Axel Muller

Metod Saniga

Alain Giorgetti
email: alain.giorgetti@femto-st.fr

Henri De Boutray

Frédéric Holweck

Multi-qubit doilies: Enumeration for all ranks and classification for ranks four and five

For N ≥ 2, an N -qubit doily is a doily living in the N -qubit symplectic polar space. These doilies are related to operator-based proofs of quantum contextuality. Following and extending the strategy of Saniga et al. (Mathematics 9 (2021) 2272) that focused exclusively on three-qubit doilies, we first bring forth several formulas giving the number of both linear and quadratic doilies for any N > 2. Then we present an effective algorithm for the generation of all N -qubit doilies. Using this algorithm for N = 4 and N = 5, we provide a classification of N -qubit doilies in terms of types of observables they feature and number of negative lines they are endowed with. We also list several distinguished findings about N -qubit doilies that are absent in the three-qubit case, point out a couple of specific features exhibited by linear doilies and outline some prospective extensions of our approach.

Introduction

The doily is a remarkable piece of finite geometry that occurs in a number of disguises. Here, we mention the most prominent ones.

1. The doily as a duad-syntheme geometry. Let us recall a famous Sylvester's construction of the doily [START_REF] Sylvester | XLIV. Elementary researches in the analysis of combinatorial aggregation[END_REF]. Given a six-element set M 6 ≡ {1, 2, 3, 4, 5, 6}, a duad is an unordered pair (ij) ∈ M 6 , i = j, and a syntheme is a set of three pairwise disjoint duads, i. e. a set {(ij), (kl), (mn)} where i, j, k, l, m, n ∈ M 6 are all distinct. The point-line incidence structure whose points are duads and whose lines are synthemes, with incidence being inclusion, is isomorphic to the doily, as also illustrated in Figure 1.

Figure 1: A duad-syntheme model of the doily.

2. The doily as the Cremona-Richmond configuration. It is a particular 15 3 -configuration, i. e. a self-dual configuration of 15 points and 15 lines, with three points on a line and, dually, three lines through a point such that it contains no triangles [START_REF] Cremona | Teoremi stereometrici dal quali si deducono le proprietà dell' esagrammo di Pascal (Stereometric theorems from which the properties of Pascal's hexagram are deduced)[END_REF][START_REF] Richmond | The figure formed from six points in space of four dimensions[END_REF]. Up to isomorphism, there are altogether 245,342 15 3 -configurations, of which only the doily enjoys the property of being triangle-free.

3. The doily as a generalized quadrangle. A generalized quadrangle GQ(s, t) of order (s, t) is an incidence structure of points and lines (blocks) where every point is on t + 1 lines (t > 0), and every line contains s + 1 points (s > 0) such that if p is a point and L is a line, p not on L, then there is a unique point q on L such that p and q are collinear. The doily is isomorphic to the unique generalized quadrangle with s = t = 2 [START_REF] Payne | Finite Generalized Quadrangles[END_REF]. [START_REF] Payne | Finite Generalized Quadrangles[END_REF]. The doily as a symplectic polar space. Given a d-dimensional projective space PG(d, 2) over the two-elements field F 2 = {0, 1} of modulo-2 arithmetic, a polar space P in this projective space consists of the projective subspaces that are totally isotropic/singular with respect to a given non-singular bilinear form [START_REF] Hirschfeld | General Galois Geometries[END_REF][START_REF] Cameron | Projective and Polar Spaces[END_REF]; PG(d, 2) is called the ambient projective space of P. A projective subspace of maximal dimension in P is called a generator; all generators have the same (projective) dimension r -1. One calls r the rank of the polar space. The symplectic polar space W(2N -1, 2), N ≥ 1, consists of all the points of PG(2N -1, 2), {(x 1 , x 2 , . . . , x 2N ) : x j ∈ {0, 1}, j ∈ {1, 2, . . . , 2N }}\{(0, 0, . . . , 0)}, together with the totally isotropic subspaces with respect to the standard symplectic form

σ(x, y) = x 1 y N +1 -x N +1 y 1 + x 2 y N +2 -x N +2 y 2 + • • • + x N y 2N -x 2N y N . (1)
Throughout the paper, the space name W(2N -1, 2) is often shortened as W N . This space features |W N | p = 4 N -1 points and

|W N | g = (2 + 1)(2 2 + 1) • • • (2 N + 1)
generators. The doily is isomorphic to the symplectic polar space of rank N = 2, W(3, 2).

5.

Multi-qubit doilies. This paper is about doilies related to Kochen-Specker operator-based proofs of quantum contextuality, to be called N -qubit doilies or multi-qubit doilies. We follow the terminology and notation of Section 2 of [START_REF] Saniga | Taxonomy of Polar Subspaces of Multi-Qubit Symplectic Polar Spaces of Small Rank[END_REF], to which the reader can refer for more finite-geometric background. Let X = 0 1 1 0 , Y = 0 -i i 0 and Z = 1 0 0 -1 be the Pauli matrices, I the identity matrix, '⊗' the tensor product of matrices and I N ≡ I (1) ⊗ I (2) ⊗ . . . ⊗ I (N ) , and let

S N = {G 1 ⊗ G 2 ⊗ • • • ⊗ G N : G j ∈ {I
, X, Y, Z}, j ∈ {1, 2, . . . , N }}\{I N }. The 4 N -1 N -qubit observables of S N can be bijectively identified with the 4 N -1 points of W(2N -1, 2) in such a way that any two commuting observables are represented by collinear points and the product of the three observables lying on a line of W(2N -1, 2) is +I N or -I N (see, for example, [START_REF] Holweck | Geometric constructions over C and F 2 for Quantum Information[END_REF]Section 5.3.2]). If the symplectic form in the ambient space PG(2N -1, 2), defining W(2N -1, 2), is given by Eq. ( 1), then the corresponding bijection reads

G j ↔ (x j , x j+N ), j ∈ {1, 2, . . . , N }, (2)
with the assumption that I ↔ (0, 0), X ↔ (0, 1), Y ↔ (1, 1), and Z ↔ (1, 0).

To briefly illustrate this property, let us consider the three-qubit W(5, 2) and one of its lines, say (0, 1, 1; 1, 1, 0), (1, 0, 0; 0, 0, 1) and (1, 1, 1; 1, 1, 1). Using the correspondences ( 2) and ( 3) we find that the corresponding observables are X ⊗ Y ⊗ Z, Z ⊗ I ⊗ X and Y ⊗ Y ⊗ Y , respectively; these observables indeed pairwise commute and their product is +I ⊗ I ⊗ I.

In what follows, W N will always be understood as having its points labeled by the N -qubit observables as described above, and any doily lying in it, together with the inherited labeling, will be called an N -qubit doily (N ≥ 2). Slightly rephrased, an N -qubit doily is a doily whose points are bijectively identified with 15 specific observables from S N , such that any two commuting observables share the same line, and, given any line, the product of (any) two observables lying on it is, up to a sign, equal to the remaining observable on it. A line of an N -qubit doily will be called positive (resp. negative) if the product of its three observables is +I N (resp. -I N ). To avoid any possible misunderstanding, it is worth mentioning that the product of observables is the (ordinary) matrix product, denoted by a dot (.), induced by the following multiplication table of Pauli matrices.

. X Y Z X I iZ -iY Y -iZ I iX Z iY -iX I
From here on, the geometrical points are considered to be finite words on the four-letter alphabet {I, X, Y, Z} that encode the observables

G 1 ⊗ G 2 ⊗ • • • ⊗ G N ,
while omitting the symbol ⊗ for the tensor product and forgetting in the sequel about the matrix nature of I, X, Y and Z.

