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\(t\)-design

Alternative names: Quadrature, Cubature, Averaging set.

Description

A code whose codewords are uniformly distributed in a way that is useful for determining averages of polynomials over the code's underlying space \(X\). In that way, the codewords form an approximation of the space. A code is a design on \(X\) of strength \(t\), i.e., a \(t\)-design on \(X\), if the average of any polynomial of degree up to \(t\) over its codewords is equal to the uniform average over all of \(X\).

As such, a design can be used to determine the average of degree-\(\leq t\) polynomials \(p\) over \(X\), \begin{align} \int_{X}\textnormal{d}xp(x)={\textstyle \frac{1}{|D|}}\sum_{x\in D}p(x)~, \tag*{(1)}\end{align} where the integral is over \(X\) (given some measure \(d x\)), while the sum is over the design \(D\subset X\). A weighted design is a design for which each term \(p(x)\) in the above sum must be multiplied by a weight \(w(x)\) in order to be equal to the left-hand side. The most well-known examples of weighted designs are exact Gaussian quadrature or cubature formulas for integration over the reals [1–5], \(X = \mathbb{R}^n\) (with appropriate measure); these tend to be weighted designs.

Fixed-weight codewords of a binary code can form a design on \(X\) being a Johnson space \(J^n_w\), i.e., the space of length-\(n\) binary strings of weight \(w\). Such a design is called a combinatorial design (a.k.a. block design or covering design) [6], which includes Steiner systems as a special case. Designs on the full space of binary strings (Hamming space) are called orthogonal arrays.

More generally, designs exist when \(X\) is \(q\)-ary Hamming space, ordered Hamming space [7,8], \(q\)-Johnson space [9,10] (where they are called subspace designs), a sphere [11] (where they are called spherical designs), or a compact connected two-point homogeneous space [12–14] (the sphere or the real, complex, quaternionic, or octonionic projective spaces [15]).

Complex projective designs are designs on the space of all quantum states [16–18]. Symmetric informationally complete quantum measurements (SIC-POVMs) [16,19] and mutually unbiased bases (MUBs) [20–25] are important examples of such designs. A limit of infinite dimensions yields rigged designs or, more colloquially, continuous-variable (CV) designs [26], which can be used as operator-valued measures for the space of bosonic quantum states (i.e., Schwartz space over the reals).

Designs also exist on groups. Designs on the unitary (projective unitary) group are called strong unitary (unitary) designs [27–30], while \(t\)-designs on the permutation group are called permutation \(t\)-designs [31] (a.k.a. \(t\)-wise independent permutations).

Other notable designs include torus designs [32,33], simplex designs [34–37], Grassmanian designs [38–40], quantum-channel designs [41], and designs on vertex operator algebras (a.k.a. conformal designs) [42]. Existence has been proven for combinatorial designs [43–47], subspace designs [48,49], as well as designs on continuous topological spaces [50–52].

Notes

See books [1,53] for tables of various designs.

Cousins

  • Barnes-Wall (BW) lattice— BW lattices support Grassmanian 6-designs [39].
  • Kerdock code— Kerdock codes correspond to cluster states, and the corresponding Clifford-group automorphisms of this set form a particular group [54] that is a unitary 2-design [55]. As such, cluster states form complex projective 2-designs. These are useful in matrix-vector multiplication [56].
  • Coherent-state constellation code— Coherent-state constellation codes consisting of points from a Gaussian quadrature rule can be concatenated with quantum polar codes to achieve the Gaussian coherent information of the thermal noise channel [57,58].
  • Number-phase code— Pegg-Barnett phase states undergoing Kerr evolution, together with Fock states, form a rigged 2-design for a single mode [26].
  • Bosonic code— Gaussian states, under a particular measure, do not form rigged two-designs [59].
  • Gottesman-Kitaev-Preskill (GKP) code— GKP states on \(n\) modes and their displaced versions for all possible lattices form a rigged 2-design for all \(n\) [60].
  • Haar-random qubit code— Approximating the random projections through \(t\)-designs is necessary in order to make the protocol practical. Replacing with random Clifford gates is especially convenient since the Clifford group forms a unitary 2-design and produces stabilizer codes.
  • Local Haar-random circuit qubit code— Local Haar-random circuits of polynomial depth form approximate unitary designs [61].
  • Qubit stabilizer code— Stabilizer states on \(n\) qubits form complex projective 3-designs [62], while the Clifford group is a unitary 3-design [63,64]. The \([[4,2,2]]\) code obstructs the Clifford group from being a 4-design [65].
  • Kitaev surface code— Unitary \(t\)-designs can be generated via coherent errors, syndrome extraction, and correction [66].
  • Modular-qudit stabilizer code— Stabilizer states on \(n\) prime-dimensional qubits form complex projective 2-designs [62], while the prime-qudit Clifford group is a unitary 2-design [67].
  • Galois-qudit stabilizer code— Stabilizer states on \(n\) Galois qubits form complex projective 2-designs [68].
  • Twisted \(1\)-group code— Twisted unitary \(t\)-groups [69] generalize the idea of unitary \(t\)-groups [70–72], which are subgroups of the unitary group that form unitary \(t\)-designs.

Member of code lists

Primary Hierarchy

Classical Domain
Error-correcting code (ECC)
Block code
Parents
Block code
\(t\)-design
Children
Sharp configuration
Sharp configurations attain a universal bound expressed in terms of the minimal distance, the number of distances between codewords, and the strength of the design formed by the codewords.
Orthogonal array (OA)Perfect binary MDS GRS
Orthogonal arrays are designs on Hamming space \(GF(q)^n\) (a.k.a. the Hamming association scheme) [6,13,73][12; Exam. 1]; see also Ref. [74].
Subspace designCombinatorial design
Subspace designs are designs on a space of fixed-weight \(q\)-ary strings (a.k.a. \(q\)-Johnson association scheme) [73].
Spherical design
Spherical designs are designs on real or complex spheres.

References

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Zoo Code ID: t-designs

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“\(t\)-design”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/t-designs
BibTeX:
@incollection{eczoo_t-designs, title={\(t\)-design}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/t-designs} }
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“\(t\)-design”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/t-designs

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/properties/block/universally_optimal/t-designs.yml.