Systems of Imprimitivity for the Clifford Group
D. M. Appleby, Ingemar Bengtsson, Stephen Brierley, Åsa Ericsson, Markus Grassl, Jan-Åke Larsson
aa r X i v : . [ qu a n t - ph ] O c t Systems of Imprimitivity for the Clifford Group
D.M. Appleby
Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada
Ingemar Bengtsson
Stockholms Universitet, AlbaNova, Fysikum, S-106 91 Stockholm, Sweden
Stephen Brierley
Heilbronn Institute for Mathematical Research, Department of Mathematics,University of Bristol, Bristol BS8 1TW, UK ˚Asa Ericsson
Matematiska institutionen, Link¨opings Universitet, S-581 83 Link¨oping, Sweden
Markus Grassl
Centre for Quantum Technologies, National University of Singapore,Singapore 117543
Jan-˚Ake Larsson
Institutionen f¨or Systemteknik, Link¨opings Universitet, S-581 83 Link¨oping,Sweden
Abstract
It is known that if the dimension is a perfect square the Clifford group can berepresented by monomial matrices. Another way of expressing this result is tosay that when the dimension is a perfect square the standard representation ofthe Clifford group has a system of imprimitivity consisting of one dimensionalsubspaces. We generalize this result to the case of an arbitrary dimension. Let k be the square-free part of the dimension. Then we show that the standardrepresentation of the Clifford group has a system of imprimitivity consistingof k -dimensional subspaces. To illustrate the use of this result we apply it tothe calculation of SIC-POVMs (symmetric informationally complete positiveoperator valued measures), constructing exact solutions in dimensions 8 (hand-calculation) as well as 12 and 28 (machine-calculation). Introduction
The Clifford group plays a major role in many areas of quantum information.When expressed in terms of the standard basis many of the Clifford unitaries areHadamard matrices, with the property that every matrix element has the sameabsolute value = 1 / √ N (where N is the dimension). It is therefore somewhatremarkable that, in the special case when the dimension is a perfect square, asimple change of basis will cause every Clifford unitary to become what could beregarded as the opposite of a Hadamard matrix, namely a monomial matrix, withonly one non-zero element in each row and each column [5]. In the following we willgeneralize this result to arbitrary dimensions. Given any dimension N there existunique integers k, n such that k is square-free ( i.e. k is not divisible by the squareof any prime number) and N = kn . We will show that the Clifford group admitsa representation entirely in terms of matrices with the property that, when writtenout in block form, each row and each column contains exactly one non-zero k × k block. We refer to such matrices as k -nomial. It will be seen that the result provedin Ref. [5] is simply the special case of this for k = 1.This result can be re-phrased. Let H be N -dimensional Hilbert space. Then ourresult is that H can be written as a direct sum H = n − M r,s =0 H r,s (1)where the subspaces H r,s are all k -dimensional and where, for every Clifford unitary U and every pair r, s U H r,s = H r ′ ,s ′ (2)for some r ′ , s ′ . Such a set of subspaces is called a system of imprimitivity [6, 7].Systems of imprimitivity play an important role in the theory of group represen-tations as they can be used to construct an induced representation for the wholegroup starting from a representation of a subgroup. No such application is in ques-tion here since we already have a representation for the group. There are, however,other applications, as we will see.Many interesting physical properties are revealed by a deeper understanding ofthe relevant group structures and properties. The Clifford group in its various formsis a good example of this, having applications to fault tolerant quantum comput-ing [8,9], measurement based quantum computing [10,11], describing the boundarybetween classically simulable and universal quantum computing [12, 13], quantumnon-locality [14], mutually unbiased bases (or MUBs) [2, 4, 15–20], and symmetricinformationally complete measurements (commonly referred to as SIC-POVMs or,as here, simply SICs) [4, 5, 18, 19, 21–42]. Although the result proved in this pa-per is potentially relevant to all these areas we will illustrate its use in connectionwith the SIC existence problem, as that is the route by which we were led to it.SICs are important practically, with applications to quantum tomography [43–46],quantum cryptography [47–51], quantum communication [52–56], Kochen-Speckerarguments [57], high precision radar [58–60] and speech recognition [61] (the lasttwo applications being classical). They are also important from a mathematical We should remark that there are several versions of the Clifford group [1–4]. In this paperthe focus will be on the version defined to be the normalizer of a single copy of the non-Galoisianversion of the Weyl-Heisenberg group. However, we believe our results are likely to be relevant tothe other versions. point of view, as giving insight into the geometrical structure of quantum statespace [62], and from a foundational point of view, as playing a central role in theqbist approach to the interpretation of quantum mechanics [63–65]. SICs have beenrealized experimentally [50, 66]. They have been calculated numerically for everydimension ≤
67 and exact solutions have been constructed for dimensions 2–16, 19,24, 35 and 48 (see Ref. [36] for a comprehensive listing of solutions known in 2010and Ref. [5] for the exact solution in dimension 16). There are therefore grounds forconjecturing that SICs actually exist in every finite dimension. However, in spiteof strenuous efforts by many investigators over a period of more than 10 years thequestion is still undecided. The connection with the Clifford group comes from thefact that the overwhelming majority of known SICs are covariant with respect to theWeyl-Heisenberg group (or generalized Pauli group). Moreover, Weyl-Heisenbergcovariant SICs have been found in every dimension for which SICs have been foundat all. It is therefore tempting to conjecture that, not just SICs, but specificallyWeyl-Heisenberg covariant SICs exist in every finite dimension.The investigation reported here was motivated by two striking facts concern-ing Weyl-Heisenberg covariant SICs. Firstly, the defining equations are massivelyover-determined. Specifically, one has d equations for only 2 d − >