Contributions and paper outline. Our contributions start in Section 2, with a presentation of several facts about N -qubit doilies that motivates the design of an effective algorithm to enumerate N -qubit doilies for any rank N (Section 4). By geometric considerations, we first establish in Section 3 closed formulas for the numbers of N -qubit doilies. As these numbers increase rapidly with N , the enumeration algorithm can in practice only be executed for small numbers of qubits. We use it in Section 5 to classify N -qubit doilies for N = 4 and N = 5, according to their types of observables and their configurations of negative lines. We thus produce precise tables for the number of doilies in each category/class, reproduced in the appendices of this paper. Section 5 also analyzes these results and points out various findings about N -qubit doilies that are absent in the known three-qubit case. Section 6 concludes and outlines some prospective extensions of our approach.

2 Some basic facts about multi-qubit doilies

Patterns formed by negative lines

It is a straightforward task to work out possible types of patterns of negative lines an N -qubit doily can be endowed with. This classification follows readily from the facts that each grid in the doily must contain an odd number of negative lines and that two different grids have two intersecting lines in common. And as a grid has an even number of lines the types of configurations come in complementary pairs, as depicted in Figure 2. Let us give a brief description of the individual types of configurations. In Type 3 the three negative lines are pairwise disjoint and lie in a grid; that is, their dual is a tricentric triad. Type 4 features three pairwise disjoint lines not belonging to a grid and their unique transversal. In Type 5 the five negative lines form a pentagon. Type 6 contains the three lines from Type 3 plus three concurrent lines, whose point of concurrence is not lying on any of the three former lines. Type 7A contains six lines forming a hexagon and a unique line disjoint from any of the six. Type 7B is a particular union of two Types 4 and an extra line or, equivalently, is composed of the five lines of a grid and two disjoint lines. A two-qubit doily features just a Type 3 pattern, while in a three-qubit doily we can find all the patterns from Type 3 to Type 7A inclusive [START_REF] Saniga | Taxonomy of Polar Subspaces of Multi-Qubit Symplectic Polar Spaces of Small Rank[END_REF].

Linear and quadratic doilies

Following [START_REF] Saniga | Taxonomy of Polar Subspaces of Multi-Qubit Symplectic Polar Spaces of Small Rank[END_REF], we will also distinguish between two kinds of doilies, referred to as linear and quadratic. A linear N -qubit doily spans a PG(3, 2) of the ambient PG(2N -1, 2). This means that the three lines of a perp-set of such a doily are coplanar, i. e. lie in a PG(2, 2) of the PG(3, 2), a tricentric triad corresponds to a line of the PG(3, 2) and the plane defined by a unicentric triad of the doily passes through its center. Figure 3 serves as a graphical illustration of these features for N = 4. In the doily we selected a perp-set (blue) and colored the remaining lines red. The model of PG [START_REF] Richmond | The figure formed from six points in space of four dimensions[END_REF][START_REF] Cremona | Teoremi stereometrici dal quali si deducono le proprietà dell' esagrammo di Pascal (Stereometric theorems from which the properties of Pascal's hexagram are deduced)[END_REF] is based on a 3-D tetrahedral model of Polster [START_REF] Polster | A Geometrical Picture Book[END_REF]; our version features all the points but not all the lines of the model in order to avoid too crowded appearance of the figure. The two red points at the side lie on the line passing via IY ZI that would be perpendicular to the plane of the drawings. Each black line of the PG(3, 2) is non-isotropic and corresponds to a tricentric triad in the doily. this PG [START_REF] Payne | Finite Generalized Quadrangles[END_REF][START_REF] Cremona | Teoremi stereometrici dal quali si deducono le proprietà dell' esagrammo di Pascal (Stereometric theorems from which the properties of Pascal's hexagram are deduced)[END_REF]. This quadric, as any other parabolic quadric in PG(4, 2), has a remarkable property that all its tangent hyperplanes pass through the same point J, called the nucleus (see, e. g., [START_REF] Hirschfeld | General Galois Geometries[END_REF]). Any tricentric triad of such a doily defines a plane in the PG(4, 2) that contains J; a unicentric triad also defines a plane, this plane passing through the remaining third point lying on the line defined by J and the (unique) center of the triad. Moreover, all the 15 PG(3, 2)s passing through J intersect our quadric in three concurrent lines that form a perp-set of the doily. Figure 4 offers a pictorial illustration of some of these properties. We again take a four-qubit doily, where we highlighted a perp-set (blue). Now the three lines of the perp-set are not coplanar as in the case of linear doily, but span a PG [START_REF] Richmond | The figure formed from six points in space of four dimensions[END_REF][START_REF] Cremona | Teoremi stereometrici dal quali si deducono le proprietà dell' esagrammo di Pascal (Stereometric theorems from which the properties of Pascal's hexagram are deduced)[END_REF]. We colored the remaining eight points (and the totally-isotropic lines) of the PG [START_REF] Richmond | The figure formed from six points in space of four dimensions[END_REF][START_REF] Cremona | Teoremi stereometrici dal quali si deducono le proprietà dell' esagrammo di Pascal (Stereometric theorems from which the properties of Pascal's hexagram are deduced)[END_REF] in yellow in order to stress the property that the only points shared by the doily and this PG [START_REF] Richmond | The figure formed from six points in space of four dimensions[END_REF][START_REF] Cremona | Teoremi stereometrici dal quali si deducono le proprietà dell' esagrammo di Pascal (Stereometric theorems from which the properties of Pascal's hexagram are deduced)[END_REF] are the (blue) points of the perp-set. There are two "distinguished" points of the PG(3, 2), namely ZY XX and ZY IX, which lie on the remaining seventh line passing via IIXI; the point ZY XX is nothing but the nucleus of the parabolic quadric our particular doily is located on. Given a perp-set, we know that there are four tricentric and four unicentric triads contained in it. In our particular perp, the four tri centric triads are {XXIX, XZII, ZIXI}, {XXIX, XZXI, ZIII}, {XZXI, XXXX, ZIXI} and {XXXX, ZIII, XZII}; one can readily check that the product of the three observables in any of them is ZY XX (the nucleus). The four uni centric triads of our perp-set are {XZXI, XXXX, ZIII}, {XZXI, ZIXI, XXIX}, {ZIII, XZII, XXIX} and {XXX, XZII, ZIXI}; the product of the observables in any of them is ZY IX, i. e. the second distinguished point. By this construction we get a (different) PG [START_REF] Richmond | The figure formed from six points in space of four dimensions[END_REF][START_REF] Cremona | Teoremi stereometrici dal quali si deducono le proprietà dell' esagrammo di Pascal (Stereometric theorems from which the properties of Pascal's hexagram are deduced)[END_REF] for any of the 15 perp-sets of the doily; and because in any of these perp-sets the four tricentric triads always define the nucleus, ZY XX, we get altogether 15 PG(3, 2)s that share the point ZY XX, these 15 spaces lying in that particular PG(4, 2) of the ambient PG(7, 2) that contains the quadric of our selected doily. In a recent paper [START_REF] Saniga | Taxonomy of Polar Subspaces of Multi-Qubit Symplectic Polar Spaces of Small Rank[END_REF], four of the authors have thoroughly analyzed and classified three-qubit doilies. To this end, they first explicitly computed all 63 perp-sets, 36 hyperbolic quadrics and 28 elliptic quadrics living in W(5, 2). Then, employing the fact that a linear doily is isomorphic to the intersection of two perp-sets with non-collinear nuclei, they computed and classified all 63 × 32/3! = 336 linear doilies of the W(5, 2). In the next step, making use of the property that a quadratic doily is isomorphic to the intersection of an elliptic quadric and a hyperbolic quadric, they generated and classified all 36×28 = 1 008 quadratic doilies of the W(5, 2). The procedure described above is, however, not a viable one for N > 3, as we would first need to compute all W(5, 2)s living in a particular W(2N -1, 2), N > 3, and then in each of them compute 336 linear and 1 008 quadratic doilies following the strategy of [START_REF] Saniga | Taxonomy of Polar Subspaces of Multi-Qubit Symplectic Polar Spaces of Small Rank[END_REF]. Instead, we shall follow (in Section 4) a different, and reasonably faster, approach that makes use of some properties of an ovoid of a doily. In particular, we shall start with a particular N -qubit ovoid, i. e. a set of five mutually anticommuting N -qubit observables whose product is ±I N , and introduce a unique algebro-geometrical recipe with the help of which one can find all the N -qubit doilies having this particular ovoid in common. Before embarking on this path, however, we shall introduce several general formulas for the number of both linear and quadratic doilies of W(2N -1, 2), valid for any N ≥ 2, so that we already have certain important numbers at hand to validate some of our subsequent, mostly computer-assisted, results.

