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Ψ0 denotes the initial state of the walk. The period of a walk defined by U is the smallest |positivei integer T for which U T = I, if such a T does not exist then the walk is called not periodic or the period is said to be . Thus a quantum walk with finite period means that the walk returns to the initial state∞ within a finite number of steps [12]. This property of a discrete quantum walk on cycle distinguishes itself from its classical counterpart since classical random walk on a cycle return to its starting state at irregular, unpredictable times [35]. Determining periodicity of quantum walks is one of the problems of paramount interest due to the fact that periodicity is closely related to perfect state transfer [13][15][21][31] [17]. Indeed, periodicity is a necessary condition for perfect state transfer over symmetric graphs for discrete time walks [4]. Periodicity of a discrete quantum walk on cycle was first observed in [34]. Periodicity of two state quantum walks on regular graphs with Fourier coin is investigated in [31]. Periodicity also plays an important role in experimental realization of quantum walks on a cycle [31][27]. Recently, quantum walks with high dimensional coin operator are introduced in literature [5][16][25][20][32]. One immediate application of such coin operators is that coin degrees of freedom can offer controls over the evolution of the walk [35]. Often, for three state quantum walk, the coin operator is chosen as the Grover matrix or Fourier matrix of dimension 3 3. For example, one of the well-known 3-state quantum walks is called lively quantum walk× on cycle, is introduced in [30]. In this walk, depending on the state of the coin, the particle moves to a neighbor of the vertex or can make a jump to a vertex up to a certain distance. The distance of the jump is called the liveliness parameter of the walk. If the value of this parameter is zero then the particle does not make the jump and remains in its position, called the lazy quantum walk. It is also well known that the lively quantum walk has applications for the detection of link failures in a network. Besides, the lively walk enables to avoid trapping of the walker for any coin operator [30]. Periodicity of lively and lazy quantum walks are derived in [17] when the coin operator is Grover or Fourier matrix, and it is proved that the walk has a finite period if and only if the cycle has length 3. Here we mention that, attempts are made in literature to define new coin operators by exploiting the eigenvector properties of the Grover matrix. For instance, a family of three- state quantum walks on line is proposed in [33] such that the coin operators of the walk posses the localization property of the Grover walk. In this case, the coin operators are specific one-parameter unitary matrices which are linear sum of the projectors corresponding to the eigenvectors of the Grover matrix. In this work, we consider lively discrete-time quantum walks on a cycle and determine its periodicity when the underlying coin operator is a generalization of the Grover Matrix. Our motivation for this is two fold: Firstly, this would help in an improved understanding of the underlying quantum dynamics and how it differs from its classical counterpart. Secondly, from a mathematical point of view, the Grover matrix is a sum of permutation matrices. We consider as coin operator any 3 3 real orthogonal matrix that is linear sum of permutation matrices, thereby generalizing the× well known Grover coin operator. Moreover, characterizing unitary matrices that are linear sum of permutation matrices has been an unsolved problem. Only a necessary condition is known for a linear sum of permutation matrices to be orthog- onal [18]. In this article, we are able to completely characterize all complex, real, rational orthogonal matrices of dimension 3 3, which are linear sum of permutation matrices. We establish that a linear sum of 3 3× permutation matrices is orthogonal if and only if it is a permutative orthogonal matrix.× In contrast to the earlier results on periodicity of lively quantum walk on cycle corresponding to Grover or Fourier coin that the period can be finite if and only if the length of the cycle is 3, we show that the period can be finite for cycles of any length when specific coin operators are chosen from the orthogonal group of linear sum of permutation matrices.