2. The fact that solutions actually do exist (atleast numerically) in every dimension up to 67, suggests the presence of some kindof concealed symmetry or other algebraic feature of the equations which forces asolution notwithstanding the over-determination; and it further suggests that iden-tifying that feature might be the key to the existence problem. The second strikingfact is that the known exact solutions all turn out to be expressible in terms ofradicals. Again, this could not be anticipated in advance of actually calculatingthe solutions, and seeing that they are so expressible. The defining equations of aWeyl-Heisenberg covariant SIC fiducial vector are quartic polynomials in the realand imaginary parts of the components. The standard way to solve such a systemof equations (except in the simplest cases) is to construct a Gr¨obner basis [67],thereby reducing the problem to that of solving polynomial equations in a singlevariable. It turns out [24, 26, 31, 36] that the single variable equations are all ofdegree ≫ [5, 36, 42] that in all the known cases the Galois group has a particularlysimple form even amongst the class of solvable groups, having a subnormal series ofthe form { e } ⊳ H ⊳ G where e is the identity, H and G/H are Abelian, and
G/H is order 2 ( G being the Galois group of the smallest normal extension of Q con-taining all the components of the fiducial together with √− As originally noticed by Jon Yard [68], although Ref. [36] gives correct expressions for theexact fiducials in dimension 14, there is an error in the calculation of the Galois group. Thecorrect Galois group for this dimension can be found in Ref. [42]. produce considerable simplification for N = 4, 9, 16. In the following we show thatthe k -nomial representation is similarly efficacious. We perform a hand calculationof the Weyl-Heisenberg fiducials for N = 8 and we find that the expressions whichresult are indeed much simpler than the standard basis expressions obtained bymachine calculation [36]. The fact that a hand calculation is even possible may betaken as further evidence that the k -nomial basis is better adapted to the problem.We also describe a machine calculation of a fiducial in dimension 28. Althoughexact fiducials have previously been calculated in dimensions 24, 35 and 48, theserelied on the existence of additional symmetries of order 6, 12 and 24, respectively.So far, N = 28 is the largest dimension in which it has been possible to find anexact solution only with the help of a symmetry of order 6, while N = 16 in Ref. [5]is the largest dimension for which the computation relied only on a symmetry oforder three, as conjectured by Zauner. This may be taken to confirm the hypothesisthat the k -nomial basis is indeed better adapted to the SIC problem. As furtherconfirmation of this point we revisited the calculation of an exact fiducial in di-mension 12 first reported in Ref. [31]. We found that the use of a k -nomial basisreduced the computation time by more than 3 orders of magnitude. We also foundthat the expressions for the fiducial are much more compact than the ones given inRefs. [31, 36]Finally, let us observe that the advantage of a hand calculation, such as thesolution for N = 8 presented here, or the solution for N = 9 presented in Ref. [5],is that it gives us a degree of insight into the algebraic intricacies of the problemwhich a machine calculation cannot provide. We suggested above that the fact thatsolutions exist notwithstanding the massive degree of over-determination, and thestriking simplicity of the Galois group, both hint at the presence of some underlyingsymmetry, or other algebraic feature of the equations which has so far eluded us. Itmay be that a study of hand-constructed solutions will lead us to the secret whichwill enable us finally to crack the problem.The plan of the paper is as follows. In Section 2 we describe the k -nomialrepresentation of the Clifford group. In Section 3 we briefly review a few basicfacts about SICs. In Section 4 give a hand calculation of the Weyl-Heisenberg SICfiducials in dimension 8 . In Section 5 we obtain, by machine calculation, a verycompact solution for dimension 12. In Section 6 we obtain, by machine calculation,exact fiducials in dimension 28.2. The k -nomial Representations To fix notations we begin by describing the standard basis representation of theClifford group. For more details see (for example) Ref. [25]. Let | i , . . . | N − i bethe standard basis in dimension N . Define the operators X and Z by X | u i = | u + 1 i (3) Z | u i = ω u | u i (4)where addition of ket-labels is mod N and ω = e πiN . The Weyl-Heisenberg dis-placement operators are then defined by D p = τ p p X p Z p (5) where p = ( p p ), τ = − e πiN and p , p run from 0 to ¯ N −
1, where¯ N = ( N N odd2
N N even (6)With this definition D p D q = τ h p , q i D p + q (7)where the symplectic form h· , ·i is defined by h p , q i = p q − p q (8)The symplectic group SL(2 , Z ¯ N ) consists of all matrices F = (cid:18) α βγ δ (cid:19) (9)such that α , β , γ , δ ∈ Z ¯ N and Det F = 1 (mod ¯ N ). To each such matrix therecorresponds a unitary U F , unique up to a phase, such that U F D p U † F = D F p (10)If β is relatively prime to ¯ N we have the explicit formula U F = e iθ √ N N − X u,v =0 τ β − ( δu − uv + αv ) | u ih v | (11)where β − is the multiplicative inverse of β (mod ¯ N ) and e iθ is an arbitrary phase.Observe that it follows from this formula that U F is a Hadamard matrix in thestandard basis whenever β is relatively prime to ¯ N . If β is not relatively prime to¯ N we use the fact [25] that F has the decomposition F = F F = (cid:18) α β γ δ (cid:19) (cid:18) α β γ δ (cid:19) (12)where β , β both are relatively prime to ¯ N so that U F , U F can be calculatedusing Eq. (11). U F is then given by U F = U F U F (13)up to an arbitrary phase. The Clifford group then consists of all products D p U F .The extended Clifford group is also important. Enlarge SL(2 , Z ¯ N ) to the groupESL(2 , Z ¯ N ) by including the anti-symplectic matrices F = (cid:18) α βγ δ (cid:19) (14)with Det F = − N ). Each such matrix has the unique decomposition F = ˜ F