Contextuality degree

All multi-qubit doilies are observable-based proofs of the Kochen-Specker theorem, that establishes that no Non-Contextual Hidden Variables (NCHV) model can reproduce the outcomes of quantum mechanics. This contextuality property is related to a linear problem, as follows. Let A be the incidence matrix of the points on the lines of a finite geometry, such as the doily. Its coefficients are in the two-elements field F 2 = {0, 1}, its l rows correspond to the geometric lines and its p columns to the geometric points (for the doily, l = p = 15). The positive (resp. negative) nature of a line is encoded by a 0 (resp. 1) for the corresponding coefficient of the valuation vector E in F l 2 . Then a quantum geometry is contextual iff there is no vector x such that Ax = E. The contextuality degree is the minimal Hamming distance between a vector Ax and the vector E [START_REF] De Boutray | Contextuality degree of quadrics in multi-qubit symplectic polar spaces[END_REF]. The contextuality degree is the minimal number of line valuations that one should change to make the quantum geometry satisfiable by an NCHV model. Proposition 1. All multi-qubit doilies have a contextuality degree of 3.

Proof. All multi-qubit doilies have the same incidence matrix A. Accordingly, the only parameter that is changing between all the doilies is the vector E, which only depends on the configuration of their negative lines. We have seen that there are only 12 such configurations. For each of these 12 configurations, we have computed the Hamming distance between Ax and E, for all vectors x in F 15 2 . It turns out that the minimal Hamming distance is always 3.

In practice, we did not write by hand the 12 possible E vectors, but we computed these vectors from the 5-qubit doilies, because, as described later, we have checked by enumeration that these doilies present all the configurations.

Numbers of multi-qubit doilies

This section proposes and justifies closed formulas for the numbers of linear and quadratic doilies in W(2N -1, 2). Before all we introduce some well-known formulas. First, we introduce the Gaussian (binomial) coefficient

n k q = k i=1 q n-k+i -1 q i -1 = (q n -1) . . . (q n-k+1 -1) (q k -1) . . . (q -1) (4)
where 0 ≤ k ≤ n and q is a power of a prime, which gives the number of subspaces of dimension1 k -1 in a projective space PG(n -1, q) of dimension n -1 over F q . More generally, the number of (k -1)-dimensional spaces of PG(n -1, q) that pass through a fixed (l -1)-dimensional space is

n -l k -l q . ( 5
)
Next, for a symplectic polar space W(2N -1, q) embedded in a projective space PG(2N -1, q), the number of its k-dimensional spaces is given by (see, e. g., [START_REF] Boeck | Cameron-Liebler sets of generators in finite classical polar spaces[END_REF]Lemma 2.10])

N k + 1 q k+1 i=1 q N +1-i + 1 (6)
and the number of k-dimensional spaces through a fixed m-dimensional space [START_REF] Boeck | Cameron-Liebler sets of generators in finite classical polar spaces[END_REF]Corollary 2.11] equals

N -m -1 k -m q k-m i=1 q N -m-i + 1 . (7)
Further, let ⊥ be a symplectic polarity of PG(n, q) and let denote by S ⊥ the polar space of a subspace S.

If S is of dimension k, then S ⊥ has dimension n -k -1. A projective subspace S of PG(n, q) is called isotropic if S ∩ S ⊥ = ∅ and non-isotropic if S ∩ S ⊥ = ∅. An isotropic S is called totally isotropic if S ⊆ S ⊥ .
It is easy to see that if S is a totally isotropic subspace, then every subspace contained in S is also totally isotropic. Moreover,

S ⊆ T ⊥ ⇒ T ⊆ S ⊥ . (8)
In order to prove the two theorems below, we will need a couple of lemmas.

Lemma 2. If a PG(3, 2) of the ambient PG(5, 2) equipped with a symplectic polarity ⊥ contains a totally-isotropic PG(2, 2), then it contains exactly three such PG(2, 2)s, passing through a common (totally-isotropic) PG(1, 2).

Proof. First, there are no totally-isotropic PG(3, 2)s in the PG [START_REF] Hirschfeld | General Galois Geometries[END_REF][START_REF] Cremona | Teoremi stereometrici dal quali si deducono le proprietà dell' esagrammo di Pascal (Stereometric theorems from which the properties of Pascal's hexagram are deduced)[END_REF]. Given a totally-isotropic PG(1, 2) of PG(5, 2), S, there are (see Eq. ( 7) for q = 2, N = 3, k = 2 and m = 1) three totallyisotropic PG(2, 2)s passing through it. Denoting these as T ⊥ i (i = 1, 2, 3), the lemma then follows from the fact that S ⊥ ∼ = PG(3, 2), T ⊥ i = T i , and property (8).

Remark 3. For N > 3, PG(2N -1, 2) features also totally-isotropic PG(3, 2)s; any other of its PG(3, 2)s endowed with totally-isotropic PG(2, 2)s has the property as described in Lemma 2.

Lemma 4. If a PG(4, 2) of the ambient PG(7, 2) equipped with a symplectic polarity ⊥ contains a totally-isotropic PG(3, 2), then it contains exactly three such PG(3, 2)s, passing through a common (totally-isotropic) PG(2, 2).

Proof. The proof parallels that of the preceding lemma. First, there are no totally-isotropic PG(4, 2)s in the PG [START_REF] Saniga | Taxonomy of Polar Subspaces of Multi-Qubit Symplectic Polar Spaces of Small Rank[END_REF][START_REF] Cremona | Teoremi stereometrici dal quali si deducono le proprietà dell' esagrammo di Pascal (Stereometric theorems from which the properties of Pascal's hexagram are deduced)[END_REF]. Given a totally-isotropic PG(2, 2) of PG(7, 2), S, there are (see Eq. ( 7) for q = 2, N = 4, k = 3 and m = 2) three totally-isotropic PG(3, 2)s passing through it. Denoting these as T ⊥ i (i = 1, 2, 3), the lemma then follows from the fact that S ⊥ ∼ = PG(4, 2), T ⊥ i = T i , and property [START_REF] Holweck | Geometric constructions over C and F 2 for Quantum Information[END_REF].

Remark 5. For N > 4, PG(2N -1, 2) features also totally-isotropic PG(4, 2)s; any other of its PG(4, 2)s endowed with totally-isotropic PG(3, 2)s has the property as described in Lemma 4.