2 2 Preliminaries
In this section we briefly discuss some of the fundamental concepts of algebra for the com- pleteness of the presentation of our results and that will be used in sequel. We follow the notations and terminologies given in [3]. A field E is said to be an extension of a field F if F E. We denote it by E/F. Then it follows that E is a vector space over the field F, and if the⊆ dimension of this vector space is finite, denoted by [E : F ] then E is called a finite field extension of F. If any f F [x] can be written as f(x)= λ(x α ) ... (x αk) for some ∈ − 1 − α1,...,αk E, λ F then we say that f splits over E. The field E is called a splitting field for an f ∈F [x] if∈f splits over E but not over any proper subfield of E that contains F. An element∈ α E is said to be algebraic over F if α is a root of a nonconstant polynomial f(x) F [x]. If∈ every element of E is algebraic over F then E is called an algebraic extension of F.∈An algebraic extension field E/F is called normal if every irreducible f F [x] which has at least one root in E, splits over E. ∈ Given an element α E/F that is algebraic over F, the set of polynomials f F [x] such that f(α) = 0 forms∈ an ideal in F [x]. Further, since F [x] is aI principal ideal domain∈ , is generated by a single polynomial, say m(x). If m(x) is monic then it is called the minimalI polynomial of α over F. An element α E is called separable over F if α is algebraic over F and the minimal polynomial of α over∈ F is a separable polynomial. A field E/F is called a separable extension of F if every element of E is separable over F. An extension E/F is called a Galois extension if E/F is normal and separable. Let E/F be a Galois extension. Then the set of all field isomorphisms f : E E such that f(x) = x, x F forms a group under the operation of composition of functions.→ This group is called the∈ Galois group, denoted by Gal(E/F ). An extension E/F is called cyclic extension if E/F is a Galois extension and Gal(E/F ) is cyclic. Further, since E is a vector space over F when E/F, we can define F -linear transformation M : E E. Then norm of an element x E is defined as N(x) = det(M). For→ example, Q(√d) is extension of Q,∈ where d is a square-free integer. Then the minimal polynomial of √d over Q is m(x)= x2 d. Besides, Q(√d)/Q is Galois and elements of the Gal(Q(√d)/Q) are σ : Q(√d) Q(√d)− defined by σ(a + √db) = a √db, and the identity function. Besides, N(a + √db)=→ a2 b2d. − − Theorem 2.1 (Hilbert’s theorem 90). If E/F is a cyclic extension with [E : F ] = n and Galois group G = 1,σ,...,σn 1 generated by σ, and x E, then: N(x)=1 if and only if there exists y E {such that x =− y/σ} (y). ∈ ∈ An extension E/F is called cyclotomic if E is a splitting filed for f(x)= xn 1 F [x]. The − ∈ roots of f(x) are called (primitive) nth roots of unity, denoted by ζn. We denote a cyclotomic field by F [ζn] for any positive integer n. Let R be a commutative ring and S be a subring of R. Then an element x R is said to be integral over S if f(x) = 0 for some monic polynomial f S[x]. If R = C∈and x C is integral over Z then x is called an algebraic integer. Let A be∈ the set of algebraic integers.∈ Then we denote Z[ζn]= A Q[ζn] where C/Q. The Euler’s totient function∩ φ(m) defines the number of positive integers that are less φ(n) 1 i than m and relatively prime to m. Any τ Z[ζn] can be written as − ζ [17]. ∈ i=0 n P 3 3 3 orthogonal matrices which are linear sum of per- mutations×
In this section we provide a complete characterization of 3 3 orthogonal matrices which are linear sum of permutation matrices. First we characterize× permutative orthogonal matrices of order 3 3, and then we show that linear sum of permutation matrices are orthogonal if and only if× they are orthogonal permutative matrices. Besides, we characterize all rational permutative orthogonal matrices of order 3 3. ×
