Coherent error threshold for surface codes from Majorana delocalization
- URL: http://arxiv.org/abs/2211.00655v2
- Date: Thu, 29 Jun 2023 17:57:59 GMT
- Title: Coherent error threshold for surface codes from Majorana delocalization
- Authors: Florian Venn, Jan Behrends, Benjamin B\'eri
- Abstract summary: Existing mappings assume incoherent noise, thus ignoring coherent errors due to spurious gate rotations.
We map the surface code with coherent errors, taken as $X$- or $Z$-rotations (trivial bit or phase), to a two-dimensional (2D) Ising model with complex couplings, and further to a 2D Majorana scattering network.
For both, the error-correcting phase maps explicitly show by linking 2D networks to 1D fermions, to a $mathbbZ$-trivial 2D insulator.
- Score: 0.0
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: Statistical mechanics mappings provide key insights on quantum error
correction. However, existing mappings assume incoherent noise, thus ignoring
coherent errors due to, e.g., spurious gate rotations. We map the surface code
with coherent errors, taken as $X$- or $Z$-rotations (replacing bit or phase
flips), to a two-dimensional (2D) Ising model with complex couplings, and
further to a 2D Majorana scattering network. Our mappings reveal both
commonalities and qualitative differences in correcting coherent and incoherent
errors. For both, the error-correcting phase maps, as we explicitly show by
linking 2D networks to 1D fermions, to a $\mathbb{Z}_2$-nontrivial 2D
insulator. However, beyond a rotation angle $\phi_\text{th}$, instead of a
$\mathbb{Z}_2$-trivial insulator as for incoherent errors, coherent errors map
to a Majorana metal. This $\phi_\text{th}$ is the theoretically achievable
storage threshold. We numerically find $\phi_\text{th}\approx0.14\pi$. The
corresponding bit-flip rate $\sin^2(\phi_\text{th})\approx 0.18$ exceeds the
known incoherent threshold $p_\text{th}\approx0.11$.
Related papers
- Provably learning a multi-head attention layer [55.2904547651831]
Multi-head attention layer is one of the key components of the transformer architecture that sets it apart from traditional feed-forward models.
In this work, we initiate the study of provably learning a multi-head attention layer from random examples.
We prove computational lower bounds showing that in the worst case, exponential dependence on $m$ is unavoidable.
arXiv Detail & Related papers (2024-02-06T15:39:09Z) - Interplay between depth and width for interpolation in neural ODEs [0.0]
We examine the interplay between their width $p$ and number of layer transitions $L$.
In the high-dimensional setting, we demonstrate that $p=O(N)$ neurons are likely sufficient to achieve exact control.
arXiv Detail & Related papers (2024-01-18T11:32:50Z) - 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 show that independently estimating thresholds for the $X_L$ (phase-flip), $Y_L$, and $Z_L$ (bit-flip) logical failure rates can give a more confident threshold estimate.
arXiv Detail & Related papers (2024-01-08T16:38:56Z) - Precision of quantum simulation of all-to-all coupling in a local
architecture [0.0]
We find an analytic relation between the values $J_ij$ of the desired interaction and the parameters of the 2d circuit.
For the relative error to be a constant $epsilon$, one requires an energy scale growing as $n6$ in the number of qubits.
Our proof is based on the Schrieffer-Wolff transformation and generalizes to any hardware.
arXiv Detail & Related papers (2023-02-05T18:54:28Z) - Surface codes, quantum circuits, and entanglement phases [0.0]
We map 2D surface codes under a class of incoherent or coherent errors.
We find a topologically non-trivial threshold for incoherent errors and logarithmic coherent error.
Results can be generalized to other fermionic circuits and may be independent interest.
arXiv Detail & Related papers (2022-12-15T19:00:02Z) - Cryptographic Hardness of Learning Halfspaces with Massart Noise [59.8587499110224]
We study the complexity of PAC learning halfspaces in the presence of Massart noise.
We show that no-time Massart halfspace learners can achieve error better than $Omega(eta)$, even if the optimal 0-1 error is small.
arXiv Detail & Related papers (2022-07-28T17:50:53Z) - Mean Estimation in High-Dimensional Binary Markov Gaussian Mixture
Models [12.746888269949407]
We consider a high-dimensional mean estimation problem over a binary hidden Markov model.
We establish a nearly minimax optimal (up to logarithmic factors) estimation error rate, as a function of $|theta_*|,delta,d,n$.
arXiv Detail & Related papers (2022-06-06T09:34:04Z) - Beyond the Berry Phase: Extrinsic Geometry of Quantum States [77.34726150561087]
We show how all properties of a quantum manifold of states are fully described by a gauge-invariant Bargmann.
We show how our results have immediate applications to the modern theory of polarization.
arXiv Detail & Related papers (2022-05-30T18:01:34Z) - Threshold Phenomena in Learning Halfspaces with Massart Noise [56.01192577666607]
We study the problem of PAC learning halfspaces on $mathbbRd$ with Massart noise under Gaussian marginals.
Our results qualitatively characterize the complexity of learning halfspaces in the Massart model.
arXiv Detail & Related papers (2021-08-19T16:16:48Z) - Hardness of Learning Halfspaces with Massart Noise [56.98280399449707]
We study the complexity of PAC learning halfspaces in the presence of Massart (bounded) noise.
We show that there an exponential gap between the information-theoretically optimal error and the best error that can be achieved by a SQ algorithm.
arXiv Detail & Related papers (2020-12-17T16:43:11Z) - 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.