aa r X i v : . [ qu a n t - ph ] F e b Decomposition of Clifford Gates
Tefjol Pllaha, Kalle Volanto, Olav Tirkkonen
Aalto University, Helsinki, Finlande-mails: { tefjol.pllaha, kalle.volanto, olav.tirkkonen } @aalto.fi Abstract —In fault-tolerant quantum computation and quan-tum error-correction one is interested on Pauli matrices thatcommute with a circuit/unitary. We provide a fast algorithm thatdecomposes any Clifford gate as a minimal product of Cliffordtransvections. The algorithm can be directly used for findingall Pauli matrices that commute with any given Clifford gate.To achieve this goal, we exploit the structure of the symplecticgroup with a novel graphical approach.
I. I
NTRODUCTION
The Clifford group is of central importance in quantum in-formation and computation. This paper is primarily motivatedby its importance in fault-tolerant quantum computation andquantum error-correction [1], [2]. Traditionally, the Cliffordgroup is studied via its connection with the binary symplecticgroup [3] and the associated decompositions of the latter. TheBruhat decomposition of the symplectic group [4] gives a stan-dard generating set made of qubit permutation (16), diagonalgates (17), and partial Hadamard gates (18). Alternatively, theClifford group can be studied via the transvection decom-position of the symplectic group, which we briefly describein Section III. It is well-known [5], [6] that the symplecticgroup is generated by symplectic transvections (19). Althoughthese references give a constructive proof, the decompositionprimarily relies on exhaustive search. In this paper we givea simple and fast algorithm that decomposes any symplecticmatrix as a minimal product of symplectic transvections. Onthe other hand, the Clifford gates (45) correspond to symplec-tic transvections, and for this reason we will refer to themas Clifford transvections. By definition, Clifford transvectionsare sparse (in fact, they are the most sparse Cliffords otherthan Paulis and diagonal Cliffords), and given their simpleconjugation action, they are also easy to implement. Thisyields directly a decomposition of any m -qubit Clifford gateas a minimal product of Clifford transvections.We exploit the structure of symplectic matrices with anovel graphical approach. We associate to a symplectic matrix,written as a minimal product of transvections, a (binary,symmetric) Gram-type matrix (20) that captures the com-mutativity relations of the defining transvections. Viewed asan adjacency matrix, it yields a graph whose directed pathscompletely determine the given symplectic matrix; see Theo-rem 1. These directed paths can be counted with an invertibleupper-triangular matrix (26), and this allows us to reduce thedecomposition problem to a matrix triangulation problem overthe binary field. For the latter we make use of the results of [7].In [8], the authors studied the Clifford group via thesupport (44) of a unitary matrix. In that language, Cliffordtransvections are precisely those Cliffords that have a supportof size two, which is smallest support among non-PauliCliffords. On top of being a useful algebraic tool, the support of a unitary encodes valuable information about the Paulis thatcommute with the given unitary. In [8, Prop. 9], the authorscompute the support of standard Clifford gates (16)-(18). Theresults of this paper provide a fast algorithm for computingthe support of any Clifford gate. Heuristically, we expect ourresults to have applications in designing flag gadgets [9], [10]for stabilizer circuits.II. P
RELIMINARIES
A. The binary symplectic group