J (15)where ˜ F is symplectic and J = (cid:18) − (cid:19) . (16)To J we associate the anti-unitary U J which acts by complex conjugation in thestandard basis: U J N − X u =0 ψ u | u i ! = N − X u =0 ψ ∗ u | u i (17) for all | ψ i = P N − u =0 ψ u | u i . We then associate to F the anti-unitary U F = U ˜ F U J (18)The extended Clifford group consists of all products of the form D p U F , with p ∈ Z N and F ∈ ESL(2 , Z N ).These preliminaries completed we now turn to the construction of the k -nomialrepresentations. Let N = kn (one would usually choose k to be square-free, soas to make the non-zero blocks in the representation matrices as small as possible;however this is not necessary). Then it can be seen from Eq. (7) that X kn and Z kn commute, and are therefore simultaneously diagonalizable. It is easily verified thata joint eigenbasis is | r, s, j i = 1 √ n n − X t =0 λ − rt | ( s + jn ) + tkn i (19)where λ = e πin , where r, s ∈ Z n label the eigenspaces according to X kn | r, s, j i = λ r | r, s, j i Z kn | r, s, j i = λ s | r, s, j i (20)and where j ∈ Z k labels the basis vectors within each eigenspace. The action of X , Z in this basis is X | r, s, j i = | r, s + 1 , j i s = n − | r, s + 1 , j + 1 i s = n − , j = k − λ r | r, s + 1 , j + 1 i s = n − , j = k − Z | r, s, j i = ω s + nj | r − , s, j i (22)from which we see that the displacement operators are k -nomial in this basis (infact monomial) just as they are in the standard basis. From Eq. (22) it follows thatnot only X kn and Z kn are diagonal in this basis, but also Z n .The advantage of this basis is that in it, not only the displacement operators,but also the symplectic unitaries are represented by k -nomial matrices. In fact, let F = (cid:18) α βγ δ (cid:19) (23)be an arbitrary matrix ∈ SL(2 , Z ¯ N ). Then U † F X kn U F = ( ( − γδ X knδ Z − knγ k odd, n even X knδ Z − knγ otherwise (24) U † F Z kn U F = ( ( − αβ X − knβ Z knα k odd, n even X − knβ Z knα otherwise (25)where we used the fact that τ k n = ( N odd( − k N even (26)to calculate the signs. Consequently X kn U F | r, s, j i = λ r ′ U F | r, s, j i (27) Z kn U F | r, s, j i = λ s ′ U F | r, s, j i (28) where r ′ = δr − γs + mγδ (29) s ′ = − βr + αs + mαβ (30)with m = ( n k odd, n even,0 otherwise. (31)So U F takes the ( r, s ) eigenspace to the ( r ′ , s ′ ) eigenspace: U F | r, s, j i = k − X j ′ =0 ( M F,rs ) jj ′ | r ′ , s ′ , j ′ i (32)for some family of k × k matrices M F,rs . It is thus k -nomial, as claimed. Note thatthis also shows that the eigenspaces labeled by ( r, s ) form a system of imprimitivity(see Eq. (2)).To calculate the matrices M F,rs suppose, first, that β is relatively prime to ¯ N .Then using Eq. (11) one finds, after a certain amount of algebra, that, up to anoverall phase,( M F,rs ) jj ′ = 1 √ k τ β − ( δ ( s ′ + j ′ n ) − s ′ + j ′ n )( s + jn )+ α ( s + jn ) ) (33)where s ′ is given in terms of r , s by Eq. (30) (note that in this formula it is essentialthat s ′ be reduced mod n , so that it is an integer in the range 0 ≤ s ′ < n ). The casewhen β is not relatively prime to ¯ N can then be handled using the decompositionof Eq. (12).It is perhaps interesting to note that when β is relatively prime to ¯ N the non-zeroblocks in the k -nomial representation are Hadamard (in contrast to the standardrepresentation where the whole matrix is Hadamard).Finally, we can calculate the anti-symplectic anti-unitaries using Eq. (18) to-gether with the fact U J | r, s, j i = | − r, s, j i (34)3. SICs
The k -nomial representations described in the last section are a very general con-struction which is potentially relevant to any area where the Clifford and extendedClifford groups play a role. In the remainder of this paper we illustrate their useby showing how they can be used to simplify the calculation of SICs. We begin, inthis section, by reviewing a few basic facts (for more details see Refs. [22, 23, 25, 36]). A SIC in dimension N is a family of N operators E r = 1 N | ψ r ih ψ r | (35)such that the vectors | ψ r i satisfy (cid:12)(cid:12) h ψ r | ψ s i (cid:12)(cid:12) = ( r = s, √ N +1 otherwise. (36) The fact that the matrix Tr( E r E s ) is non-singular means that the E r are a basisfor operator space and N X r =1 E r = I. (37)So the E r form an informationally complete POVM (positive operator valued mea-sure).The vast majority of SICs which have been constructed to date are covariantunder the action of the Weyl-Heisenberg group. That is, they have the property D q E p D † q = E p + q (38)for all p , q (where, instead of labelling the POVM elements by the single index r ,we label them by p ∈ Z N ). For a SIC of this kind we can generate the entire SIC byacting on E with the displacement operators D p . The vector | ψ i correspondingto E is called the fiducial vector. We usually omit the subscript and simply denotethe fiducial vector | ψ i . The problem of constructing a Weyl-Heisenberg covariantSIC thus reduces to the problem of finding a single vector | ψ i such that (cid:12)(cid:12) h ψ | D p | ψ i (cid:12)(cid:12) = ( p = mod N √ N +1 p = mod N (39)It is a striking, so far unexplained fact that every known Weyl-Heisenberg SICfiducial vector is an eigenvector of a Clifford unitary D q U F for which Tr F = − N and F = I (note that the requirement F = I is automatic if N = 3).It can be shown [22, 25, 36] that such unitaries are always order 3 up to a phase.Conversely, it appears that every such unitary has a Weyl-Heisenberg SIC fiducialas one of its eigenvectors. In the following sections we exploit this fact by lookingfor fiducials which are eigenvectors of U F z , where F Z = (cid:18) − − (cid:19) (40)is the Zauner matrix [22]. We refer to U F z as the Zauner unitary.4. Dimension fiducial We now perform a hand-calculation of the fiducials in dimension 8 which arecovariant under the version of the Weyl-Heisenberg group considered in this paper(for fiducials covariant with respect to a different version of the group see Refs. [19,21]). Of course exact expressions for the standard basis components have alreadybeen found by Scott and Grassl [36] (by means of a machine-calculation of a Gr¨obnerbasis). So if we only wanted to make the point that the expressions become muchsimpler in the k -nomial basis it would be enough just to transform the expressionsin Ref. [36]. However, as we stated in the Introduction, we feel that a hand-calculation gives a greater degree of insight into the problem than one gets from amachine-calculation. In particular we entertain the hope that a close inspection ofhand-calculated solutions may enable us to spot the “secret ingredient” responsiblefor the fact that SICs exist (at least numerically for dimensions ≤
