The XYZ$^2$ hexagonal stabilizer code

• URL: http://arxiv.org/abs/2112.06036v2
• Date: Mon, 25 Apr 2022 08:57:01 GMT
• Title: The XYZ$^2$ hexagonal stabilizer code
• Authors: Basudha Srivastava, Anton Frisk Kockum, Mats Granath
• Abstract summary: The "XYZ$2$" code is a simple realization of a "matching code" discussed by Wootton. The code possesses distinctive syndrome properties with unidirectional pairs of plaquette defects along the three directions of the triangular lattice.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We consider a topological stabilizer code on a honeycomb grid, the "XYZ$^2$" code. The code is inspired by the Kitaev honeycomb model and is a simple realization of a "matching code" discussed by Wootton [J. Phys. A: Math. Theor. 48, 215302 (2015)], with a specific implementation of the boundary. It utilizes weight-six ($XYZXYZ$) plaquette stabilizers and weight-two ($XX$) link stabilizers on a planar hexagonal grid composed of $2d^2$ qubits for code distance $d$, with weight-three stabilizers at the boundary, stabilizing one logical qubit. We study the properties of the code using maximum-likelihood decoding, assuming perfect stabilizer measurements. For pure $X$, $Y$, or $Z$ noise, we can solve for the logical failure rate analytically, giving a threshold of 50%. In contrast to the rotated surface code and the XZZX code, which have code distance $d^2$ only for pure $Y$ noise, here the code distance is $2d^2$ for both pure $Z$ and pure $Y$ noise. Thresholds for noise with finite $Z$ bias are similar to the XZZX code, but with markedly lower sub-threshold logical failure rates. The code possesses distinctive syndrome properties with unidirectional pairs of plaquette defects along the three directions of the triangular lattice for isolated errors, which may be useful for efficient matching-based or other approximate decoding.

Related papers

• Far from Perfect: Quantum Error Correction with (Hyperinvariant) Evenbly Codes [38.729065908701585]
  We introduce a new class of qubit codes that we call Evenbly codes. Our work indicates that Evenbly codes may show promise for practical quantum computing applications.
  arXiv  Detail & Related papers  (2024-07-16T17:18:13Z)
• Exact results on finite size corrections for surface codes tailored to biased noise [0.0]
  We study the XY and XZZX surface codes under phase-biased noise. We find exact solutions at a special disordered point. We show that calculating thresholds based not only on the total logical failure rate, but also independently on the phase- and bit-flip logical failure rates, gives a more confident estimate.
  arXiv  Detail & Related papers  (2024-01-08T16:38:56Z)
• Logical Error Rates of XZZX and Rotated Quantum Surface Codes [9.69910104594168]
  We present theoretical formulas based on recent advancements in understanding the weight distribution of stabilizer codes. We observe that the logical error rate approaches $p_mathrmL to 10 p2$ for the rotated $[9,1,3]]$ XZZX code and $p_mathrmL to 18.3 p2$ for the $[13,1,3]]$ surface code. Our findings demonstrate that implementing both rotation and XZZX modifications simultaneously can lead to suboptimal performance.
  arXiv  Detail & Related papers  (2023-12-28T15:09:48Z)
• Performance Analysis of Quantum CSS Error-Correcting Codes via MacWilliams Identities [9.69910104594168]
  We analyze the performance of stabilizer codes, one of the most important classes for practical implementations. We introduce a novel approach that combines the knowledge of WE with a logical operator analysis. For larger codes our bound provides $rho_mathrmL approx 1215 rho4$ and $rho_mathrmL approx 663 rho5$ for the $[85,1,7]]$ and the $[181,1,10]]$ surface codes.
  arXiv  Detail & Related papers  (2023-05-02T10:19:02Z)
• Near-optimal fitting of ellipsoids to random points [68.12685213894112]
  A basic problem of fitting an ellipsoid to random points has connections to low-rank matrix decompositions, independent component analysis, and principal component analysis. We resolve this conjecture up to logarithmic factors by constructing a fitting ellipsoid for some $n = Omega(, d2/mathrmpolylog(d),)$. Our proof demonstrates feasibility of the least squares construction of Saunderson et al. using a convenient decomposition of a certain non-standard random matrix.
  arXiv  Detail & Related papers  (2022-08-19T18:00:34Z)
• Variance-Aware Sparse Linear Bandits [64.70681598741417]
  Worst-case minimax regret for sparse linear bandits is $widetildeThetaleft(sqrtdTright)$. In the benign setting where there is no noise and the action set is the unit sphere, one can use divide-and-conquer to achieve an $widetildemathcal O(1)$ regret. We develop a general framework that converts any variance-aware linear bandit algorithm to a variance-aware algorithm for sparse linear bandits.
  arXiv  Detail & Related papers  (2022-05-26T15:55:44Z)
• Tailored XZZX codes for biased noise [60.12487959001671]
  We study a family of codes having XZZX-type stabilizer generators. We show that these XZZX codes are highly qubit efficient if tailored to biased noise.
  arXiv  Detail & Related papers  (2022-03-30T17:26:31Z)
• Near-Optimal Regret Bounds for Contextual Combinatorial Semi-Bandits with Linear Payoff Functions [53.77572276969548]
  We show that the C$2$UCB algorithm has the optimal regret bound $tildeO(dsqrtkT + dk)$ for the partition matroid constraints. For general constraints, we propose an algorithm that modifies the reward estimates of arms in the C$2$UCB algorithm.
  arXiv  Detail & Related papers  (2021-01-20T04:29:18Z)
• Mitigating Coherent Noise by Balancing Weight-2 $Z$-Stabilizers [2.4851820343103035]
  Physical platforms such as trapped ions suffer from coherent noise where errors manifest as rotations about a particular axis and can accumulate over time. We investigate passive mitigation through decoherence free subspaces, requiring the noise to preserve the code space of a stabilizer code. By adjusting the size of these components, we are able to construct a large family of QECC codes, oblivious to coherent noise.
  arXiv  Detail & Related papers  (2020-10-31T06:09:40Z)
• Optimal Robust Linear Regression in Nearly Linear Time [97.11565882347772]
  We study the problem of high-dimensional robust linear regression where a learner is given access to $n$ samples from the generative model $Y = langle X,w* rangle + epsilon$ We propose estimators for this problem under two settings: (i) $X$ is L4-L2 hypercontractive, $mathbbE [XXtop]$ has bounded condition number and $epsilon$ has bounded variance and (ii) $X$ is sub-Gaussian with identity second moment and $epsilon$ is
  arXiv  Detail & Related papers  (2020-07-16T06:44:44Z)
• Curse of Dimensionality on Randomized Smoothing for Certifiable Robustness [151.67113334248464]
  We show that extending the smoothing technique to defend against other attack models can be challenging. We present experimental results on CIFAR to validate our theory.
  arXiv  Detail & Related papers  (2020-02-08T22:02:14Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.