The binary symplectic group, denoted Sp ( m ; 2 ) , consistsof m × m matrices over the binary field F that preserve thematrix Ω = [ m I m I m m ] , (1)under congruence. That is, F ∈ Sp ( m ; 2 ) iff FΩF T = Ω .Equivalently, symplectic matrices are precisely those matricesthat preserve the symplectic inner product over F m ⟨ ( a , b ) ∣ ( c , d ) ⟩ s = ad T + bc T = ( a , b ) Ω ( c , d ) T . (2)We will denote by GL ( n ; 2 ) and Sym ( n ; 2 ) the groups of n × n invertible and symmetric matrices over the binary field F , respectively. A matrix F = [ A BC D ] ∈ Sp ( m ; 2 ) satisfies FΩF T = Ω , which in turn is equivalent with AB T , CD T ∈ Sym ( m ; 2 ) and AD T + BC T = I m . In Sp ( m ; 2 ) we distin-guish two subgroups: F D ∶= { F D ( P ) = [ P m m P − T ] ∣ P ∈ GL ( m ; 2 )} , (3) F U ∶= { F U ( S ) = [ I m S m I m ] ∣ S ∈ Sym ( m ; 2 )} . (4)Above, ( ● ) − T denotes the inverse transposed, and directly bydefinition we have F D ≅ GL ( m ; 2 ) and F U ≅ Sym ( m ; 2 ) .Together with matrices F Ω ( r ) = [ I m ∣− r I m ∣ r I m ∣ r I m ∣− r ] , (5)with I m ∣ r being the block matrix with I r in upper left cornerand 0 elsewhere, and I m ∣− r = I m − I m ∣ r , these two groups arethe building blocks of the Bruhat decomposition with manyapplications in quantum computation [4], [11]. A symplecticmatrix F ∈ Sp ( m ; 2 ) is said to be an involution if F = I m and is said to be hyperbolic if ⟨ v ∣ vF ⟩ s = for all v ∈ F m .It is straightforward to verify that a hyperbolic map is also aninvolution. We will denote Fix ( F ) ∶= ker ( I + F ) ∶= { v ∈ F m ∣ v = vF } , (6) Res ( F ) ∶= rs ( I + F ) ∶= { v + vF ∣ v ∈ F m } , (7)here ker ( ● ) and rs ( ● ) denote the null space and the rowspace of a matrix, respectively. By definition, these spacessatisfy dim Res ( F ) + dim Fix ( F ) = m. (8)Involutions have the nice property that Res ( F ) ⊆ Fix ( F ) . Ad-ditionally, for an involution we have ⟨ x ∣ yF ⟩ s = ⟨ xF ∣ y ⟩ s andthus ⟨ x + xF ∣ y + yF ⟩ s = for all x , y ∈ F m . This meansthat Res ( F ) is self-orthogonal (or self-dual if dim Res ( F ) = m ) with respect to (2). B. The Heisenberg-Weyl group
The bit-flip and the phase-flip gates are given by X ∶= [ ] and Z ∶= [ − ] , (9)respectively. For vectors a , b ∈ F m we will denote D ( a , b ) ∶= X a Z b ⊗ ⋯ ⊗ X a m Z b m . (10)The Heisenberg-Weyl group is defined as H W N ∶= { i k D ( a , b ) ∣ a , b ∈ F m , k ∈ Z } ⊂ U ( N ) , (11)where N = m . We will denote by PHW N ∶ =HW N /{± I N , ± i I N } the projective Heisenberg-Weyl group.Hermitian elements of HW N are given (and denoted) by E ( a , b ) ∶ = i ab T D ( a , b ) . C. The Clifford group
The
Clifford group
Cliff N is defined to be the normalizerof HW N in U ( N ) , that is, Cliff N ∶ = { G ∈ U ( N ) ∣ G HW N G † ⊂ HW N } . (12)In order to obtain a finite group, (12) is meant modulo U ( ) .Let { e , . . . , e m } be the standard basis of F m , and con-sider G ∈ Cliff N . Let c i ∈ F m be such that GE ( e i ) G † = ± E ( c i ) . (13)Then the matrix F G whose i th row is c i is a symplectic matrixsuch that GE ( c ) G † = ± E ( cF G ) (14)for all c ∈ F m . We thus have a group homomorphism Φ ∶ Cliff N Ð → Sp ( m ; 2 ) , G z→ F G . (15)In addition, Φ is surjective with kernel ker Φ = PHW N [12],and thus Cliff N / PHW N ≅ Sp ( m ; 2 ) . It follows that Cliff N is generated by preimages of symplectic matrices (3),(4),(5).Here a preimage Φ − ( F ) is meant up to HW N . Thesepreimages are, respectively, G D ( P ) ∶ = ∣ v ⟩ z→ ∣ vP ⟩ , (16) G U ( S ) ∶ = diag ( i vSv T mod 4 ) v ∈ F m , (17) G Ω ( r ) ∶ = ( H ) ⊗ r ⊗ I m − r , (18)where H is the Hadamard gate.Since Φ is a homomorphism we have that Φ ( G † ) = F − G . Itfollows that if G ∈ Cliff N is Hermitian then F G is a symplecticinvolution. Conversely, if F is a symplectic involution then G = Φ − ( F ) satisfies G ∈ HW N . As mentioned, a specialclass of involutions are the hyperbolic maps. If G ∈ Cliff N corresponds to a hyperbolic F ∈ Sp ( m ; 2 ) then (14) impliesthat GEG † commutes with E for all E .III. T RANSVECTION D ECOMPOSITION OF S YMPLECTIC M ATRICES A symplectic transvection is a symplectic map with one-dimensional residue space. It is easily seen that if Res ( F ) = ⟨ v ⟩ then the matrix F ∈ Sp ( m ; 2 ) must act as T v ∶ = I + Ωv T v , x z→ x + ⟨ x ∣ v ⟩ s v . (19)We will call two transvections T v , T w independent if thedefining v , w are independent. Otherwise, we will call thetransvections dependent. Note also that T v , T w commute , thatis, T v ⋅ T w = T w ⋅ T v iff ⟨ v ∣ w ⟩ s = , that is, iff v , w areorthogonal (with respect to (2) of course).It is well-known that Sp ( m ; 2 ) is generated by transvec-tions. It is shown in [5], [6] that a non-hyperbolic map F can be written as a product of r independent transvections T v , . . . , T v r , where r = r ( F ) ∶ = dim Res ( F ) = m − dim Fix ( F ) and Res ( F ) = ⟨ v , . . . , v r ⟩ . The strategy of [5]is to find v such that ⟨ x ∣ xF ⟩ s = (which exists for non-hyperbolics), and consider FT v , v = x + xF ∈ Res ( F ) , forwhich r ( FT v ) = r ( F ) − . One then repeats the processaccordingly until a one-dimensional residue space is reached.The following result will enable us to restrict without lossof generality to non-hyperbolic maps. Lemma 1 ([5, 2.1.8]) . Let F ∈ Sp ( m ; 2 ) be hyperbolic. Thenthere exists v ∈ F m such that FT v is non-hyperbolic and Res ( F ) = Res ( FT v ) .Proof. Fix any ≠ v = x + xF ∈ Res ( F ) . Then any y suchthat ⟨ y ∣ v ⟩ s = = ⟨ yF ∣ v ⟩ s (which of course exists) satisfies ⟨ y ∣ yFT v ⟩ s = , and thus FT v is non-hyperbolic. Next, bythe choice of v , Res ( FT v ) ⊆ Res ( F ) holds trivially, andequality is due to equal cardinalities.It follows from Lemma 1 that a hyperbolic map F is aproduct of r + transvections, r of which form a basis for Res ( F ) , and the additional transvection is dependent of thefirst r .For involutions (hyperbolic or not) we have the followingnicer result. Proposition 1.
Any involution is a product of commutingtransvections. The converse is also true, that is, any productof commuting transvections yields and involution.Proof.
The result follows immediately by the fact that twotransvections commute iff their defining vectors are orthogo-nal, along with the fact that the residue space of an involutionis self-orthogonal.
A. A Gram-type matrix
In this section F will be a generic symplectic matrix.We associate to a minimal transvection decomposition F = T v ⋯ T v r a Gram-type matrix A ( v , . . . , v r ) ∶ = [⟨ v i ∣ v j ⟩ s ] i,j = VΩV T , (20)here V is the r × m matrix formed by stacking v , . . . , v r .Obviously, A is symmetric and has zero diagonal. Since aminimal transvection decomposition is given by some basisof the residue space, we will assume that v i ∈ Res ( F ) . Notethat A = iff F is an involution iff V is self-orthogonal. Onthe other hand, ⟨ v i ∣ v j ⟩ s = ⟨ x i + x i F ∣ x j + x j F ⟩ s (21) = ⟨ x i F ∣ x j ⟩ s + ⟨ x i ∣ x j F ⟩ s (22) = x i FΩx T j + x j ΩF T x T j (23) = x i ( F + F − ) Ωx T j (24) = ⟨ x i ( F + F − ) ∣ x j ⟩ s . (25)Obviously, F is an involution iff F = F − , and thus A alsocaptures how far is F from being an involution, or equivalently,how far is V from being self-orthogonal. In what follows wewill denote A u ∶ = triu ( A ) the upper triangular part of A and B ( v , . . . , v r ) ∶ = r − ∑ ℓ = A ℓ u . (26)By definition, it follows that B is upper triangular with all-ones diagonal for any symplectic F , and is the identity matrixfor any involution (since in this case A = ) . Moreover, A u is r × r upper triangular with all-zero diagonal. This yields A r u = , and thus B = ( I r + A u ) − and A u = I r + B − . (27)The matrices A and B have a natural graphical interpretation.Let us start with A , which can be thought as the adjacencymatrix of the graph with vertices v i and edges ( v i , v j ) iff ⟨ v i ∣ v j ⟩ s = . On the other hand, its upper triangular part A u can be thought as the adjacency matrix of the corresponding directed graph with edges ( v i , v j ) iff ⟨ v i ∣ v j ⟩ s = and i < j .As for the matrix B , note first that entry ( i, j ) of A ℓ u countsdirected paths from v i to v j of length ℓ . Thus, entry ( i, j ) (always for i < j ) counts the number of directed paths from v i to v j .Before providing an example of the notions introduced, wepoint out that the matrix B also captures the number of distinct transvection decompositions of a given symplectic matrix F .However, this treatment goes beyond the scope of this paperand will be presented in future work. Example 1.
