A Variant of the Bravyi-Terhal Bound for Arbitrary Boundary Conditions

• URL: http://arxiv.org/abs/2502.04995v2
• Date: Mon, 10 Feb 2025 13:23:04 GMT
• Title: A Variant of the Bravyi-Terhal Bound for Arbitrary Boundary Conditions
• Authors: François Arnault, Philippe Gaborit, Wouter Rozendaal, Nicolas Saussay, Gilles Zémor,
• Abstract summary: We consider a quotient $mathbbZD/Lambda$ of $mathbbZD$ of cardinality $n$ on a $D$-dimensional lattice quotient. We prove that if all stabilizer generators act on qubits whose indices lie within a ball of radius $rho$, the minimum distance $d$ of the code satisfies $d leq msqrtgamma_D(sqrtD + 4rho)nfracD-1D$ whenever $n1/D
• Score: 10.560637835517094
• License:
• Abstract: We present a modified version of the Bravyi-Terhal bound that applies to quantum codes defined by local parity-check constraints on a $D$-dimensional lattice quotient. Specifically, we consider a quotient $\mathbb{Z}^D/\Lambda$ of $\mathbb{Z}^D$ of cardinality $n$, where $\Lambda$ is some $D$-dimensional sublattice of $\mathbb{Z}^D$: we suppose that every vertex of this quotient indexes $m$ qubits of a stabilizer code $C$, which therefore has length $nm$. We prove that if all stabilizer generators act on qubits whose indices lie within a ball of radius $\rho$, then the minimum distance $d$ of the code satisfies $d \leq m\sqrt{\gamma_D}(\sqrt{D} + 4\rho)n^\frac{D-1}{D}$ whenever $n^{1/D} \geq 8\rho\sqrt{\gamma_D}$, where $\gamma_D$ is the $D$-dimensional Hermite constant. We apply this bound to derive an upper bound on the minimum distance of Abelian Two-Block Group Algebra (2BGA) codes whose parity-check matrices have the form $[\mathbf{A} \, \vert \, \mathbf{B}]$ with each submatrix representing an element of a group algebra over a finite abelian group.

Related papers

• Sparsifying Suprema of Gaussian Processes [6.638504164134713]
  We show that there is an $O_varepsilon(1)$-size subset $S subseteq T$ and a set of real values $c_s_s in S$. We also use our sparsification result for suprema of centered Gaussian processes to give a sparsification lemma for convex sets of bounded geometric width.
  arXiv  Detail & Related papers  (2024-11-22T01:43:58Z)
• The Communication Complexity of Approximating Matrix Rank [50.6867896228563]
  We show that this problem has randomized communication complexity $Omega(frac1kcdot n2log|mathbbF|)$. As an application, we obtain an $Omega(frac1kcdot n2log|mathbbF|)$ space lower bound for any streaming algorithm with $k$ passes.
  arXiv  Detail & Related papers  (2024-10-26T06:21:42Z)
• Subspace Controllability and Clebsch-Gordan Decomposition of Symmetric Quantum Networks [0.0]
  We describe a framework for the controllability analysis of networks of $n$ quantum systems of an arbitrary dimension $d$, it qudits Because of the symmetry, the underlying Hilbert space, $cal H=(mathbbCd)otimes n$, splits into invariant subspaces for the Lie algebra of $S_n$-invariant elements in $u(dn)$, denoted here by $uS_n(dn)$.
  arXiv  Detail & Related papers  (2023-07-24T16:06:01Z)
• 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)
• Non-asymptotic spectral bounds on the $\varepsilon$-entropy of kernel classes [4.178980693837599]
  This topic is an important direction in the modern statistical theory of kernel-based methods. We discuss a number of consequences of our bounds and show that they are substantially tighter than bounds for general kernels.
  arXiv  Detail & Related papers  (2022-04-09T16:45:22Z)
• Uncertainties in Quantum Measurements: A Quantum Tomography [52.77024349608834]
  The observables associated with a quantum system $S$ form a non-commutative algebra $mathcal A_S$. It is assumed that a density matrix $rho$ can be determined from the expectation values of observables. Abelian algebras do not have inner automorphisms, so the measurement apparatus can determine mean values of observables.
  arXiv  Detail & Related papers  (2021-12-14T16:29:53Z)
• Spectral properties of sample covariance matrices arising from random matrices with independent non identically distributed columns [50.053491972003656]
  It was previously shown that the functionals $texttr(AR(z))$, for $R(z) = (frac1nXXT- zI_p)-1$ and $Ain mathcal M_p$ deterministic, have a standard deviation of order $O(|A|_* / sqrt n)$. Here, we show that $|mathbb E[R(z)] - tilde R(z)|_F
  arXiv  Detail & Related papers  (2021-09-06T14:21:43Z)
• Determining when a truncated generalised Reed-Solomon code is Hermitian self-orthogonal [0.7614628596146599]
  We prove that there is a Hermitian self-orthogonal $k$-dimensional truncated generalised Reed-Solomon code of length $n$ over $mathbb F_q2$. We also provide examples of Hermitian self-orthogonal $k$-dimensional Reed-Solomon codes of length $q2+1$ over $mathbb F_q2$, for $k=q-1$ and $q$ an odd power of two.
  arXiv  Detail & Related papers  (2021-06-18T15:16:44Z)
• Linear Bandits on Uniformly Convex Sets [88.3673525964507]
  Linear bandit algorithms yield $tildemathcalO(nsqrtT)$ pseudo-regret bounds on compact convex action sets. Two types of structural assumptions lead to better pseudo-regret bounds.
  arXiv  Detail & Related papers  (2021-03-10T07:33:03Z)
• Tight Quantum Lower Bound for Approximate Counting with Quantum States [49.6558487240078]
  We prove tight lower bounds for the following variant of the counting problem considered by Aaronson, Kothari, Kretschmer, and Thaler ( 2020) The task is to distinguish whether an input set $xsubseteq [n]$ has size either $k$ or $k'=(1+varepsilon)k$.
  arXiv  Detail & Related papers  (2020-02-17T10:53:50Z)

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.