67) in spite ofthe massive degree of overdetermination, and for the striking features of the Galoisgroup—thereby, perhaps, leading to a solution of the existence problem.It may be worth remarking that we found it surprising that we were able to solvethe equations at all (by hand, that is). One of us (DMA) well recalls that he could only obtain the dimension 7 fiducials in Ref. [25] by a process of trial and error,and that in order to be sure that he had found all the solutions he had to appealto the numerical work of Renes et al. [23].We look for SIC fiducial vectors which are eigenvectors of the Zauner unitary, U F z . Note that 8 is one of the special dimensions [36] for which fiducials can befound in all three eigenspaces of U F z . The first step in the calculation is to choosea basis which diagonalizes U F Z . Since there are only 3 eigenspaces there are manyways to do this, which are far from being equivalent from the point of view ofcalculational complexity. One natural way to proceed, which causes the equationsto be solved to take a particularly simple form, is to consider the normalizer N Z of the cyclic subgroup generated by F Z — i.e. the set of matrices G ∈ ESL(2 , Z ¯ N )such that GF Z = F rZ G (41)for r = 1 or 2. The significance of N Z is that the corresponding unitaries andanti-unitaries permute the eigenvectors of U F Z . In particular they permute thefiducials which are eigenvectors of U F Z . Having constructed N Z one then picksout a maximal subset containing F Z such that the corresponding unitaries/anti-unitaries commute. The joint eigenvectors of the operators in this subgroup giveus our desired basis. As we will see the fact that the matrices are all k -nomialconsiderably facilitates the calculation.The group N Z is generated by the matrices K = (cid:18) (cid:19) A = (cid:18) − (cid:19) P = (cid:18) − − (cid:19) (42)Here K is order 2 anti-symplectic, A is order 24 anti-symplectic and P is order 2symplectic. We have F Z = A (43)and P A = AP P K = KP AK = KA − P (44)Define symplectic matrices˜ K = (cid:18) −
11 0 (cid:19) ˜ A = (cid:18) − − − (cid:19) (45)We then have K = ˜ KJ A = ˜ AJ (46)and, consequently, U K = U ˜ K U J U A = U ˜ A U J (47)where J is the matrix defined in Eq. (16).We now calculate the matrices corresponding to these operators. For dimension8 we have k = n = 2. If we order the basis vectors | r, s, j i lexicographically | , , i , | , , i , | , , i , . . . , | , , i (48)the matrices become (with a given choice for the arbitrary phase factors) U ˜ K = K K K K U ˜ A = A A
00 0 0 A A (49) and U P = I σ x σ z
00 0 0 − σ x (50)where σ x , σ z are the Pauli matrices and K = 1 √ (cid:18) − (cid:19) A = 1 √ (cid:18) − i − − i (cid:19) (51) K = 1 √ (cid:18) i − i (cid:19) A = τ − √ (cid:18) − (cid:19) (52) K = 1 √ (cid:18) i − i (cid:19) A = τ √ (cid:18) − − i − i (cid:19) (53) K = τ √ (cid:18) ii (cid:19) A = τ − √ (cid:18) − ii − (cid:19) (54)We fix the arbitrary phase in the definition of U F Z by making the choice U F Z = U A (55)It is readily confirmed that U K and U P are order 2. However U A , unlike A , is onlyorder 12. This is because [25] the mapping G U G takes A = ( ) to a multipleof the identity.We seek the joint eigenvectors of the commuting unitaries U P and U A . We have U A = U ˜ A U ∗ ˜ A = A A A A (56)where A rs = A r A ∗ s (57)The 2 × A is easily diagonalized. To diagonalize the other, 6 × U A observe that A A A = A A A = A A A = I (58)Consequently A A A u u u = ξ u u u (59)with ξ = 0 if and only if ξ is a cube root of unity and u = ξA A u u = ξ A u (60)If we impose the further requirement that u u u (61) be an eigenvector of U P we must have that u is a multiple of ( ) or ( ). Weconclude that a complete set of joint eigenvectors is | , i = 1 √ (cid:0) , , η , η , √ η , , − , (cid:1) t (62) | , − i = 1 √ (cid:0) , , η , η , , √ η , η , η (cid:1) t (63) | , i = 16 (cid:16) η q √ , η q − √ , , , , , , (cid:17) t (64) | , i = 1 √ (cid:0) , , η , η , √ η , , − , (cid:1) t (65) | , − i = 1 √ (cid:0) , , η , η , , √ η, η , η (cid:1) t (66) | , i = 1 √ (cid:0) , , η, η, √ η , , − , (cid:1) t (67) | , − i = 1 √ (cid:0) , , η , η , , √ η , η , η (cid:1) t (68) | , i = 16 (cid:16) η q − √ , η q √ , , , , , , (cid:17) t (69)where η = e πi , U A | r, s i = η r | r, s i U P | r, s i = s | r, s i (70)for all r, s , and where the phases have been chosen so that U K | r, s i = | r, s i (71)for all r, s . Note that the fact that U K is an anti-unitary means that Eq. (71) fixesthe phase of | r, s i up to a sign (this is an important point: if one chooses the phasesdifferently the fiduciality conditions are much harder to solve). The action of theanti-unitary U A is: U A | , i = η | , i U A | , − i = η | , − i (72) U A | , i = η | , i (73) U A | , i = η | , i U A | , − i = η | , − i (74) U A | , i = η | , i U A | , − i = η | , − i (75) U A | , i = η | , i (76)The eigenspaces of U F Z = U A are as in Table 1. As we remarked above N = 8 is eigenspace eigenvalue dimension basis S | , i , | , − iS η | , i , | , i , | , − iS η | , i , | , i , | , − i Table 1. one of the dimensions for which fiducials exist in all 3 eigenspaces of the Zauner unitary. Since U A toggles the eigenspaces S and S we only need to calculatefiducials in S and S . We thus need to solve the equations |h ψ | D p | ψ i| = 19 (77)for all p = 0 (mod 8) and | ψ i of the form | ψ i = cos θ | , i + sin θ cos φe iχ | , i + sin θ sin φe iχ | , − i (78)with 0 ≤ θ, φ ≤ π and 0 ≤ χ , χ < π , or | ψ i = cos θ | , i + sin θe iχ | , − i (79)with 0 ≤ θ ≤ π and 0 ≤ χ < π . The fact that | ψ i is assumed to be an eigenvectorof U F Z means that it is enough to require that it satisfy the 11 equations (cid:12)(cid:12) h ψ | D p ψ i (cid:12)(cid:12) = 19 (80) (cid:12)(cid:12) h ψ | D A r p ψ i (cid:12)(cid:12) = 19 r = 0 , (cid:12)(cid:12) h ψ | D A r p ψ i (cid:12)(cid:12) = 19 r = 0 , . . . , (cid:12)(cid:12) h ψ | D A r p ψ i (cid:12)(cid:12) = 19 r = 0 , . . . , p = (0 , p = (0 , p = (0 , p = (1 ,