Let us consider an example with m = and F = T v T v T v T v T v , where V = ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣ v v v v v ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦ = ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦ . Then one computes A = ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦ and B = ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦ . The graphical description of this scenario is given in Figure 1.For instance, entry b , = and there are precisely threedirected paths from v to v , namely, ( v , v ) , ( v , v , v ) ,and ( v , v , v , v ) . v v v v v Fig. 1. The directed graph with adjacency matrix A u . Theorem 1.
For any symplectic matrix F = T v ⋯ T v r wehave F = I + ΩV T BV . (28) As a consequence, if F is an involution then F = I + ΩV T V Proof.
By the definition of transvections, the action of F on x is given by some linear combination of v j added to x , thatis xF = x + r ∑ j = w j v j , (29)where w j depends on ⟨ x ∣ v i ⟩ s for i < j . We claim that w j = xΩV T B j where B j is the j th column of B . This in turn willcomplete the proof. In order to prove the claim, note that theinput of T v j is xT v ⋯ T v j − . Thus ⟨ x ∣ v i ⟩ s contributes to w j only if i < j and there is a directed path form v i to v j ,which could be of length ≤ ℓ ≤ j − i . This information isprecisely encoded by B j .To the best of our knowledge, Theorem 1 constitutes a novelstructural result about symplectic matrices, and comparing itwith (19), should come as no surprise. This structure is themain building block of what follows. Based on Theorem 1, itis imperative to consider the residue matrix ̂ F ∶ = Ω ( I + F ) = V T BV = ∑ i,j b i,j v i T v j . (30)The terminology comes from the obvious fact that rs (̂ F ) = Res ( F ) . Note that ̂ F is symmetric iff B = I (recall that B islower triangular) iff F is an involution. Moreover, since ̂ F T has all-zero diagonal iff ̂ F does, and since x ̂ F T x T = xΩ ( I + F T ) x T = xΩx T + xΩF T x T = ⟨ x ∣ xF ⟩ s , (31)we conclude that ̂ F has all-zero diagonal iff F is hyperbolic.In such case F is also an involution, and thus ̂ F is alternating (that is, symmetric and all-zero diagonal). It follows byLemma 1 that we may restrict ourselves to non-hyperbolicmaps, and thus we will assume that ̂ F is not alternating. . Decomposition of Symplectic Involutions In this subsection we will present a simple algorithm for thedecomposition of (non-hyperbolic) symplectic involutions, andprovide intuition for the much more delicate decomposition ofgeneral symplectic matrices.
Theorem 2 (Transvection Decomposition of Involutions) . Let F be a non-hyperbolic involution, so that the residue matrix ̂ F is non-alternating. Then there exists P ∈ GL ( m ; 2 ) suchthat F = T v ⋯ T v r , where r = dim Res ( F ) and v j is the j throw of P ̂ F for ≤ j ≤ r .Proof. Let R be the matrix of row operations that transforms ̂ F into Row-Reduced Echelon form. Let E be the r × r upperleft block of R ̂ FR T , which is invertible by construction. Itwill also be symmetric and have non-zero diagonal since F is non-hyperbolic involution. Then there exists Q ∈ GL ( r ; 2 ) such that QEQ T = I r ; see [5, 2.1.14] for instance. Now put P = blkdiag ( Q , I m − r ) R . Then P ̂ FP T = [ I r
00 0 ] . (32)We will consider the nonzero rows of P ̂ F , that is, [ QE 0 ] R − T . For ≤ j ≤ r let w j denote the j th rowof P ̂ F , that is, w j = e j P − T , where e j ∈ F m is the j thstandard basis vector. Put F ′ = T w ⋯ T w r . Since w j ’s arelinear combinations of v j ’s and since F is an involutionit follows that A ( w , . . . , w r ) = A ( v , . . . , v r ) = r and B ( w , . . . , w r ) = B ( v , . . . , v r ) = I r . Then (30) yields ̂ F ′ = r ∑ j = w j T w j = r ∑ j = P − e j T e j P − T = P − [ I r
