIC17.13-14-7
url: "https://arxiv.org/abs/1703.00612v1"
claim: "Majorana fermion codes are self-orthogonal binary codes; with the fermion parity in the stabilizer group, no code on at most 12 modes has distance 4 and a logical qubit, codes on 16 modes do, and the paper believes that 16 is the minimum."
strata_touched:
  - D5/S3/Quantum/Codes/MajoranaFourteenModeDistanceFour
license: citation-only
triage: anchor

Hastings, Small Majorana fermion codes

M. B. Hastings, Quantum Inf. Comput. 17(13–14), 1191–1205 (2017); arXiv:1703.00612v1. Quotations are from the v1 TeX source.

A Majorana fermion code on $n_{\mathrm{phys}}$ modes corresponds to a self-orthogonal subspace of ${\mathbb{F}}_2^{n_{\mathrm{phys}}}$ (bit strings of the stabilizers), with $n_{\log }=n_{\mathrm{phys}}/2-n_{\mathrm{stab}}$ logical qubits; the distance is the minimum weight of a product of Majorana operators that commutes with every stabilizer but is not one. The paper restricts to codes whose stabilizer group contains the fermion parity. It proves that no such code with $n_{\mathrm{phys}}\le 12$ has distance 4 and $n_{\log }>0$, constructs codes with $n_{\mathrm{phys}}=16$, and states:

For $n_{\mathrm{phys}}=14$, we performed a numerical search (described in the next section) for a code with $n_{\mathrm{stab}}=6$ and did not succeed. Hence, we believe that no such code exists, i.e., we believe that $n_{\mathrm{phys}}=16$ is the minimum number of modes to have $d=4,n_{\log }>0$.

Verified locator

Source: https://arxiv.org/abs/1703.00612v1

• DOI: https://doi.org/10.26421/QIC17.13-14-7 (Quantum Information and Computation 17(13–14), 1191–1205; title, author, journal, volume, issue and pages from Crossref).
• URL: https://arxiv.org/abs/1703.00612v1. The e-print member c.tex has 510 lines and md5 fa535fa72b2e45da4d6ea551fe901001. Locations:
  • codes as self-orthogonal subspaces, l. 86–97; $n_{\log }$, l. 99–104; logical operators and distance, l. 107–113; the parity convention, l. 115–117;
  • the weight-2 reduction, l. 131–139; the belief, l. 272–273.