2) (84)or, equivalently, the following linear combinations of these equations: (cid:12)(cid:12) h ψ | D p ψ i (cid:12)(cid:12) + (cid:12)(cid:12) h ψ | D A p ψ i (cid:12)(cid:12) = 29 (E1)2 (cid:12)(cid:12) h ψ | D p ψ i (cid:12)(cid:12) − (cid:12)(cid:12) h ψ | D p ψ i (cid:12)(cid:12) − (cid:12)(cid:12) h ψ | D A p ψ i (cid:12)(cid:12) = 0 (E2) X r =0 k (1) r (cid:12)(cid:12) h ψ | D A r p ψ i (cid:12)(cid:12) + X r =0 k (1) r (cid:12)(cid:12) h ψ | D A r p ψ i (cid:12)(cid:12) = 89 (E3) (cid:12)(cid:12) h ψ | D p ψ i (cid:12)(cid:12) − (cid:12)(cid:12) h ψ | D A p ψ i (cid:12)(cid:12) = 0 (E4) X r =0 k (2) r (cid:12)(cid:12) h ψ | D A r p ψ i (cid:12)(cid:12) − X r =0 k (2) r (cid:12)(cid:12) h ψ | D A r p ψ i (cid:12)(cid:12) = 0 (E5) X r =0 ( √ k (2) r − k (1) r ) (cid:12)(cid:12) h ψ | D A r p ψ i (cid:12)(cid:12) + X r =0 ( √ k (2) r + k (1) r ) (cid:12)(cid:12) h ψ | D A r p ψ i (cid:12)(cid:12) = 0 (E6) X r =0 ( k (2) r + √ k (1) r ) (cid:12)(cid:12) h ψ | D A r p ψ i (cid:12)(cid:12) + X r =0 ( k (2) r − √ k (1) r (cid:12)(cid:12) h ψ | D A r p ψ i (cid:12)(cid:12) = 0 (E7) X r =0 k (3) r (cid:12)(cid:12) h ψ | D A r p ψ i (cid:12)(cid:12) = 0 (E8) X r =0 k (3) r (cid:12)(cid:12) h ψ | D A r p ψ i (cid:12)(cid:12) = 0 (E9) X r =0 k (4) r (cid:12)(cid:12) h ψ | D A r p ψ i (cid:12)(cid:12) = 0 (E10) X r =0 k (4) r (cid:12)(cid:12) h ψ | D A r p ψ i (cid:12)(cid:12) = 0 (E11)where k (1) = k (2) = − − k (3) = − k (4) = − (85)We begin with the fiducials in S . Substituting Eq. (78) into these equations wefind that Eqs. (E1)–(E3) are each equivalent to the single conditioncos 2 θ (1 + cos 2 θ ) = 0 (86)The solution θ = π is inconsistent with the remaining equations. On the other handif one sets θ = π in Eqs. (E4)–(E11) it is straightforward (though perhaps a littletedious) to show that the resulting equations are equivalent to the four conditionssin φ cos 2( χ − χ ) = − √ φ (87)cos φ cos 2 χ = 14 √ (cid:0) cos φ + 2 cos 2 φ − (cid:1) (88)sin φ cos 2 χ = − √ (cid:0) cos φ − φ − (cid:1) (89)cos φ − φ − φ = s − √
52 (91) e iχ = s (cid:18)q √ − √
30 + is q − √ √ (cid:19) (92) e iχ = s e − s πi (cid:18)q √ − √
30 + is q − √ √ (cid:19) (93)where s , s , s = ± S we find on substituting Eq. (79) into Eqs. (E1)–(E11) that the equations reduce to the two conditionssin θ cos 2 φ = 0 (94)1 + cos 2 θ − cos θ = 0 (95)whose solution iscos θ = 12 q − √ e iφ = e (2 r +1) πi (96)where r = 0 , . . . , To conclude: there are 8 fiducials in the S eigenspace given by | ψ i = 12 √ η p − √ η p √ + s s − √ e is χ η η √ η − + s s √ − e is ( χ − π ) η η √ ηη η (97)with s , s , s = ± e iχ = 14 (cid:18)q √ − √
30 + i q − √ √ (cid:19) (98)another 8 fiducials in the S eigenspace obtained by acting on these with the anti-unitary U A , and 4 fiducials in the S eigenspace given by | ψ i = 12 s − √ η η √ η − + i r s √ η η √ η η η (99)where r = 0 , . . . ,
3. The fiducials in S , S are on Scott-Grassl [36] orbit 8 a ; thefiducials in S are on Scott-Grassl orbit 8 b .Comparing these k -nomial basis expressions with the standard basis expressionsin Ref. [36] we see that the degree of simplification achieved is very considerable.5. Dimension fiducial Before we present our new solution for dimension 28, we revisit dimenion N =12 = 2 × X and Z are diagonal, the representation ofthe Clifford group will be 3-nomial. In this basis, the Weyl-Heisenberg group is generated by X = ω ω ω ω ω ω ω ω ω ω
00 0 0 0 0 0 0 0 0 0 0 ω ω (100)and Z = ω ω ω ω ω
00 0 0 0 0 0 0 0 0 0 0 ω ω ω ω ω ω ω , (101)where ω denotes a primitive 24th root of unity.Like in the case of dimension N = 8 discussed in detail in the previous section,we consider a maximal Abelian subgroup of the normalizer N Z of the Zauner sym-metry F Z in the Clifford group when choosing the basis of the eigenspace of F Z ofdimension 5. We find the non-normalized orthogonal basis { v , v , v , v , v } withthe vectors v = (0 , , , , , , , , , , ,
1) (102) v = (1 , , , , , , , , , , ,
0) (103) v = (1 , , ω , ω , ω , ω , ω , , , , ω ,
0) (104) v = (0 , , , , , , , , , ω , , ω ) (105) v = (1 , ω , ω , , ω , ω , , , ω , , ω ,
0) (106)In the approach for computing a solution with Zauner’s symmetry reported in[31], computing a single modular Gr¨obner basis took about 40 hours using morethan 17 GB of memory. Using the k -nomial representation of the Clifford groupand an adapted basis of the eigenspace of Zauner’s matrix, the corresponding stepuses less than 100 MB of memory and takes less than 40 seconds on the samehardware–reduced to less than 25 seconds on current computers. Moreover, instead of more than 300 different primes, now only 21 suffice. Overall, we get a speed-upby more than 3 orders of magnitude.A non-normalized fiducial vector | e ψ i = X i =0 x i | b i i (107)with respect to the representation of the Weyl-Heisenberg group generated by thematrices X and Z given in Eqs. (100) and (101), respectively, can be expressed inthe number field K = Q ( √ , √ , √ , t , s , s , √−
1) (108)of degree 192, where s = q ( √ − / , s = q (3 √
13 + 9) / , and t = 12 t + 10 . (109)In this field, a primitive 24th root of unity is given by ω = √ (cid:16) (1 − √
3) + ( √ √− (cid:17) . (110)The coefficients of | e ψ i in Eq. (107) are as follows: x = (( − √ − s + (24 √ t − √ t − √ − t − t − √−
1+ (26 √ − √ s + 28 √ t − √ t − √
39 + 96 √ t − √ t − √ x = (24 √ t − √ t − √ − t − t − √−
1+ 28 √ t − √ t − √
39 + 96 √ t − √ t − √ x = ((24 √ − s − √ t + 138 √ t + 375 √ − t + 142 t + 667) √−
1+ (28 √ − √ s − √ t − √ t − √ − √ t − √ t + 219 √ x = (( − √ − s − √ t + 138 √ t + 315 √ − t + 142 t + 43) √−
1+ (26 √ − √ s − √ t − √ t + 93 √ − √ t − √ t + 1077 √ x = (30 √ t − √ t − √
13 + 190 t + 16 t − √− − √ t + 80 √ t + 168 √ − √ t + 100 √ t − √ x = ((24 √ − s + (30 √ t − √ t − √
13 + 190 t + 16 t − √−
1+ (28 √ − √ s − √ t + 80 √ t + 111 √ − √ t + 100 √ t − √ x = (( − √ − s + (30 √ t − √ t − √
13 + 190 t + 16 t − √−
1+ (26 √ − √ s − √ t + 80 √ t + 285 √ − √ t + 100 √ t + 341 √ x = 488 √ s x = ((24 √ − s + (24 √ t − √ t − √ − t − t + 619)) √−
1+ (28 √ − √ s + 28 √ t − √ t − √
39 + 96 √ t − √ t − √ x = ((85 √
39 + 91 √ s s − √ s ) √−
1+ (23 √ − s s + 122 √ s x = ( − √ t + 138 √ t + 84 √ − t + 142 t − √− − √ t − √ t − √ − √ t − √ t + 336 √ x = ((31 √