3 3.1 Complex and real matrix groups of linear sum of permutation matrices Recall that a matrix is permutative if the rows of the matrix are permutations of each other [29]. Thus any 3 3 permutative matrix is of the form × x A(x; P,Q)= xP (1) xQ where x = x y z is a row vector in R3 and P,Q , which is the group of permutation matrices of order 3 3. The following theorem provides∈P a characterization of all permutative orthogonal matrices× of order 3 3 which we denote by . Obviously . The elements of are matrices corresponding× to the permutationsPO in the symmetricP ⊂PO group S . If π S P 3 ∈ 3 then the permutation matrix corresponding to π is defined as P = [pij ] where pij = 1 if and only if π(i)= j. Thus can be described as P = P = I = id, P = (123), P = (132), P = (23), P = (12), P = (13) (2) P { 1 2 3 4 5 6 } where id denotes the identity element of the symmetric group. Theorem 3.1. A matrix A(x; P,Q) if and only if it is either xP + yP + zP or ∈ PO 1 2 3 xP4 + yP5 + zP6 where (x,y,z) belongs to either of the affine varieties x + y + z =1 x + y + z = 1 S+ = 2 2 2 and S = 2 2 2− (3) x + y + z =1 − x + y + z =1 and Pi,i =1,..., 6 are described in (2). Proof: The ‘if’ part is obvious and easy to check. To prove the ‘only if’ part consider the following cases. First assume that A(x; P,Q) and the symbolic matrix A(x; P,Q) has no repetition of entries in any of the columns.∈ Then PO A(x; P,Q) can be either of the following forms. Besides, since rows and columns are orthogonal, P = Q and P,Q are not the identity matrix. Thus 6 x y z A(x; P2, P3) = z x y = xP1 + yP2 + zP3, and y z x x y z A(x; P3, P2) = y z x = xP4 + yP5 + zP6. z x y Further, due to the orthonormality condition of the rows, it follows that xy + yz + xz =0 and x2 + y2 + z2 = 1 which further imply that x + y + z 1 . Thus the desired result follows. ∈ {± } Next, consider the symbolic orthogonal permutative matrices A(x; P,Q), in which, one entry is repeated in at least one column. For any such matrix, the following system of polynomial equations hold due to the orthonormality condition of the rows. 2 2 2 x1 + x2 + x3 = 1 2 2 2xi + xj = 1 2 2 2xk + xj = 1, where xi, xj , xk x,y,z and xi = xj = xk. Then solving these equations, it follows that ∈ { } 6 6 xi = xk. Now, if xi = xk then either xj 1 , xi = xk =0, or xj = 1/3, xi = xk = 2/3. ± ∈ {± } ± ∓ Finally, if xi = xk then xj 1 and xi = xk =0 or xj = 1/3, xi = xk = 2/3. Then it follows that A(x−; P,Q) has the∈ {± desired} form. This completes the± proof. ∓ Now we prove the main theorem of this section that characterizes all orthogonal matrices which are linear combination of permutation matrices.
4 Theorem 3.2. A linear combination of 3 3 permutation matrices is orthogonal if and only if it is a permutative orthogonal matrix. × 6 Proof: Let M = i=1 αiPi be a linear combination of permutation matrices. Then M can be written as a sum of symmetric permutative matrix A and a permutative matrix B, where P α4 α5 α6 α1 α2 α3 A = α α α , B = α α α . 5 6 4 3 1 2 α6 α4 α5 α2 α3 α1 Then it follows that AT B = BT A, ABT = BAT and BT B = BBT . Suppose that M is orthogonal. Then MM T = M T M = I implies AAT + ABT + BAT + BBT = AT A + AT B + BT A + BT B = I. (4) Which further implies that AT B ABT =0. Consequently, − α6 α5 α4 α6 α5 α4 T − − − A(B B ) = (α2 α3) α4 α6 α5 α4 α6 α5 =0. − − α − α α − α α − α 5 − 4 6 − 5 4 − 6 Then two cases arise: either α2 = α3 or α4 = α5 = α6. If α2 = α3 then α4 + α1 α5 + α2 α6 + α2 M = α + α α + α α + α . 5 2 6 1 4 2 α6 + α2 α4 + α2 α5 + α1 Since each row has unit 2-norm, αi,i =1,..., 6 must satisfy the following system of polynomial equations.
2 2 2 (α4 + α1) + (α5 + α2) + (α6 + α2) = 1 (5) 2 2 2 (α5 + α2) + (α6 + α1) + (α4 + α2) = 1 (6) 2 2 2 (α6 + α2) + (α4 + α2) + (α5 + α1) =1. (7) Then (5)-(6), (5)-(7) and (6)-(7) imply (α α )(α α )=0, (α α )(α α )=0, and (α α )(α α )=0 4 − 6 1 − 2 4 − 5 1 − 2 5 − 6 1 − 2 respectively. Thus if α = α then α = α = α . This yields, 1 6 2 4 5 6 α4 + α1 α4 + α2 α4 + α2 M = α + α α + α α + α 4 2 4 1 4 2 α4 + α2 α4 + α2 α4 + α1 which is a permutative matrix. Similarly, if α1 = α2 then M is a permutative matrix. Finally, let α2 = α3. Then α4 = α5 = α6 which implies that M = α4J + B. Thus M is a permutative matrix.6 Now consider the converse part. Let M be a 3 3 permutative orthogonal matrix. Then by Theorem 3.1 it follows that M is a linear combination× of permutation matrices. This completes the proof. Setting z =1 x y and eliminating z from the equation x2 + y2 + z2 =1, the (x,y,z) − − which satisfy the system of polynomials S+ are given by (x, y) which lie on the ellipse given by x2 + y2 x y + xy =0. (8) − − Similarly, the points (x,y,z) which satisfy the system S are given by z = 1 x y and (x, y) which lie on the ellipse − − − − x2 + y2 + x + y + xy =0. (9) Hence we have the following corollary.
