Spherical codes and designs

Delsarte, P.; Goethals, J.-M.; Seidel, J.J.

doi:10.1007/bf03187604

Cited by 845 publications

(801 citation statements)

References 18 publications

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“…(where µ is uniform measure on Ω d normalized to have total measure 1) holds for all polynomials f of degree ≤ t ( [10]; [13]; [6, §3.2]). In the present paper we are concerned only with the case…”

Section: Introductionmentioning

confidence: 99%

McLaren’s improved snub cube and other new spherical designs in three dimensions

Hardin

Sloane

1996

Discrete Comput Geom

289

223

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Section: Introductionmentioning

confidence: 99%

McLaren’s improved snub cube and other new spherical designs in three dimensions

Hardin

Sloane

1996

Discrete Comput Geom

289

223

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“…However, the beautiful recent work on the theory of 2-designs [9,10,11,12] provides the means for doing so. Delsarte [13] showed how integrating polynomials on the sphere could be reduced to averaging the integrand on a finite set of points coined spherical designs (the important fact is that one can use the same set of points to evaluate the average of any polynomial -of a fixed degree-). The same idea can be extended to integrals over the entire Hilbert space.…”

Section: Computing Averages In Hilbert Space Using 2-designsmentioning

confidence: 99%

Selective and efficient quantum process tomography

Bendersky¹,

Pastawski

Paz³

2009

Phys. Rev. A

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In this paper we describe in detail and generalize a method for quantum process tomography that was presented in [1]. The method enables the efficient estimation of any element of the χ-matrix of a quantum process. Such elements are estimated as averages over experimental outcomes with a precision that is fixed by the number of repetitions of the experiment. Resources required to implement it scale polynomically with the number of qubits of the system. The estimation of all diagonal elements of the χ-matrix can be efficiently done without any ancillary qubits. In turn, the estimation of all the off-diagonal elements requires an extra clean qubit. The key ideas of the method, that is based on efficient estimation by random sampling over a set of states forming a 2-design, are described in detail. Efficient methods for preparing and detecting such states are explicitly shown.

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“…Proof: of corollary 1. 4 The claim of the corollary is obtained using (5), (6), and (8), and simplifying.…”

Section: A Lower Bound On M Lp (N θ)mentioning

confidence: 99%

On Linear Programming Bounds for Spherical Codes and Designs

Samorodnitsky

2004

Discrete and Computational Geometry

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We investigate universal bounds on spherical codes and spherical designs that could be obtained using Delsarte's linear programming methods. We give a lower estimate for the LP upper bound on codes, and an upper estimate for the LP lower bound on designs.Specifically, when the distance of the code is fixed and the dimension goes to infinity, the LP upper bound on codes is at least as large as the average of the best known upper and lower bounds.When the dimension n of the design is fixed, and the strength k goes to infinity, the LP bound on designs turns out, in conjunction with known lower bounds, to be proportional to k n−1 .

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Spherical codes and designs

Cited by 845 publications

References 18 publications

McLaren’s improved snub cube and other new spherical designs in three dimensions

McLaren’s improved snub cube and other new spherical designs in three dimensions

Selective and efficient quantum process tomography

On Linear Programming Bounds for Spherical Codes and Designs

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