39 + 481 √ s s + 122 √ s ) √−
1+ (139 √ − s s + 122 √ s (111) This expression for the fiducial vector is much more compact than the page fillingones given in Ref. [31] and the supplementary material of Ref. [36].6.
Dimension fiducial In order to compute a fiducial vector for dimension N = 28 = 2 × X and Z are diagonal. It turns out that such a change of basis isgiven by the matrix T = 1 √ (cid:18) − (cid:19) ⊗ , (112)which is a tensor product of a 2 × c in [36] indicates that there is a solution which possesses an order-twoanti-unitary symmetry F c = (cid:18)
11 650 17 (cid:19) , (113)in addition to the symmetry F z of order three conjectured by Zauner. In orderto compute a basis of the corresponding eigenspace, we represent the anti-unitarytransformation U F c F z as a real orthogonal matrix O F c F z ∈ O (2 N ) using the isomor-phism C → R . We obtain ten linearly independent eigenvectors v ′ i ∈ R whichare then mapped back to ten vectors | v i i ∈ C . Then we write the fiducial vectoras | ψ i = X i =0 y i | v i i , (114)with real variables y i , i = 0 , . . . ,
9. The defining equations for the SIC are given as h ψ | ψ i = 1 and (115) |h ψ | D p | ψ i| = 129 , for p = 0. (116)We solved these equations via computing a Gr¨obner basis for the correspondingpolynomial ideal, using similar techniques to those described in [31]. First, theequations are reformulated as equations over the rationals, adding an auxiliary variable. This allows us to compute a modular GB, i.e., performing all computationsmodulo a large prime.Using Magma V2.18 on a Linux PC with 3 GHz clock speed and 32 GB RAM,the first step took about 60 hours and used up to 45 GB of memory. One of thepolynomials in the modular Gr¨obner basis could be lifted to the rationals. Addingthis polynomial, the computation for three more primes took only approximatelysix hours. Having computed four modular Gr¨obner bases, six polynomials could belifted to the rationals. Adding those polynomials reduced the time to compute amodular Gr¨obner basis to only five minutes. Then, with five different primes, welifted 20 polynomials to the rationals. The following Gr¨obner basis computationstook less than one second each. In total we used 18 different primes with about 23bits each to obtain a Gr¨obner basis over the rationals. As a next step, the Gr¨obnerbasis is converted to a Gr¨obner basis with respect to lexicographic order. Thisyields a polynomial system of equations in triangular form. The coefficients of thepolynomials have numerators and denominators with more than 50 digits. It turnsout that the ideal is zero-dimensional, i.e., there are finitely many solutions. Notethat we are only interested in solutions where all variables y i assume real valuesInserting one of these solution into Eq. (114), we find that a non-normalizedfiducial vector | e ψ i = X i =0 x i | b i i (117)in the adapted bases given by the matrix T in Eq. (112) can be expressed in thenumber field K = Q ( √ , √ , √ , r , r , s , s , √−
1) (118)of degree 576, where r = q √
29 + 5 , r = q √
29 + 1 , s = 21 s − , and s = 14 s − . (119)Note that even though we do not explicitly use a 56th root of unity, it can beexpressed as ω = 1252 (cid:16) ( s − s − √− √ − (7 s − s − − √− √ (cid:17) . (120)The Galois group of K is isomorphic to C × (( C × C × C × C ) ⋊ C ), and K isan Abelian extension of Q ( √ | e ψ i in Eq. (117) are as follows: x = (((2 s s + (4 s − s − s + 42) √
29+ (( − s − s + 112) s + 203 s + (210 s + 315 s − √ r + ((1 / s + 28 s ) s + (8 s + 20 s − s − s − s + 168) √
29+ (1 / s − s − s + ( − s − s + 224) s − s + 282 s + 952)) √ r √−
1+ (((9 s + 9 s ) √ − s + 18 s + 27 s − √ r + (( − s + (18 s + 126 s − √
29 + (270 s − s − r x = ((( − s s + ( − s + 21) s + (35 s + 84)) √
29+ (( − s − s + 56) s − s + 42 s + 63 s − √ r + ((1 / s + 28 s − s + (8 s + 26 s − s − s − s + 336) √
29+ (1 / − s − s + 168) s − s s − s + 184 s + 840)) √ r √− + (((( s − s + (2 s − s − s + 28) √
29+ (( − s − s + 38) s + (8 s + 12 s − s + (10 s + 15 s − √ r + ((( − s − s ) s + (8 s + 14 s − s + (46 s + 112 s − √
29+ (( − s + 20 s + 56) s + ( − s − s + 448) s − s − s + 1400))) r x = ((2 s s − s s − s ) √ r + ((1 / s + 14 s − s + (4 s + 16 s − s − s − s + 392) √
29+ (1 / − s − s + 448) s + (12 s − s − s + (204 s + 16 s − √ r √−
1+ ((( − s − s + 70) √
29+ ((4 s + 6 s − s + (8 s + 12 s − s − s − s − √ r + ((( − s − s + 56) s + (4 s + 28 s − s + (116 s + 308 s − √
29+ ((12 s + 18 s − s + (60 s − s − s − s − s + 2576))) r x = ((1 / − s − s + 168) s + (4 s + 16 s ) s + (68 s + 128 s − √
29+ (1 / s + 28 s − s + (12 s − s − s − s − s + 3472)) √ r √−
1+ ((( − s − s + 56) s + ( − s − s + 448) s + (172 s + 448 s − √
29+ ((24 s + 36 s − s + (228 s + 168 s − s − s − s + 4144)) r x = (( − s s + 8 s s + 28 s ) √ r + ((1 / s + 14 s − s + (4 s + 16 s − s − s − s + 392) √
29+ (1 / − s − s + 448) s + (12 s − s − s + (204 s + 16 s − √ r √−
1+ (((2 s + 4 s − √
29+ (( − s − s + 18) s + ( − s − s + 36) s + (32 s + 48 s + 378))) √ r + ((( − s − s + 56) s + (4 s + 28 s − s + (116 s + 308 s − √
29+ ((12 s + 18 s − s + (60 s − s − s − s − s + 2576))) r x = ((( s s + (2 s − s − s − √
29+ ((6 s + 9 s − s + 203 s − s − s + 1204)) √ r + ((1 / s + 28 s − s + (8 s + 26 s − s − s − s + 336) √
29+ (1 / − s − s + 168) s − s s − s + 184 s + 840)) √ r √−
1+ (((( − s + 2) s + ( − s + 1) s + (17 s − √
29+ ((2 s + 3 s − s + ( − s − s + 65) s − s − s + 364)) √ r + ((( − s − s ) s + (8 s + 14 s − s + (46 s + 112 s − √
29+ (( − s + 20 s + 56) s + ( − s − s + 448) s − s − s + 1400))) r x = ((( − s s + ( − s + 21) s + (7 s − √
29+ ((12 s + 18 s − s − s − s − s + 2366)) √ r + ((1 / s + 28 s ) s + (8 s + 20 s − s − s − s + 168) √
29+ (1 / s − s − s + ( − s − s + 224) s − s + 282 s + 952)) √ r √−
1+ ((( − s − s ) √
29 + (87 s − s − s + 168)) √ r + (( − s + (18 s + 126 s − √
29 + (270 s − s − r x = (((1 / s + 28 s − s + (2 s + 2 s ) s − s − s + 336) √
29+ (1 / − s − s + 168) s + ( − s + 2 s + 168) s + (54 s + 226 s − √
7+ ((( − s − s + 28) s + ( − s − s + 224) s + (34 s + 154 s − √
29+ (( − s + 20 s + 56) s + (6 s + 154 s − s + (174 s − s − √ r √−
1+ (((1 / s + 28 s − s + (2 s + 2 s ) s − s − s + 336) √
29+ (1 / − s − s + 168) s + ( − s + 2 s + 168) s + (54 s + 226 s − √
7+ (((2 s + 8 s − s + (22 s + 70 s − s − s − s + 560) √