5 Corollary 3.3. An orthogonal matrix M is a linear combination of permutation matrices of order 3 3 if and only if ×
M = xP1 + yP2 + zP3 or M = xP4 + yP5 + zP6 where (x,y,z) are given by
z =1 x y z = 1 x y or (10) x2 + y−2 −x y + xy =0 x2 +−y2 −+ x −+ y + xy =0. − − Moreover, it can further be noted that the ellipses given by equations (8) and (9) are inherently different. Observe that
x2 + y2 x y + xy = XT Q X = 0 and (11) − − + x2 + y2 + x + y + xy = XT Q X =0, (12) − where 1 1/2 1/2 1 1/2 1/2 x − Q+ = 1/2 1 1/2 ,Q = 1/2 1 1/2 ,X = y . 1/2 1/2− 0 − 1/2 1/2 0 1 − − 2 In addition, since coefficient of y det(Q+) > 0, the equation (11) represents a real ellipse, whereas (12) represents− an imaginary× ellipse [23]. It follows from the proof of Theorem 3.1 that the permutative orthogonal matrices which have repeated entries in their columns are given by permutation matrices and matrices of the form 2 2 J P and J + P 3 − − 3 where J is the all-one matrix and P . In particular, setting P = I, the identity matrix, we obtain the Grover matrix given by∈ P
1 2 2 3 3 3 2 −2 1 2 J I = 3 3 3 . 3 − 2 −2 1 3 3 − 3 Now we have the following definition.
2 Definition 3.4. (Grover-type matrix) A permutative orthogonal matrix of the form 3 J P, P is called a Grover-type matrix. The set of all Grover-type matrices is denoted by − ∈P 2 Grov = J P : P . 3 − ∈P It may also be noted that matrices in Grov can be obtained by permuting the rows of the Grover matrix. Besides, permutation matrices, Grover-type matrices, and matrices of 2 the form 3 J + P, P belong to the classes of matrices given by Corollary 3.3. Hence we consider− the sets of∈ all P parametric orthogonal matrices which are linear combinations of permutation matrices, and investigate the algebraic structures of these sets.
6 Set x y 1 x y − − = 1 x yx y : x2 + y2 x y + xy =0 (13) X − y − 1 x y x − − − − x y 1 x y − − − = 1 x yx y : x2 + y2 + x + y + xy =0 (14) Y − −y − 1 x y x − − − x y 1 x y − − = y 1 x y x : x2 + y2 x y + xy =0 (15) Z 1 x yx− − y − − − − x y 1 x y = y 1 x y− − x − : x2 + y2 + x + y + xy =0 . (16) W 1 x yx− − − y − − − It may be noted that = P4A : A and = P4A : A . Then we have the following proposition whichZ describes{ group∈ X} structuresW of{ the above∈ sets. Y} Proposition 3.5. Let , , and be defined as above. Then the following are true. X Y Z W (a) det(M)=1 if M , ; and det(M)= 1 if M , . ∈ X Z − ∈ Y W (b) , , , , = are complex orthogonal matrix groups withX X respect ∪ Y X to ∪ Z matrixX ∪W multiplication,X∪Y∪Z∪W and POis a normal subgroup of all these groups. X 1 (c) is a real orthogonal matrix group, denoted by R if and only if x 1. X X −3 ≤ ≤ 1 (d) R R is a real orthogonal matrix group where R = A : 1 x . X ∪ Y Y { ∈ Y − ≤ ≤ 3} 1 (e) R R is a real orthogonal