Next, through a (totally-isotropic) point of PG(5, 2), S, there pass 15 totally-isotropic PG(1, 2)s, T ⊥ j (j = 1, 2, 3, . . . , 15) and the same number of PG(2, 2)s. Given the facts that S ⊥ ∼ = PG(4, 2) and T j ∼ = PG(3, 2), a PG(4, 2) of PG(5, 2) will contain 15 PG(3, 2)s of type defined by Lemma 2 concurring at a point, namely the pole of this particular PG(4, 2). As PG(4, 2) contains altogether 31 PG(3, 2)s, each of the remaining 16 PG(3, 2)s does not contain totally-isotropic PG(2, 2)s and so hosts a unique linear doily. As each such doily can be viewed as a projection of a quadratic doily from the pole, a PG(4, 2) is found to be spanned by 16 quadratic doilies. Remark 6. If a PG(4, 2) of the ambient PG(2N -1, 2), N > 3, is devoid of totally-isotropic PG(3, 2)s, then it is of the type described above, i. e. it entails 16 quadratic doilies.

Number of linear doilies

Theorem 7. For any N ≥ 2 the number of linear doilies in W(2N -1, 2) is

D l (N ) = 2N 4 2 - N 4 2 4 i=1 2 N +1-i + 1 -7 N 3 2 2 2N -6 3 i=1 2 N +1-i + 1 /3. (9)
Proof. A linear doily of W(2N -1, 2) spans a particular PG(3, 2) of the ambient PG(2N -1, 2) that does not contain any totally-isotropic PG(2, 2). And since any such PG(3, 2) is spanned by a single linear doily, the number of linear doilies of W(2N -1, 2) is thus equal to the number of PG(3, 2)s that are devoid of totally-isotropic planes. To find this number, from Eq. ( 4) we first note that there are altogether 2N 4 2 (10)

PG(3, 2)s in PG(2N -1, 2), out of which N 4 2 4 i=1 2 N +1-i + 1 (11)
(Eq. ( 6) with k = 3 and q = 2) are totally isotropic.

To ascertain the cardinality of the remaining PG(3, 2)s that feature totally-isotropic PG(2, 2)s, we proceed as follows. We first observe that by Eq. ( 6) with k = 2 and q = 2 there are

N 3 2 3 i=1 2 N +1-i + 1 (12)
totally-isotropic PG(2, 2)s in PG(2N -1, 2). Next, with k = 3 and m = 2 in [START_REF] Saniga | Taxonomy of Polar Subspaces of Multi-Qubit Symplectic Polar Spaces of Small Rank[END_REF], it follows that there are N -3 1

2 2 N -3 + 1 = 2 2(N -3) -1 (13)
totally-isotropic PG(3, 2)s passing through a totally-isotropic PG(2, 2). And since the total number of PG(3, 2)s passing via a PG(2, 2

) of PG(2N -1, 2) is 2N -3 4 -3 2 = 2 2N -3 -1 (14)
(as stemming from Eq. ( 5) for n = 2N , k = 4, l = 3 and q = 2), through a totally-isotropic PG(2, 2) there pass

2 2N -3 -1 -2 2(N -3) -1 = 7 × 2 2N -6 (
2 N +1-i + 1 -7 N 3 2 2 2N -6 3 i=1 2 N +1-i + 1 /3
PG(3, 2)s in the ambient PG(2N -1, 2) that are devoid of totally-isotropic PG(2, 2)s, and so the same number of linear doilies in W(2N -1, 2).

Given the fact that the three lines of a perp-set of a linear doily span a PG(2, 2), and namely that PG(2, 2) that features just three totally-isotropic PG(1, 2)s, we arrive at the interesting expression

D l (N ) = 4 15 4 N -3 Θ 2 (N ), (16)
for the number of linear doilies in W(2N -1, 2), where

Θ 2 (N ) = 1 16 2 2N 2 i=1 2 N -2+i -1 2 i -1 2 i=1 (2 N +1-i + 1) (17)
is the number of those PG(2, 2)s of the ambient PG(2N -1, 2) each of which features just three totally-isotropic PG(1, 2)s.

Number of quadratic doilies

Theorem 8. For any N ≥ 3 the number of quadratic doilies in W(2N -1, 2) is

D q (N ) = 16 2N 5 2 - N 5 2 5 i=1 2 N +1-i + 1 -15 N 4 2 2 2N -8 4 i=1 2 N +1-i + 1 /3 . (
2 N +1-i + 1 (20)
(Eq. ( 6) with k = 4 and q = 2) are totally isotropic.

To ascertain the cardinality of the remaining isotropic PG(4, 2)s, we proceed as follows. We first observe that by Eq. ( 6) with k = 3 and q = 2 there are

N 4 2 4 i=1 2 N +1-i + 1 (21)
totally-isotropic PG(3, 2)s in PG(2N -1, 2). Next, with k = 4, m = 3 and q = 2 in (7) it follows that there are N -4 1

2 2 N -4 + 1 = 2 2(N -4) -1 (22)
totally-isotropic PG(4, 2)s passing through a totally-isotropic PG [START_REF] Richmond | The figure formed from six points in space of four dimensions[END_REF][START_REF] Cremona | Teoremi stereometrici dal quali si deducono le proprietà dell' esagrammo di Pascal (Stereometric theorems from which the properties of Pascal's hexagram are deduced)[END_REF]. And since the total number of PG(4, 2)s passing via a PG(3, 2) of PG(2N -1, 2) is

2 2N -4 -1 (23)
(as stemming from Eq. ( 5) for n = 2N , k = 5, l = 4 and q = 2), through a totally-isotropic PG(3, 2) there pass

2 2N -4 -1 -2 2(N -4) -1 = 15 × 2 2N -8 (
2 N +1-i + 1 -15 N 4 2 2 2N -8
PG(4, 2)s in the ambient PG(2N -1, 2) that are not endowed with any totally-isotropic PG(3, 2)s, the number of quadratic doilies of W(2N -1, 2) being just 16 times this number.

Employing further the fact that the three lines of a perp-set of a quadratic doily span a PG(3, 2), in particular that PG(3, 2) that contains just three totally-isotropic PG(2, 2)s, we find the compact formula

D q (N ) = 48 15 4 N -3 Θ 3 (N ) (26)
for the number of quadratic doilies in W(2N -1, 2), where

Θ 3 (N ) = 7 3 2 2N -6 3 i=1 2 N -3+i -1 2 i -1 3 i=1 2 N +1-i + 1 (27)
is the number of those PG(3, 2)s of the ambient PG(2N -1, 2) each of which features just three totally-isotropic PG(2, 2)s. 1: First numbers D(N ) (resp. D l (N ), D q (N )) of (resp. linear, quadratic) N -qubit doilies.

N D l (N ) D q (N ) D(N )
Comparing expressions ( 9) and (18), one gets

D q (N ) = (4 N -2 -1)D l (N ). (28)
Consequently, the total number of doilies is

D(N ) = 4 N -2 D l (N ). ( 29
)
For 2 ≤ N ≤ 9 the numbers of N -qubit doilies are collected in Table 1. Since the quadratic doilies of W(2N -1, 2) span a PG(4, 2), it has no geometrical meaning to consider quadratic doilies for N = 2. It can nevertheless be noticed that Eq. ( 18) also holds for N = 2, and consistently gives D q (2) = 0.