29+ ((6 s − s − s + ( − s − s + 56) s − s + 406 s + 1624))) √ r x = ((((( s − s + (2 s − s − s + 70) √
29+ (( − s − s + 38) s + ( − s − s + 76) s + (16 s + 24 s − √
7+ (( s s − s s − s ) √
29+ ((6 s + 9 s − s + ( − s − s + 224) s − s − s + 784))) r + (((1 / − s − s + 28) s + (2 s + 2 s − s + (22 s + 58 s − √
29+ (1 / s + 28 s − s + ( − s + 2 s + 168) s − s − s + 504)) √
7+ ((( − s − s ) s + ( − s − s + 84) s + (20 s + 98 s − √
29+ (( − s + 10 s + 28) s + (48 s + 14 s − s + (132 s − s − √ r √−
1+ ((((( − s + 2) s + ( − s + 4) s + (8 s − √
29+ ((2 s + 3 s − s + (4 s + 6 s − s − s − s + 826)) √
7+ (( s s − s s − s ) √
29+ ((6 s + 9 s − s + ( − s − s + 224) s − s − s + 784))) r + (((1 / − s − s + 28) s + (2 s + 2 s − s + (22 s + 58 s − √
29+ (1 / s + 28 s − s + ( − s + 2 s + 168) s − s − s + 504)) √
7+ ((( s + 4 s ) s + (8 s + 14 s − s − s − s + 168) √
29+ ((3 s − s − s + ( − s − s + 448) s − s + 266 s + 1232))) √ r x = ((((( − s − s + ( s − s + (11 s + 28)) √
29+ (( − s − s + 38) s + ( − s − s + 103) s + (46 s + 69 s − √
7+ (( s s + ( − s − s + (7 s − √
29+ (( − s − s + 56) s + ( − s − s + 259) s + (126 s + 189 s − r + (((1 / − s − s − s + ( s − s − s − s − s + 224) √
29+ (1 / − s + 21 s + 140) s + ( − s + 12 s + 196) s + (93 s + 9 s − √
7+ ((( s + s ) s + (5 s + 14 s − s + (13 s + 49 s − √
29+ (( − s + s + 84) s + ( − s − s + 84) s + (27 s − s − √ r √−
1+ ((((( s + 2) s + ( − s + 1) s − s − √
29+ ((2 s + 3 s − s + (10 s + 15 s − s − s − s + 700)) √
7+ (( s s + ( − s − s + (7 s − √
29+ (( − s − s + 56) s + ( − s − s + 259) s + (126 s + 189 s − r + (((1 / − s − s − s + ( s − s − s − s − s + 224) √
29+ (1 / − s + 21 s + 140) s + ( − s + 12 s + 196) s + (93 s + 9 s − √
7+ ((( − s − s ) s + ( − s − s + 42) s − s − s + 168) √ + ((9 s − s − s + (9 s + 28 s − s − s + 119 s + 252))) √ r x = (((9 s √ − s − s − s + 336) √ s s + (2 s + 21) s − s − √
29+ ((12 s + 18 s − s + (84 s + 126 s − r + (((1 / s + 14 s − s + ( − s − s + 14) s − s − s + 112) √
29+ (1 / − s − s + 448) s + ( − s + 20 s + 56) s + (132 s − s − √
7+ ((42 s − s − s + 420) √
29 + (216 s + 63 s − √ r √−
1+ ((( − s √
29 + (87 s + (36 s + 54 s − √ s s + (2 s + 21) s − s − √
29+ ((12 s + 18 s − s + (84 s + 126 s − r + (((1 / s + 14 s − s + ( − s − s + 14) s − s − s + 112) √
29+ (1 / − s − s + 448) s + ( − s + 20 s + 56) s + (132 s − s − √
7+ (( − s + (36 s + 63 s − √
29 + ( − s − s + 2016))) √ r x = ((( − s √
29 + (87 s + (36 s + 54 s − √ − s s + ( − s − s + (56 s + 42)) √
29+ (( − s − s + 315) s − s − s + 378))) r + (((1 / s + 14 s − s + ( − s − s + 14) s − s − s + 112) √
29+ (1 / − s − s + 448) s + ( − s + 20 s + 56) s + (132 s − s − √
7+ ((42 s − s − s + 420) √
29 + (216 s + 63 s − √ r √−
1+ (((9 s √ − s − s − s + 336) √ − s s + ( − s − s + (56 s + 42)) √
29+ (( − s − s + 315) s − s − s + 378))) r + (((1 / s + 14 s − s + ( − s − s + 14) s − s − s + 112) √
29+ (1 / − s − s + 448) s + ( − s + 20 s + 56) s + (132 s − s − √
7+ (( − s + (36 s + 63 s − √
29 + ( − s − s + 2016))) √ r x = ((((( s + 2) s + ( − s + 1) s − s − √
29+ ((2 s + 3 s − s + (10 s + 15 s − s − s − s + 700)) √
7+ (( − s s + ( s + 21) s − s + 84) √
29+ ((6 s + 9 s − s + (6 s + 9 s − s − s − s + 364))) r + (((1 / − s − s − s + ( s − s − s − s − s + 224) √
29+ (1 / − s + 21 s + 140) s + ( − s + 12 s + 196) s + (93 s + 9 s − √
7+ ((( s + s ) s + (5 s + 14 s − s + (13 s + 49 s − √
29+ (( − s + s + 84) s + ( − s − s + 84) s + (27 s − s − √ r √−
1+ ((((( − s − s + ( s − s + (11 s + 28)) √
29+ (( − s − s + 38) s + ( − s − s + 103) s + (46 s + 69 s − √
7+ (( − s s + ( s + 21) s − s + 84) √
29+ ((6 s + 9 s − s + (6 s + 9 s − s − s − s + 364))) r + (((1 / − s − s − s + ( s − s − s − s − s + 224) √
29+ (1 / − s + 21 s + 140) s + ( − s + 12 s + 196) s + (93 s + 9 s − √
7+ ((( − s − s ) s + ( − s − s + 42) s − s − s + 168) √
29+ ((9 s − s − s + (9 s + 28 s − s − s + 119 s + 252))) √ r x = ((((( − s + 2) s + ( − s + 4) s + (8 s − √
29+ ((2 s + 3 s − s + (4 s + 6 s − s − s − s + 826)) √ − s s + 4 s s + 14 s ) √
29+ (( − s − s + 56) s + (24 s + 36 s − s + (84 s + 126 s − r + (((1 / − s − s + 28) s + (2 s + 2 s − s + (22 s + 58 s − √
29+ (1 / s + 28 s − s + ( − s + 2 s + 168) s − s − s + 504)) √
7+ ((( − s − s ) s + ( − s − s + 84) s + (20 s + 98 s − √
29+ (( − s + 10 s + 28) s + (48 s + 14 s − s + (132 s − s − √ r √−
1+ ((((( s − s + (2 s − s − s + 70) √
29 + (( − s − s + 38) s + ( − s − s + 76) s + (16 s + 24 s − √ − s s + 4 s s + 14 s ) √
29+ (( − s − s + 56) s + (24 s + 36 s − s + (84 s + 126 s − r + (((1 / − s − s + 28) s + (2 s + 2 s − s + (22 s + 58 s − √
29+ (1 / s + 28 s − s + ( − s + 2 s + 168) s − s − s + 504)) √
7+ ((( s + 4 s ) s + (8 s + 14 s − s − s − s + 168) √
29+ ((3 s − s − s + ( − s − s + 448) s − s + 266 s + 1232))) √ r x = (( − s √
203 + (42 s − √ √ r + (((16 s − s − √ − s − √
7+ (( − s − s − √ − s − s + 812))) √−
1+ ( − s √
203 + ( − s + 84) √ √ r + (( − s + 34 s + 248) √
29 + (174 s + 696)) √
7+ ( − s − s − √ − s − s + 812 x = ((( − s − s + 56) √
203 + (42 s + 168) √ √ r + (((40 s + 26 s − √
29 + (232 s − s − √
7+ ((60 s + 42 s − √
29 + (116 s + 406 s + 812)))) √−
1+ (( − s − s + 56) √
203 + ( − s − √ √ r + (( − s − s + 788) √ − s + 58 s + 4988) √
7+ (60 s + 42 s − √
29 + 116 s + 406 s + 812 x = ((4 s + 8 s − √ r + ((( − s − s + 452) √ − s + 1740) √ s − s ) √
29+ (116 s − s + 3248)))) √− s + 8 s − √ r + ((16 s + 68 s − √
29 + (348 s − √ s − s ) √
29 + 116 s − s + 3248 x = ((( − s − s + 192) √ − s − s + 2320) √ s + 112 s − √
29+ (348 s − √− s + 144 s − √
29 + (116 s + 928 s − √
7+ (68 s + 112 s − √
29 + 348 s − x = (( − s − s + 140) √ r + ((( − s − s + 452) √ − s + 1740) √
7+ ((60 s − s ) √
29 + (116 s − s + 3248)))) √−
1+ ( − s − s + 140) √ r + ((16 s + 68 s − √
29 + (348 s − √
7+ (60 s − s ) √
29 + 116 s − s + 3248 x = (((4 s + 2 s − √
203 + ( − s − √ √ r + (((40 s + 26 s − √
29 + (232 s − s − √ s + 42 s − √
29+ (116 s + 406 s + 812)))) √− s + 2 s − √
203 + (42 s + 168) √ √ r + (( − s − s + 788) √ − s + 58 s + 4988) √
7+ (60 s + 42 s − √
29 + 116 s + 406 s + 812 x = ((18 s √
203 + ( − s + 84) √ √ r + (((16 s − s − √ − s − √
7+ (( − s − s − √ − s − s + 812))) √−
1+ (18 s √
203 + (42 s − √ √ r + (( − s + 34 s + 248) √
29 + (174 s + 696)) √
7+ ( − s − s − √ − s − s + 812 x = ((( − s − s + 56) s + ( − s + 28 s + 112) s + (104 s + 140 s − √
29+ ((36 s − s − s + (216 s − s − s − s − s + 7392)) r √−