matrix group where R = A : x 1 . X ∪ Z Z { ∈ Z −3 ≤ ≤ } 1 (f) R R is a real orthogonal matrix group where R = A : 1 x . X ∪W W { ∈W − ≤ ≤ 3} (g) = is a real orthogonal matrix group. POR XR ∪ YR ∪ ZR ∪WR Proof: Proof of (a) is computational and easy to verify. Consider (b). First we prove that is a complex orthogonal matrix group. Clearly I . Since X∪Y∪Z∪W 1 T ∈X∪Y∪Z∪WT any A is orthogonal, A− = A . If A then A follows from exchanging∈X∪Y∪Z∪W the role of 1 x y and y. Similarly, AT if∈A X . Since ∈ Xand contain complex symmetric matrices,− − obviously AT if A ∈ Y , and∈A YT ifZ A W. Hence, is closed under inverses. The∈ Z closure∈ property Z follows∈ W by considerin∈ W g the X∪Y∪Z∪Wfollowing cases. Let A = x1P1 + y1P2 + z1P3,B = x2P1 + y2P2 + z2P3 where (xi,yi) satisfies ∈X∪Y 2 2 one of the equations (8) and (9), and zi = 1 xi yi,i 1, 2 . Note that P = P , P = − − ∈ { } 2 3 3 P2, P2P3 = I = P3P2. Then AB = x3P1 + y3P2 + z3P3 where
x3 = x1x2 + y1z2 + z1y2, y3 = x1y2 + x2y1 + z1z2, z3 = x1z2 + y1y2 + z1x2.
Then z =1 x y if zi =1 xi yi or zi = 1 xi yi,i =1, 2, and z = 1 x y if 3 − 3 − 3 − − − − − 3 − − 3 − 3
7 zi =1 xi yi and zj = 1 xj yj ,i = j, i, j 1, 2 . Hence x + y + z 1 . Further, − − − − − 6 ∈{ } 3 3 3 ∈ {± } T 2 T 2 T 2 x1 x2 x1 y2 x1 z2 x2 + y2 + z2 = y z + y x + y y 3 3 3 1 2 1 2 1 2 z1 y2 z1 z2 z1 x2 T T T T x1 x2 x2 x1 x1 y2 y2 x1 = y z z y + y x x y 1 2 2 1 1 2 2 1 z1 y2 y2 z1 z1 z2 z2 z1 T T x1 z2 z2 x1 + y y y y 1 2 2 1 z1 x2 x2 z1 T x1 a b b x1 = y b a b y , 1 1 z1 b b a z1 2 2 2 where a = x2 + y2 + z2 = 1 and b = x2z2 + x2y2 + y2z2 =0. Now, x2 + y2 + z2 1 since 2 2 2 2 2 2 ∈ {± } B . This yields x3 + y3 + z3 = x1 + y1 + z1 =1. Thus AB . ∈X∪YNow let A = x P + y P + z P ; B = x P + y P + z P ∈X∪Y where (x ,y ) 1 1 1 2 1 3 ∈ X 2 4 2 5 2 6 ∈Z∪W 1 1 satisfies the equation (8), and (x2,y2) lies on one of the ellipses given by equations (8) and (9). Thus z =1 x y and x + y + z 1, 1 . Note that P P = P , P P = P , P P = 1 − 1 − 1 2 2 2 ∈{ − } 2 4 5 2 5 6 2 6 P4, P3P4 = P6, P3P5 = P4, P3P6 = P5. Then AB = x3P4 + y3P5 + z3P6, where
x3 = x1x2 + y1z2 + y2z1,y3 = x1y2 + x2y1 + z1z2,z3 = x1z2 + y1y2 + x2z1. Hence, 1 if B x + y + z = (x + y + z )(x + y + z )= 3 3 3 1 1 1 2 2 2 1 if B∈ Z . − ∈W 2 2 2 2 2 2 Next, as above, x3 + y3 + z3 = x1 + y1 + z1 =1. Thus AB if B and AB if B . ∈X∪Z ∈ Z ∈X∪W ∈W Next let A = x1P4 + y1P5 + z1P6 and B = x2P4 + y2P5 + z2P6 such that either both A, B belong to or both belong to . Thus (xi,yi) satisfies either equation (8) or (9), and Z W 2 2 2 xi + yi + zi = 1 or 1,i = 1, 2. Now, P = P = P = P , P P = P , P P = P , P P = − 4 5 6 1 4 5 3 4 6 2 5 4 P2, P5P6 = P3, P6P4 = P3, P6P5 = P2. Then AB = x3P1 + y3P2 + z3P3, where
x3 = x1x2 + y1y2 + z1z2,y3 = x2y1 + x1z2 + y2z1,z3 = x1y2 + x2z1 + y1z2.