00 0 ] P − T = ̂ F , (33)and thus F = F ′ .The strength of Theorem 2 is that, as we will see, it can begeneralized to non-involutions. The case of involutions can bedealt separately with an alternate approach, which, however,does not generalize to non-involutions. According to [13,Thm. 4.1], an involution F is conjugate with an involutionof form F U ( S ) ≡ [ I S0 I ] , S ∈ Sym ( m ; 2 ) , (34)that is, there exists M ∈ Sp ( m ; 2 ) such that MFM − = F U ( S ) . On the other hand, the involutions of form (34) areeasy to decompose as described in [8, Prop. 9(2)]. So let usassume F U ( S ) = T v ⋯ T v r . It is straightforward to verifythat T v M = MT vM holds for any symplectic M . This yields F = M − F U ( S ) M (35) = ( M − T v M ) ⋅ ( M − T v M ) ⋯ ( M − T v r M ) (36) = T v M ⋯ T v r M . (37) C. Decomposition of Symplectic Matrices
Finding a transvection decomposition for involutions isfacilitated by the simple nature of their associated A and B matrices. As we will see, the general case is much morecomplicated. Let F be any (non-hyperbolic) symplectic matrix and consider its residue matrix ̂ F , for which rank (̂ F ) = r .Thus, a transvection decomposition of F is given by somebasis of Res ( F ) = rs (̂ F ) . The task in hand is how to findsuch basis. The main idea is to start with some fixed basis andtransform it accordingly until we reach the desired result. Wewill start with a basis of Res ( F ) in Row-Reduced Echelonform, that is, let R be a matrix of row operations so that R ̂ F = [ V0 ] , where V is a r × m basis. This can be done,for instance, via Gauss Elimination over F . Then R ̂ FR T = [ E 00 0 ] , E ∈ GL ( r ; 2 ) . (38)As mentioned, the basis V may or may not constitute atransvection decomposition of F , and the idea is to considerother bases of form QV where Q ∈ GL ( r ; 2 ) . Let us denote P = blkdiag ( Q , I m − r ) , and let B = B ( QV ) . Lemma 2.
With the same notation as above, the basis QV constitutes a transvection decomposition of F iff QEQ T = B − T .Proof. Assume QV gives a transvection decomposition for F .Then, by (30) we have ̂ F = ( QV ) T ⋅ B ⋅ ( QV ) . But with thenotation above we have QV = [ QE 0 ] R − T . Thus ̂ F = V T Q T ⋅ B ⋅ QV (39) = R − [ E T Q T ] ⋅ B ⋅ [ QE 0 ] R − T (40) = R − [ E T Q T ⋅ B ⋅ QE 00 0 ] R − T . (41)It follows by (38) that E = E T Q T ⋅ B ⋅ QE and thus QEQ T = B − T . The reverse direction follows similarly. Lemma 3.
With the same notation as above, if
QEQ T islower triangular, then QEQ T = B − T .Proof. Assume
QEQ T is lower triangular and put E ′ = E T Q T ⋅ B ⋅ QE . Then QEQ T = (( QEQ T ) T ⋅ B ) − ⋅ QE ′ Q T . (42)If QEQ T = I r , the statement is clear because in this case ̂ F is symmetric, and therefore B = I r . If QEQ T ≠ I r , thenboth QE ′ Q T and ( QEQ T ) T ⋅ B have to be invertible andlower triangular. But ( QEQ T ) T and B are both invertibleand upper triangular, meaning ( QEQ T ) T ⋅ B is also uppertriangular. Thus ( QEQ T ) T ⋅ B = I r , and QEQ T = B − T It follows by Lemmas 2 and 3 that we are seeking formatrices Q that triangularize E from (38) by congruence.For more on triangularizations by congruence and relatedalgorithms we refer the reader to [7]. It also follows byLemma 2 that ̂ F can be triangularized by congruence forany non-hyperbolic F (since for this, one would only needa transvection decomposition of F , which we know it alwaysexists). We resume everything to the following theorem. Theorem 3 (Transvection Decomposition of Symplectic Ma-trices) . Let F be a generic symplectic matrix. Then there exists lgorithm 1 Transvection Decomposition of Clifford Gates
Input:
