Characterizing Quantum Codes via the Coefficients in Knill-Laflamme Conditions

• URL: http://arxiv.org/abs/2410.07983v1
• Date: Thu, 10 Oct 2024 14:38:15 GMT
• Title: Characterizing Quantum Codes via the Coefficients in Knill-Laflamme Conditions
• Authors: Mengxin Du, Chao Zhang, Yiu-Tung Poon, Bei Zeng,
• Abstract summary: Quantum error correction (QEC) is essential for protecting quantum information against noise. We study the structure of the Knill-Laflamme (KL) coefficients $lambda_ij$ from the condition $PE_idagger E_j P = lambda_ij P$.
• Score: 3.980076328494117
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Quantum error correction (QEC) is essential for protecting quantum information against noise, yet understanding the structure of the Knill-Laflamme (KL) coefficients $\lambda_{ij}$ from the condition $PE_i^\dagger E_j P = \lambda_{ij} P$ remains challenging, particularly for nonadditive codes. In this work, we introduce the signature vector $\vec{\lambda}(P)$, composed of the off-diagonal KL coefficients $\lambda_{ij}$, where each coefficient corresponds to equivalence classes of errors counted only once. We define its Euclidean norm $\lambda^*(P)$ as a scalar measure representing the total strength of error correlations within the code subspace defined by the projector $P$. We parameterize $P$ on a Stiefel manifold and formulate an optimization problem based on the KL conditions to systematically explore possible values of $\lambda^*$. Moreover, we show that, for $((n,K,d))$ codes, $\lambda^*$ is invariant under local unitary transformations. Applying our approach to the $((6, 2, 3))$ quantum code, we find that $\lambda^*_{\text{min}} = \sqrt{0.6}$ and $\lambda^*_{\text{max}} = 1$, with $\lambda^* = 1$ corresponding to a known degenerate stabilizer code. We construct continuous families of new nonadditive codes parameterized by vectors in $\mathbb{R}^5$, with $\lambda^*$ varying over the interval $[\sqrt{0.6}, 1]$. For the $((7, 2, 3))$ code, we identify $\lambda^*_{\text{min}} = 0$ (corresponding to the non-degenerate Steane code) and $\lambda^*_{\text{max}} = \sqrt{7}$ (corresponding to the permutation-invariant code by Pollatsek and Ruskai), and we demonstrate continuous paths connecting these extremes via cyclic codes characterized solely by $\lambda^*$. Our findings provide new insights into the structure of quantum codes, advance the theoretical foundations of QEC, and open new avenues for investigating intricate relationships between code subspaces and error correlations.

Related papers

• A computational transition for detecting correlated stochastic block models by low-degree polynomials [13.396246336911842]
  Detection of correlation in a pair of random graphs is a fundamental statistical and computational problem that has been extensively studied in recent years. We consider a pair of correlated block models $mathcalS(n,tfraclambdan;k,epsilon;s)$ that are subsampled from a common parent block model $mathcal S(n,tfraclambdan;k,epsilon;s)$ We focus on tests based on emphlow-degrees of the entries of the adjacency
  arXiv  Detail & Related papers  (2024-09-02T06:14:05Z)
• Optimal Sketching for Residual Error Estimation for Matrix and Vector Norms [50.15964512954274]
  We study the problem of residual error estimation for matrix and vector norms using a linear sketch. We demonstrate that this gives a substantial advantage empirically, for roughly the same sketch size and accuracy as in previous work. We also show an $Omega(k2/pn1-2/p)$ lower bound for the sparse recovery problem, which is tight up to a $mathrmpoly(log n)$ factor.
  arXiv  Detail & Related papers  (2024-08-16T02:33:07Z)
• Block Circulant Codes with Application to Decentralized Systems [12.014314088945968]
  We develop a family of $[n,k,d]$ block circulant codes that support distributed erasure decoding. The code is ideal for use in protocols that address the data availability problem in blockchain networks.
  arXiv  Detail & Related papers  (2024-06-18T00:22:20Z)
• Hamiltonian simulation for low-energy states with optimal time dependence [45.02537589779136]
  We consider the task of simulating time evolution under a Hamiltonian $H$ within its low-energy subspace. We present a quantum algorithm that uses $O(tsqrtlambdaGamma + sqrtlambda/Gammalog (1/epsilon))$ queries to the block-encoding for any $Gamma$.
  arXiv  Detail & Related papers  (2024-04-04T17:58:01Z)
• Kernelized Normalizing Constant Estimation: Bridging Bayesian Quadrature and Bayesian Optimization [51.533164528799084]
  We show that to estimate the normalizing constant within a small relative error, the level of difficulty depends on the value of $lambda$. We find that this pattern holds true even when the function evaluations are noisy.
  arXiv  Detail & Related papers  (2024-01-11T07:45:09Z)
• Manifold learning in Wasserstein space [2.9581047417235298]
  We build the theoretical foundations for manifold learning algorithms on a compact and convex subset of $mathbbRd$, metrized with the Wasserstein-2 distance $mathrmW$. We show that the metric space $(Lambda,mathrmW_Lambda)$ can beally recovered in the sense of Gromov--Wasserstein from a graph with nodes $lambda_i_i=1N and edge weights $lambda_i,lambda_j.
  arXiv  Detail & Related papers  (2023-11-14T21:21:35Z)
• The Approximate Degree of DNF and CNF Formulas [95.94432031144716]
  For every $delta>0,$ we construct CNF and formulas of size with approximate degree $Omega(n1-delta),$ essentially matching the trivial upper bound of $n. We show that for every $delta>0$, these models require $Omega(n1-delta)$, $Omega(n/4kk2)1-delta$, and $Omega(n/4kk2)1-delta$, respectively.
  arXiv  Detail & Related papers  (2022-09-04T10:01:39Z)
• 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)
• Optimal Spectral Recovery of a Planted Vector in a Subspace [80.02218763267992]
  We study efficient estimation and detection of a planted vector $v$ whose $ell_4$ norm differs from that of a Gaussian vector with the same $ell$ norm. We show that in the regime $n rho gg sqrtN$, any spectral method from a large class (and more generally, any low-degree of the input) fails to detect the planted vector.
  arXiv  Detail & Related papers  (2021-05-31T16:10:49Z)
• Small Covers for Near-Zero Sets of Polynomials and Learning Latent Variable Models [56.98280399449707]
  We show that there exists an $epsilon$-cover for $S$ of cardinality $M = (k/epsilon)O_d(k1/d)$. Building on our structural result, we obtain significantly improved learning algorithms for several fundamental high-dimensional probabilistic models hidden variables.
  arXiv  Detail & Related papers  (2020-12-14T18:14:08Z)
• $\lambda$-Regularized A-Optimal Design and its Approximation by $\lambda$-Regularized Proportional Volume Sampling [1.256413718364189]
  We study the $lambda$-regularized $A$-optimal design problem and introduce the $lambda$-regularized proportional volume sampling algorithm. The problem is motivated from optimal design in ridge regression, where one tries to minimize the expected squared error of the ridge regression predictor from the true coefficient in the underlying linear model.
  arXiv  Detail & Related papers  (2020-06-19T15:17:57Z)

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.