1+ ((1 / − s − s + 56) s + ( − s − s + 112) s + (8 s + 92 s − √
29+ (1 / − s + 84 s + 560) s + (24 s + 36 s − s + (264 s − s − √ r x = (((( s + 2) s + (2 s + 4) s − s − √
29+ ((2 s + 3 s − s + (4 s + 6 s − s − s − s + 826)) √ r + (((2 s + 2 s ) s + ( − s − s + 168) s − s − s + 672) √
29+ (( − s + 2 s + 168) s + (36 s + 112 s − s + (540 s + 56 s − r √−
1+ ((( − s s + 4 s s + 14 s ) √
29+ ((6 s + 9 s − s + ( − s − s + 224) s − s − s + 784)) √ r + ((1 / − s − s + 56) s + (8 s + 20 s − s + (16 s + 76 s − √
29+ (1 / − s + 42 s + 280) s + ( − s − s + 224) s + (96 s − s − √ r x = ((( − s + ( − s − s + (6 s + 28)) √
29+ ((4 s + 6 s − s + (2 s + 3 s − s − s − s + 252)) √ r + ((( − s − s + 56) s + ( − s − s + 28) s + (20 s + 14 s − √
29+ ((12 s + 18 s − s + ( − s + 42 s + 280) s − s + 42 s + 1904))) r √−
1+ ((( − s s + ( − s − s + (28 s − √
29+ ((6 s + 9 s + 147) s − s − s + 1596)) √ r + ((1 / − s − s ) s + ( − s − s + 84) s + (40 s + 106 s − √
29+ (1 / s + 14 s − s + (42 s + 34 s − s − s − s + 896)) √ r x = ((( − s + 9 s ) √
29 + (87 s − s − s + 168)) √ r + (( − s − s − s + 672) √
29 + (162 s + 504 s − r √−
1+ (((2 s s + ( − s − s − s + 42) √
29+ ((12 s + 18 s − s + (12 s + 18 s + 91) s − s − s + 770)) √ r + ((1 / s + 56 s − s + (4 s + 4 s − s − s − s + 616) √
29+ (1 / − s − s + 336) s + ( − s + 4 s + 336) s + (162 s + 272 s − √ r x = ((( − s + 9 s ) √
29 + (87 s − s − s + 168)) √ r + ((84 s + (90 s + 252 s − √
29 + ( − s − s + 1512))) r √−
1+ (((2 s s + ( − s − s − s + 42) √
29+ ((12 s + 18 s − s + (12 s + 18 s + 91) s − s − s + 770)) √ r + ((1 / − s − s + 224) s + ( − s − s + 28) s + (58 s + 148 s − √
29+ (1 / s + 112 s − s + (36 s − s − s − s − s + 1512)) √ r x = ((( − s + ( − s − s + (6 s + 28)) √
29+ ((4 s + 6 s − s + (2 s + 3 s − s − s − s + 252)) √ r + (((4 s + 10 s − s + (2 s + 14 s − s − s − s + 280) √
29+ (( − s − s + 112) s + (30 s − s − s + (204 s − s − r √−
1+ ((( − s s + ( − s − s + (28 s − √
29+ ((6 s + 9 s + 147) s − s − s + 1596)) √ r + ((1 / s + 14 s ) s + (10 s + 22 s − s − s − s + 168) √
29+ (1 / − s − s + 448) s + ( − s − s + 392) s + (96 s + 202 s − √ r x = (((( s + 2) s + (2 s + 4) s − s − √
29+ ((2 s + 3 s − s + (4 s + 6 s − s − s − s + 826)) √ r + ((( − s − s ) s + (20 s + 56 s − s + (76 s + 112 s − √
29+ ((18 s − s − s + ( − s − s + 336) s − s − s + 5040))) r √−
1+ ((( − s s + 4 s s + 14 s ) √
29+ ((6 s + 9 s − s + ( − s − s + 224) s − s − s + 784)) √ r + ((1 / s + 14 s − s + ( − s − s + 56) s − s − s + 280) √
29+ (1 / s − s − s + (24 s + 36 s − s − s + 204 s + 896)) √ r Finally, the (non-normalized) fiducial vector with respect to the standard basis canbe obtained as | e ψ i = T | e ψ i .We have also tried to compute a fiducial vector for dimension N = 18 = 3 × N = 18we only have the Zauner symmetry (as indicated by the numerical solutions), whichmeans we have to search for a fiducial vector within a 7-dimensional complex vectorspace, resulting in 13 real parameters. By contrast for N = 28 the additionalsymmetry reduces the number of real parameters to only 10. It turns out that thedisadvantage, of having more parameters for N = 18, outweighs the advantage,of a more sparse group representation. Consequently we were unable to obtain asolution in the time available. 7. Conclusion
In this paper we showed that the standard representation of the Clifford grouphas a system of imprimitivity consisting of k -dimensional subspaces, where k is thesquare free part of the dimension. This means that one can choose a basis such thatthe representation matrices are all what we call k -nomial, with exactly one non-zero k × k block in each row and each column of blocks. We then used this basis toperform a hand-calculation of the (already known) exact fiducials in dimension 8, and to obtain by machine-calculation a new exact fiducial in dimension 28. Wealso revisited the machine calculation of a fiducial in dimension 12 having only theZauner symmetry, and found that the computation time was reduced by a factorof more than 10 , and that the solution obtained was much more compact. Ourresults suggest that the k -nomial basis is better adapted to the SIC-problem thanthe standard basis. As we remarked in the Introduction, the fact that SICs existat all (analytically or numerically) in every dimension ≤
67 in spite of the definingequations being greatly over-determined, and the fact that the Galois group ofthe known exact fiducials has a particularly simple form even among the class ofsolvable groups, both hint at the presence of some underlying symmetry or otheralgebraic feature of the equations which has hitherto escaped notice. Our hope isthat the work reported here will help us to discover that hidden feature and so leadto a solution to the SIC-existence problem.The Clifford group has numerous applications. It appears to us that the sparsityof the representation matrices described here means that they are likely to berelevant to other problems, apart from the SIC-existence problem.As we mentioned in the Introduction there are several different, though closelyrelated constructions which go by the name “Clifford group”. Besides the versionof the group considered here there is the version defined to be the normalizer of thetensor product of several copies of the Weyl-Heisenberg group [1, 2], and there isalso what might be called the Galoisian version defined in prime power dimensionusing a finite field [3,4]. It would be interesting to try to extend the analysis of thispaper to this more general setting.
Acknowledgements
We thank Jon Yard for valuable discussions. DMA was supported in part bythe U. S. Office of Naval Research (Grant No. N00014-09-1-0247) and by the JohnTempleton Foundation. IB was supported by the Swedish Research Council undercontract VR 621-2010-4060. ˚A.E. acknowledges support from the Wenner-Grenfoundations, and hospitality and support from Institut Mittag-Leffler where partof the research for this paper was carried out. Research at Perimeter Instituteis supported by the Government of Canada through Industry Canada and by theProvince of Ontario through the Ministry of Research & Innovation. The Centre forQuantum Technologies is a Research Centre of Excellence funded by the Ministryof Education and the National Research Foundation of Singapore.
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