A Clifford gate G .1. Compute F from (13).2. Compute v , v , ⋯ , v r from Theorem 3.3. G = G v ∏ j G v j .4. Find E = E ( v ) such that G = E G . Output: v , v j ’s. an algorithm that for any generic symplectic matrix F outputsa minimal transvection decomposition.Proof. If the residue matrix ̂ F is alternating, that is, if F is hyperbolic, then pick v as in Lemma 1 and update theinput F with the non-hyperbolic FT v , while keeping theresidue space intact. Next, perform Gauss Elimination on ̂ F with R as in (38), and let Q be such that QEQ T is lowertriangular. Then, by Lemmas 2 and 3, the r nonzero rows of blkdiag ( Q , I m − r ) R ̂ F , where r = dim Res ( F ) , along with v , yield a minimal transvection decomposition for F .IV. D ECOMPOSITION OF C LIFFORD G ATES
In [8], the authors studied the Clifford hierarchy via thesupport of the underlying gates. Every gate U ∈ U ( N ) can bewritten as U = N ∑ v ∈ F m Tr ( E ( v ) U ) E ( v ) , (43)and the support of U consist of the basis terms that appearin (43), that is,supp ( U ) ∶ = { E ( v ) ∈ HW N ∣ Tr ( E ( v ) U ) ≠ } . (44)Given the isomorphism E ( v ) ←→ v , the support can be equiv-alently though of as a subspace of F m . On the other hand, (15)assigns F ∈ Sp ( m ; 2 ) to a coset HW N G = Φ − ( F ) for any G ∈ Cliff N . It is straightforward to verify that the Clifford G v ∶ = I N ± i E ( v )√ ∈ Cliff N (45)corresponds to the transvection T v . Then, since every sym-plectic is a product of transvections, it follows that G = E k ∏ n = I N + i E n √ = E √∣ S ∣ ∑ E ∈ S α E E , (46)where E ∈ HW N , S = ⟨ E , . . . , E k ⟩ , and α E ∈ C ; see [8,Prop. 4]. From earlier discussion, it follows that the supportof any G ∈ Φ − ( F ) is given by Res ( F ) if F is non-hyperbolic,and by some subspace of Res ( F ) of index otherwise. In [8,Prop. 9], the authors determined the support of the standardClifford gates (16)-(18), while the general case remained open.The difficulty arose by the fact that the support of productsis hard to compute. This problem can now be solved with theaid of Theorem 3, as resumed in Algorithm 1. It is also worthmentioning that in this process one may lose an eighth root ofunity; see Example 3 for instance. We point out here that G istraceless iff E ∉ S . Thus the search in Step 4. of Algorithm 1can be reduced to either outside S if G is traceless or in S otherwise. Example 2.
The Hadamard gate can be written as H = √ ( X + Z ) = X I + i Y √ , (47)where Y = i XZ as usual. Consider now the m fold transversalHadamard gate H N = ( H ) ⊗ m , for which Φ ( H N ) = Ω .Additionally ̂ Ω = [ I II I ] and dim Res ( Ω ) = m . Then P = [ I 0I I ] triangularizes ̂ Ω : P ̂ ΩP T = [ I 00 0 ] . (48)The first m nonzero rows of P ̂ Ω are [ I I ] . We see that the n th row yields the gate Y n with Y in qubit n and identityelsewhere. From Step 3. of Algorithm 1 we compute G = m ∏ n = I N + i Y n √ . (49)We then find H N = X ⊗ m G . A similar result holds for partialHadamard gates H ⊗ r ⊗ I m − r , to which correspond symplecticsof form (5); see also [8, Prop. 9(3)] Example 3.
The symplectic and residue matrices correspond-ing to the CNOT gate are given by F = ⎡⎢⎢⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎥⎥⎦ and ̂ F = ⎡⎢⎢⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎥⎥⎦ , (50)from which we see that ̂ F is alternating, and thus F ishyperbolic. So first, we transform F to a non-hyperbolic mapby using the first non-zero row of ̂ F , that is, v = . Thenwe update F ← FT v , for which F = ⎡⎢⎢⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎥⎥⎦ and ̂ F = ⎡⎢⎢⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎥⎥⎦ . (51)A matrix that triangularizes ̂ F is given by P = ⎡⎢⎢⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎥⎥⎦ . (52)The non-zero rows of P ̂ F are v = , v = . Note that v = v + v , and ⟨ v ∣ v ⟩ s = as in Proposition 1. Then wecompute G = ( I + i I ⊗ X )( I − i Z ⊗ X )( I + i Z ⊗ I )√ . (53)And then we end with the observation that CNOT = ξ G ,where ξ = ( − i )/√ is an eighth root of unity.A CKNOWLEDGEMENTS
This work was funded in part by the Academy of Finland(grant 334539).
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