4 Generation of all N -qubit doilies

An N -qubit doily can be represented by an isomorphism f , sometimes called a (doily) labeling, mapping the points of W 2 to distinct points of W N , preserving commutations and anticommutations, and such that f (a.b) = ±f (a).f (b) for any two commuting points/observables a and b (the dot (.) denotes the matrix product). This section describes an algorithm for the enumeration of all N -qubit doilies, for any N ≥ 2, by construction of one of their labelings. Let us start with some definitions. An N -qubit ovoid is a 5-set of mutually anticommuting Nqubit observables whose product is the identity I N . A triad is a 3-set of mutually anticommuting N -qubit observables. A center of a triad is a point commuting with the three points of the triad. A unicentric triad is a triad that has only one center. Let ε denote the empty word. The lexicographic order < on words is such that ε < u for all non-empty word u, and a.u < b.v if and only if either a < b, or a = b and u < v, for any letters a and b and words u and v.

In order to avoid to consider several times objects that are similar but differently ordered, we define as follows a total order among letters and words, and then extend it to all tuples and sets of objects of the same nature, such as lines, sets of lines, etc. Pauli observables, encoded as words on the alphabet {I, X, Y, Z}, are totally ordered by the lexicographic order < induced by the order on letters, also denoted <, such that I < X < Z < Y . These orders are chosen so that their binary counterpart through the encoding I → 00, X → 01, Z → 10, Y → 11 is the lexicographic order on bit vectors (aka. bytes or binary words) induced by the order 0 < 1 on bits. This order < extends further to tuples (a 1 , a 2 , . . . , a n ) of words, by considering them as words a 1 a 2 • • • a n and re-using the former lexicographic order on words. It also extends to sets of words, by associating canonically to each set the tuple (a 1 , a 2 , . . . , a n ) of its elements written in increasing order (a i < a j when i < j), and so on at any level of the hierarchy of objects of the same nature, such as a point-line geometries, seen as sets of lines, that are sets of points. In the figure the third element of the tuples in this completion order is numbered from 1 to 9.

The algorithm itself is presented in Algorithm 1, where f (a) ← b denotes the assignment of b as the image of a by f . Algorithm 1 Doily generation algorithm.

1: for each ovoid O = {o 1 , o 2 , o 3 , o 4 , o 5 } in W N , with o 1 < o 2 < o 3 < o 4 < o 5 do 2: f (IX) ← o 1 || f (IZ) ← o 2 || f (XY ) ← o 3 || f (ZY ) ← o 4 || f (Y Y ) ← o 5 3:
for each center c of {o 1 , o 2 , o 3 } in W N that anticommutes with o 4 and o 5 do 4:

f (c 2 ) ← c 5:
for each line (p, q, r) in the order of the sequence S do f (r) ← |f (p).f (q)| end for 6:

if O is not the smallest ovoid of f then discard f end if . . . location for a potential treatment of f 8:

end for 9: end for On Line 2 a doily labeling f is partially defined by the choice of images for the 5 points of the ovoid O 2 of W 2 . These images are the points of some ovoid O = {o 1 , o 2 , o 3 , o 4 , o 5 } of N -qubit observables. The points are assigned in increasing order so that to avoid duplicates. As these five assignments are independent, they can be performed in parallel.

Then (on Line 3) the algorithm looks for a point c that commutes with the first three points of O and that anticommutes with its last two points o 4 and o 5 . On Line 4 this point becomes the image by f of the center c 2 of the triad

T 2 of O 2 .
The completion step on Line 5 computes one by one the images of all the other points of W 2 by f , in the order described by the sequence of lines S. At each iteration of this loop, for the line (p, q, r), the values f (p) and f (q) are known. By definition of a doily line, the image by f of the third point r is the product of the images f (p) and f (q) of the first two points, up to a possible minus sign, removed by the operation | | that denotes absolute value.

Knowing that each doily features 6 ovoids, the same doily is generated 6 times before Line 6, whose statement keeps only one of them, namely the doily d generated from the ovoid that is the smallest (according to the lexicographic order) among the 6 ovoids in d.

On Line 7 various treatments of the generated doilies f can be added, such as a storage, or the computation of classification criteria defined in Section 5.

Justification of the generation algorithm

First of all, the fact that doily labelings encode multi-qubit doilies is a direct consequence of the definition of a multi-qubit doily. Then, the properties of correctness and completeness for the doily enumeration algorithm mainly come from the following definition and proposition, whose proof is illustrated by Figure 6. 6. Finally, it is easy to check that the six 3-sets represented by the thin lines in Figure 6 indeed are geometric lines. A noticeable property is that the product of the three points on each of these lines contains twice the center c and once each point of the ovoid O. By applying the known commutation and anticommutation relations between these points, it comes that the product of both centers annihilates. So, modulo a possible minus sign, it remains the product of all observables of the ovoid, known to equal identity. Therefore, the product of the three observables on each line equals ±I N . Consequently, these 15 points and 15 lines form a doily, shown in Figure 6, so the algorithm is correct. Each multi-qubit doily features at least one ovoid and the first loop explores all ovoids in W N . So, each doily is found six times before Line 6, since each multi-qubit doily features exactly six ovoids. As the statement on this line always keeps one of them (the one that has been produced from the smallest of its ovoids), the algorithm is also complete.

Algorithmic complexity and implementation details

The enumeration algorithm explores all 4-tuples of observables likely to form an ovoid (the fifth point in the ovoid is computed as the product of the previous four), and then explores all observables to find c (on Line 3 of Algorithm 1). Therefore, the complexity of the algorithm is estimated to be O 4 5N , when the time unit is the duration to check whether two observables commute.

For efficiency reasons, we have implemented the algorithm in the C language, which allows for many optimizations. The total code is composed of about 2 300 lines and 50 functions, some of which implementing the classification process presented in Section 5. Some factors make the algorithm implementation more efficient than the former one presented in [START_REF] Saniga | Taxonomy of Polar Subspaces of Multi-Qubit Symplectic Polar Spaces of Small Rank[END_REF]: The new algorithm has a lower complexity; compared to the previously used language Magma [START_REF] Bosma | The Magma Algebra System I: The User Language[END_REF], the low-level language C allows to perform fast operations on bit vector representations of the observables, using as few CPU instructions as necessary, and to split the workload into multiple threads.

The calculations were run on Linux Ubuntu, on a PC equipped with an Intel (R) Core(TM) i7-8665U 1.90 GHz and 15 GB RAM. The code was compiled with gcc 9.3.0 with optimization Ofast and is multi-threaded with OpenMP.

Multi-doily classification process and results

This section presents our classification criteria of N -qubit doilies and the classification results for N = 4 and N = 5.

Classification criteria

The classification parameters adopted are the same as in [START_REF] Saniga | Taxonomy of Polar Subspaces of Multi-Qubit Symplectic Polar Spaces of Small Rank[END_REF]. The classification of an N -qubit doily is based on: 1) its signature, i. e. the number of its observables containing a given number of I : N -1, N -2, N -3, . . . respectively named types A, B, C, . . . ; 2) the configuration of its negative lines, as described in Section 2; and 3) its linear or quadratic character.

To find the line configuration of a doily, the first discriminatory factor is the number of negative lines, since for each number of negative lines except 7 and 8, there is only one configuration possible. Then the property used to distinguish configurations 7A from 7B and 8A from 8B is to count the number of observables contained in at least one negative line, since this number is different between A and B.

We use the following property to check whether a doily is linear or quadratic. Given an N -qubit doily, we pick up in it a tricentric triad (here we take the image of {XY, ZY, Y I}). If the product of the corresponding three observables is ±iI N , then the doily is linear, otherwise it is quadratic. This is because any tricentric triad is a line in the ambient PG(3, 2) if a doily spans a PG(3, 2).

Database of numerical results

Using the program described in Section 4, we were able to classify all doilies for N = 3 (2 016 ovoids), N = 4 (548 352 ovoids) and N = 5 (142 467 072 ovoids). This classification is a treatment added on Line 7 of the algorithm presented in Algorithm 1, that determines the complete type of each generated doily, counts the number of doilies for each type, and registers it in a result table.

The sums of the numbers of linear and quadratic doilies found in each of the above-mentioned cases correspond exactly to those stemming from eqs. ( 9) and (18), respectively, summarized in Table 1. The results of our classification are collected in Appendix A (three qubits), Appendix B (four qubits) and Appendix C (five qubits). The data for three qubits are in complete agreement with those of [START_REF] Saniga | Taxonomy of Polar Subspaces of Multi-Qubit Symplectic Polar Spaces of Small Rank[END_REF]; we found 11 different types of doilies of which five are linear and six quadratic. The 95 distinct types of four-qubit doilies split into 24 linear and 71 quadratic ones, whereas amongst 447 types of five-qubit doilies one finds 89 linear and 358 quadratic.

The structure of the classification table in each appendix is the same: the first column gives the type, the next N columns feature the numbers of observables of the corresponding types in a doily of the given type, the ν column shows the doily's character, and the remaining columns contain information about how many doilies of the given type are endowed with a particular number of negative lines (the blank space stands for zero here). The types are ordered in decreasing order of the number of observables containing no Is, in case of equality in decreasing order of the number of observables containing one I, and so on up to the number A of observables containing N -1 Is. For instance, for 4 qubits, the type 1 contains the maximal number 12 of D-type observables, and the last type 95 contains only A-and B-type observables. For a given signature, the type of quadratic doilies precedes that of linear ones.

The result tables are stored in https://quantcert.github.io/.

The C code for classification runs in 0.3 s for 4 qubits with 1.4 MB of memory and 12 min with 1.8 MB of memory for 5 qubits. The memory usage is low because the doilies are not stored, all the measurements are performed on the fly.

Remarks about five-qubit doilies

- (Y Y Y Y Y ) (9, 2) of W(9, 2). This special quadric Q - (Y Y Y Y Y ) (9, 2)
, like any non-degenerate quadric, is a geometric hyperplane of W(9, 2). As a doily is also a subgeometry of W(9, 2), it either lies fully in

Q - (Y Y Y Y Y ) (9,
2) (in which case B ∪ D = 15, such a doily will be called special), or shares with

Q - (Y Y Y Y Y ) (9,
2) a set of points that form a geometric hyperplane, in particular an ovoid (B ∪ D = 5), a perp-set (B ∪ D = 7) and/or a grid (B ∪ D = 9) and being referred to as ovoidal, perpial and/or gridal, respectively.

From Appendix C one can infer a number of interesting properties. We first notice that signatures with B ∪ C being even or odd are endowed with even or odd numbers of negative lines, respectively.

We also observe that there are 12 different signatures with A = C = E = 0, i. e., signatures featuring solely special doilies.

Further, there are 17 particular signatures such that each features observables of every type and no two types have the same cardinality. Out of them, six are ovoidal (e. g., 2-1-3-4-5), seven perpial (e. g., 1-3-2-4-5) and four gridal (e. g., 2-4-3-5-1).

If all doilies of a particular signature have just five or just six negative lines, then each doily is ovoidal; if a signature features just seven negative lines, then all of its doilies are perpial.

Among 33 distinct signatures with four negative lines only, one finds 12 ovoidal, 11 perpial and 9 gridal ones; doilies of the remaining signature, viz. 0-8-0-7-0, are special.

Next, there are 15 different signatures whose doilies are endowed with 12 (i. e., the maximum number of) negative lines. Out of them, five are ovoidal, five perpial and four gridal; the doilies of the remaining signature, namely 0-0-0-15-0, are special. Similarly, there are 35 distinct signatures whose doilies contain 11 (i. e., the maximum odd number of) negative lines; out of them, 10 are ovoidal, 11 perpial and 12 gridal, with the remaining two signatures, viz. 0-1-0-14-0 and 0-3-0-12-0, featuring solely special doilies.

Specific behavior of linear doilies

Finally, this section and the next one briefly mention some properties of linear doilies. Like the three-and four-qubit cases, a linear five-qubit doily can be either ovoidal or gridal and always contains an odd number of negative lines. Also, 75 types of linear doilies share their signatures with their quadratic siblings. However, there are 14 different signatures that are genuinely linear, of which eight cases are ovoidal.

From our results on three-, four-and five-qubit cases it follows that a linear doily (a) always features an odd number of negative lines, and (b) does not share a perp-set with the distinguished quadric.

We conjecture that Property (a) holds for any number of qubits N ≥ 2, but we have not yet found of proof of it; we surmise that it has something to do with the fact that a linear doily is "squeezed" into a PG [START_REF] Richmond | The figure formed from six points in space of four dimensions[END_REF][START_REF] Cremona | Teoremi stereometrici dal quali si deducono le proprietà dell' esagrammo di Pascal (Stereometric theorems from which the properties of Pascal's hexagram are deduced)[END_REF], compared to a quadratic doily that enjoys more degrees of freedom being stretched out in a PG [START_REF] Payne | Finite Generalized Quadrangles[END_REF][START_REF] Cremona | Teoremi stereometrici dal quali si deducono le proprietà dell' esagrammo di Pascal (Stereometric theorems from which the properties of Pascal's hexagram are deduced)[END_REF]. Property (b) can readily be proved to hold for any N ≥ 3, as follows.

Proof. Let us consider a linear doily with one of its perp-set; on Figure 7a, this perp-set is illustrated in bold font. Any perp-set of any doily features four tricentric triads; in our perp-set these triads are: {1, 2, 3}, {3, 4, 5}, {1, 5, 6} and {2, 6, 4}. Now, we know that any tricentric triad of a linear doily corresponds to a non-isotropic line in the ambient projective space, the four lines plus the three (totally isotropic) lines of the perp-set forming a Fano plane in this space, which is illustrated in Figure 7b. The assumption that our perp-set also lies on the distinguished quadric would mean that the whole plane would lie in the distinguished quadric and so would be totally isotropic, a contradiction.

A distinguished hexad of (linear) doilies

One knows that given an ovoid, there is a unique linear doily containing this ovoid. Now, take any quadratic doily. As each of its six ovoids defines a unique linear doily, we have a unique hexad of doilies tied to each quadratic doily. This holds for any N ≥ 3. Figure 8 illustrates this property for N = 3. It features a quadratic doily in the middle, its six ovoids depicted explicitly as pentads of points located on bold gray lines and the corresponding six linear doilies; for better readability, the points of the corresponding ovoids are illustrated by double-circles.

Conclusion

There are a number of intriguing extensions and generalizations of the ideas and findings presented in this paper. We shall mention a few of them.

An interesting situation that will be worth addressing occurs in the case of N = 4. Given a PG [START_REF] Richmond | The figure formed from six points in space of four dimensions[END_REF][START_REF] Cremona | Teoremi stereometrici dal quali si deducono le proprietà dell' esagrammo di Pascal (Stereometric theorems from which the properties of Pascal's hexagram are deduced)[END_REF] of the ambient PG(7, 2) of W(7, 2), its polar space is another PG [START_REF] Richmond | The figure formed from six points in space of four dimensions[END_REF][START_REF] Cremona | Teoremi stereometrici dal quali si deducono le proprietà dell' esagrammo di Pascal (Stereometric theorems from which the properties of Pascal's hexagram are deduced)[END_REF]. Hence, PG(3, 2)s in PG [START_REF] Saniga | Taxonomy of Polar Subspaces of Multi-Qubit Symplectic Polar Spaces of Small Rank[END_REF][START_REF] Cremona | Teoremi stereometrici dal quali si deducono le proprietà dell' esagrammo di Pascal (Stereometric theorems from which the properties of Pascal's hexagram are deduced)[END_REF] come in polar pairs. Taking into account the fact that a non-isotropic PG(3, 2) features a unique linear doily of W(7, 2), the above property means that also linear doilies of W(7, 2) occur in pairs. That is, picking up any linear four-qubit doily, there exists a unique linear doily such that each of its 15 observables commutes with each observable of the selected doily. This observation raises several interesting questions. For example, it would be interesting to ascertain which signatures are/can be paired, or which cardinalities of negative lines can occur in such self-polar pairs; we have already checked by hand a few examples where both doilies in a pair have the same signature and feature the same number of negative lines. There are (see Appendix B) altogether 24 different signatures featured by linear four-qubit doilies. We can then create a graph on 24 vertices such that its two edges are connected if there exists a pair of linear doilies exhibiting the corresponding signatures; we can even add a weight to an edge showing how many pairs of doilies feature this particular pair of signatures. This graph, as it follows from the examples checked, will also have edges joining a vertex to itself when the two paired signatures are identical. So, being an interesting graph of its own, it will also reveal some finer traits of the relation between individual linear doilies in W(7, 2)! A particular case deserving closer attention is N = 6. Here, let us formally view any six-qubit observable as a 'syntheme' partitioned into three two-qubit observables ('duads'). Given a partition, we find a set of linear doilies such that any doily in the set features 15 particular observables such that when restricted to the same duad we get a two-qubit doily; that is, any such doily can formally be regarded as being composed of three two-qubit doilies. Moreover, each partition features a prominent doily having all the three duads identical. The next worth-exploring case in this respect is N = 9, as W(17, 2) hosts not only composites comprising three doilies having the same number of qubits (namely three), but also those whose compounds feature different numbers of qubits (namely four, three and two).

Another prospective, but much more challenging, task will be to count and classify all rank-three spaces, W(5, 2)s, living in a particular W N , for N ≥ 4. The case N = 4 was already briefly examined in [START_REF] Saniga | Taxonomy of Polar Subspaces of Multi-Qubit Symplectic Polar Spaces of Small Rank[END_REF]. To address higher rank cases, we plan to employ the strategy that is the direct and natural generalization of the ovoid-based algorithm for doilies described in this paper. Geometrically, an N -qubit ovoid is a set of five points lying on a certain elliptic quadric of a PG [START_REF] Richmond | The figure formed from six points in space of four dimensions[END_REF][START_REF] Cremona | Teoremi stereometrici dal quali si deducono le proprietà dell' esagrammo di Pascal (Stereometric theorems from which the properties of Pascal's hexagram are deduced)[END_REF] in the ambient PG(2N -1, 2). Hence, its analogue will be a set of 27 N -qubit observables lying on an elliptic quadric of PG [START_REF] Hirschfeld | General Galois Geometries[END_REF][START_REF] Cremona | Teoremi stereometrici dal quali si deducono le proprietà dell' esagrammo di Pascal (Stereometric theorems from which the properties of Pascal's hexagram are deduced)[END_REF] in the ambient PG(2N -1, 2), and a triad of the ovoid will have its counterpart in a quadratic doily located on the quadric. A root of an N -qubit W(5, 2) will thus comprise an elliptic quadric and an off-quadric point such that its associated observable commutes with each of the 15 observables of a doily located in the quadric and anticommutes with the remaining 12 observables. It is obvious that this task will require a more elaborate generation algorithm and a more complex computer code to be successfully accomplished. 19 440 378 3 3 3 6 0 q 19 440 379 4 3 2 6 0 l 1620 380 0 9 0 6 0 q 55 080 9720 1620 381 0 0 10 5 0 q 6480 19 440 382 1 0 9 5 0 q 29 160 9720 383 1 0 9 5 0 l 12 960 9720 384 0 2 [START_REF] Holweck | Geometric constructions over C and F 2 for Quantum Information[END_REF] 155 520 19 440 402 3 3 5 4 0 q 38 880 403 2 5 4 4 0 q 19 440 19 440 404 2 5 4 4 0 l 7290 405 4 5 2 4 0 q 2430 406 0 11 0 4 0 q 4860 407 0 2 10 3 0 q 136 080 32 400 19 440 19 440 408 1 2 9 3 0 q 51 840 19 440 409 1 2 9 3 0 l 38 880 410 0 4 8 3 0 q 77 760 123 120 19 440 411 2 2 8 3 0 q 77 760 412 1 4 7 3 0 q 142 560 136 080 19 440 413 3 2 7 3 0 l 3240 414 0 6 6 3 0 q 97 200 19 440 415 2 4 6 3 0 q 97 200 416 1 6 5 3 0 q 38 880 19 440 417 1 6 5 3 0 l 19 440 418 3 4 5 3 0 q 19 440 419 0 3 10 2 0 q 383 940 19 440 420 0 3 10 2 0 l 3240 9720 421 1 3 9 2 0 q 38 880 6480 422 0 5 8 2 0 q 332 910 58 320 19 440 423 2 3 8 2 0 q 29 160 19 440 424 2 3 8 2 0 l 9720 425 1 5 7 2 0 q 38 880 426 0 7 6 2 0 q 43 740 29 160 4860 427 0 7 6 2 0 l 9720 428 2 5 6 2 0 q 29 160 429 2 7 4 2 0 q 9720

A Taxonomy of 3-qubit doilies

Figure 2 :

 2 Figure 2: Generic representatives of the twelve different types of configurations of negative lines (bold) that can be found in a multi-qubit doily.

Figure 3 :Figure 4 :

 34 Figure3: A linear four-qubit doily with one of its perp-sets highlighted in blue color (a) and the corresponding PG(3, 2) of PG(7, 2) it spans (b). One can readily see that the three lines of the perp-set lie in a plane of the PG(3, 2) and the three points on a non-isotropic line of the space (black) correspond to a tricentric triad of the doily.

8 Figure 5 :

 85 Figure 5: The 2-qubit doily W 2 , the ovoid O 2 (framed), the triad T 2 (framed and dashed), its center c 2 (circled and dashed) and the completion order (subscripted). The negative lines are doubled. The algorithm relies on the following predefined elements, depicted in Fig. 5: the 2-qubit doily W 2 , the ovoid O 2 ≡ {IX, IZ, XY, ZY, Y Y } in W 2 , the unicentric triad T 2 ≡ {IX, IZ, XY } in O 2 , the center c 2 ≡ XI of T 2 , and the sequence of lines S ≡ (XI, IX, XX), (XI, IZ, XZ), (XI, XY, IY ), (ZY, XX, Y Z), (ZY, XZ, Y X), (ZY, IY, ZI), (Y Y, XX, ZZ), (Y Y, XZ, ZX), (Y Y, IY, Y I).

Definition 9 ( 3 ±c.o 3 ±c.o 1 o 4 ± 2 ± 1 ±o 2 o 5 ±(c.o 2 ).o 5 Figure 6 :

 9334215256 Figure 6: Expression of each observable according to the completion order followed by the generation algorithm, from the doily root ({o 1 , o 2 , o 3 , o 4 , o 5 }, c). The thick lines are the lines used to compute these expressions.

  Let us have a closer look at the five-qubit case. The 3 2 × 5 2 = 90 observables of type B and 3 4 × 5 4 = 405 observables of type D lie on an elliptic quadric Q

Figure 7 :

 7 Figure 7: Graphical arguments for the property that a linear doily cannot share a perp-set with the distinguished quadric.

Figure 8 :

 8 Figure 8: A particular hexad of linear doilies in the three-qubit symplectic polar space.

All dimensions in this section are projective dimensions.

i=1 2 N +1-i + 1 /3 (25)

Acknowledgments

This project is supported by the EIPHI Graduate School (contract ANR-17-EURE-0002). This work was also supported by the Slovak VEGA Grant Agency, Project # 2/0004/20. We thank our friend Zsolt Szabó for the help in preparation of several figures.

29 160 81 4 3 0 2 6 l 1620 82 0 6 2 1 6 q 58 320 83 0 5 4 0 6 q 4860 84 0 7 2 0 6 q 19 440 14 580 85 0 9 0 0 6 q 810 2430 86 0 0 1 9 5 q 9720 116 640 524 880 379 080 388 800 58 320 87 1 0 0 9 5 q 58 320 194 400 58 320 88 0 1 1 8 5 q 583 200 2 682 720 2 332 800 933 120 699 840 116 0 4 3 3 5 q 524 880 816 480 58 320 349 920 117 2 2 3 3 5 q 233 280 291 600 58 320 118 0 3 5 2 5 q 758 160 583 200 174 960 119 1 3 4 2 5 q 58 320 116 640 58 320 120 1 3 4 2 5 l 58 320 121 0 5 3 2 5 q 116 640 122 2 3 3 2 5 q 58 320 123 0 4 5 1 5 q 58 320 58 320 124 0 0 2 9 4 q 1 224 720 58 320 176 2 3 4 2 4 l 14 580 177 3 3 3 2 4 q 58 320 178 0 7 2 2 4 q 58 320 58 320 179 2 5 2 2 4 q 29 160 29 160 180 4 5 0 2 4 q 2430 181 0 7 4 0 4 q 7290 182 0 9 2 0 4 q 7290 183 0 0 3 9 3 q 907 200 3 032 640 1 720 440 5 452 920 3 936 600 1 720 440 184 0 0 3 [START_REF] Polster | A Geometrical Picture Book[END_REF] 699 840 116 640 230 0 6 3 3 3 q 116 640 58 320 231 0 3 7 2 3 q 524 880 213 840 58 320 232 1 3 6 2 3 q 291 600 291 600 58 320 233 1 3 6 2 3 l 29 160 234 0 5 5 2 3 q 524 880 58 320 235 2 3 5 2 3 q 145 800 236 1 5 4 2 3 q 58 320 29 160 237 3 3 4 2 3 q 29 160 238 0 4 7 1 3 q 58 320 239 0 6 5 1 3 q 58 320 240 0 0 4 9 2 q 1 283 040 349 920 288 1 4 5 3 2 q 291 600 116 640 289 0 3 8 2 2 q 349 920 262 440 29 160 290 0 3 8 2 2 l 58 320 291 1 3 7 2 2 q 174 960 292 0 5 6 2 2 q 379 080 145 800 29 160 293 2 3 6 2 2 q 72 900 294 0 7 4 2 2 q 58 320 295 2 5 4 2 2 q 29 160 296 0 0 5 [START_REF] Polster | A Geometrical Picture Book[END_REF] 583 200 326 2 4 3 5 1 q 58 320 327 0 1 9 4 1 q 991 440 622 080 58 320 58 320 328 1 1 8 4 1 q 349 920 291 600 329 1 1 8 4 1 l 58 320 330 0 3 7 4 1 q 2 507 760 408 240 331 1 3 6 4 1 q 874 800 262 440 332 0 5 5 4 1 q 349 920 333 1 5 4 4 1 q 145 800 334 3 3 4 4 1 q 29 160 335 0 2 9 3 1 q 913 680 787 320 204 120 58 320 336 0 2 9 3 1 l 58 320 337 1 2 8 3 1 q 233 280 338 0 4 7 3 1 q 816 480 408 240 58 320 339 2 2 7 3 1 q 116 640 340 0 6 5 3 1 q 58 320 341 0 3 9 2 1 q 87 480 342 1 3 8 2 1 q 116 640 343 1 5 6 2 1 q 58 320 344 0 0 0 15 0 q 311 040 155 520 136 080 641 520 186 624 103 680 345 0 1 0 14 0 q 534 600 1 603 800 831 060 2 741 040 1 759 320 554 040 346 0 2 0 13 0 q 3 207 600 4 432 320 1 399 680 3 965 760 349 920 347 0 3 0 12 0 q 2 227 500 3 936 600 1 383 480 4 169 880 926 640 29 160 348 0 4 0 11 0 q 3 790 800 2 099 520 136 080 874 800 349 0 5 0 10 0 q 1 526 040 1 648 512 330 480 592 920 9720 350 0 0 6 9 0 q 58 320 77 760 58 320 58 320 351 1 0 5 9 0 q 12 960 3240 352 1 0 5 9 0 l 58 320 353 2 0 4 9 0 q 58 320 354 3 0 3 9 0 l 3960 355 0 6 0 9 0 q 939 600 142 560 356 0 1 6 8 0 q 165 240 174 960 311 040 262 440 58 320 4860 357 0 1 6 8 0 l 14 580 43 740 358 1 1 5 8 0 q 466 560 97 200 359 2 1 4 8 0 q 38 880 9720 360 2 1 4 8 0 l 19 440 4860 361 0 7 0 8 0 q 340 200 126 360 19 440 362 0 0 8 7 0 q 58 320 97 200 58 320 58 320 363 1 0 7 7 0 q 58 320 58 320 364 0 2 6 7 0 q 816 480 777 600 174 960 97 200 365 1 2 5 7 0 q 505 440 476 280 136 080 9720 366 1 2 5 7 0 l 77 760 38 880 367 2 2 4 7 0 q 174 960 19 440 368 3 2 3 7 0 q 38 880 9720 369 0 8 0 7 0 q 19 440 370 0 1 8 6 0 q 304 560 281 880 213 840 106 920 58 320 371 1 1 7 6 0 q 213 840 90 720 58 320 372 0 3 6 6 0 q 1 373 760 952 560 340 200 291 600 42 120 373 0 3 6 6 0 l 81 000 9720 24 300 58 320 3240 1620 374 2 1 6 6 0 q 77 760 77 760 9720 375 1 3 5 6 0 q 349 920 162 000 58 320

Configuration of negative lines TypeA B C D E ν 3 4 5 6 7A 7B 8A 8B 9 10 11 12 430 1 4 9 1 0 q 19 440 19 440 431 1 4 9 1 0 l 9720 432 0 6 8 1 0 q 19 440 433 1 6 7 1 0 q 19 440 434 3 4 7 1 0 q 9720 435 0 5 10 0 0 q 12 960 436 1 5 9 0 0 q 1080 437 0 7 8 0 0 q 32 400 810 438 2 5 8 0 0 l 1620 439 3 5 7 0 0 q 3240 440 0 9 6 0 0 q 21 330 441 0 9 6 0 0 l 90 270 442 2 7 6 0 0 q 2160 1620 443 4 5 6 0 0 l 540 444 2 9 4 0 0 l 810 445 4 7 4 0 0 q 810 446 0 15 0 0 0 q 360 447 6 9 0 0 0 l 10