2 2 2 2 2 2 Hence, x3 + y3 + z3 = (x1 + y1 + z1)(x2 + y2 + z2)=1, and x3 + y3 + z3 = x1 + y1 + z1 =1 as above, and hence AB . Therefore both ∈and X are closed under matrix multiplication. Similarly, it can be proved that if A,X B ∪Z X∪Wor A, B then AB . Besides, it can similarly be shown that∈Y∪ZAB if ∈Y∪WA, B . Thus∈X∪Y∪Z∪W the desired results follow from the fact that matrix multiplication∈X∪Y is associative.∈Z∪W Now consider (c). It is enough to determine for what real values of x the value of y is real such that (x, y) R2 which satisfy equation (8). Consider the equation (8) as a polynomial equation in which∈ y is the indeterminate. Then 1 y = (1 x) (1 x)2 4(x2 x) . 2 − ± − − − h p i Then it follows that y is real if and only if x lies in the closed interval [ 1/3, 1]. Indeed, y [ 1/3, 1]. The proof of (d) (g) follow similarly. This completes the proof.− ∈From− the construction of real− orthogonal matrix groups discussed in the above theorem it follows that the matrices in R ultimately depend on the parameter x which acts as a free X
8 parameter in a bounded interval. Then y, and hence z can be written as functions of x. Based on this observation we parametrize the values of x in the domain of the interval. Thus we have the following corollary.
Corollary 3.6. The following sets are one-parameter orthogonal matrix groups. (a)
2 cos θ+1 (1 cos θ) 1 (1 cos θ) 1 − + sin θ − sin θ 3 3 √3 3 − √3 (1 cos θ) 1 2 cos θ+1 (1 cos θ) 1 θ = − sin θ − + sin θ :0 θ< 2π . X 3 − √3 3 3 √3 ≤ (1 cos θ) 1 (1 cos θ) 1 2 cos θ+1 − − 3 + √ sin θ 3 √ sin θ 3 3 − 3 (b) θ θ where X ∪ Y (2 cos θ 1) (1+cos θ) 1 (1+cos θ) 1 − + sin θ sin θ 3 − 3 √3 − 3 − √3 (1+cos θ) 1 (2 cos θ 1) (1+cos θ) 1 θ = sin θ − + sin θ :0 θ < 2π . Y − 3 − √3 3 − 3 √3 ≤ (1+cos θ) 1 (1+cos θ) 1 (2 cos θ 1) − 3 + √ sin θ 3 √ sin θ 3 − 3 − − 3 (c) θ θ where θ = P4A : A θ , that is, matrices in θ are matrices in θ with secondX ∪ Z and thirdZ row exchanged.{ ∈ X } Z X
(d) θ θ where θ = P4A : A θ , that is, matrices in θ are matrices in θ with Xsecond∪W and thirdW row exchanged.{ ∈ Y } W Y
(e) θ := θ θ θ θ. PO X ∪ Y ∪ Z ∪W Proof: The one parameter representations of R, R, R and R are obtained as follows. θ X Y1+2Z cos θ W (1 cos θ) 1 X follows from the construction of R by setting x = , and hence y = − + sin θ. X 3 3 √3 R is the set all matrices in in which the parameter x lies in the interval [ 1, 1/3]. Then Y 2 cos θ 1 Y − setting x = 3 − , θ = R. Similarly, the expressions of θ, θ can be obtained. The rest follows from PropositionY 3.5Y. Z W
3.2 Rational matrix group of linear sum of permutation matrices In this section we determine rational orthogonal matrices which are linear sum of permutation matrices. Thus we need to determine values of (x, y) Q Q which lie on the ellipses given by equations (8) and (9). ∈ × First we determine the set of rational matrices, denoted by Q R. An equation of the form x2 dy2 = k, where k is an integer and d is a nonsquare positiveX ⊂ X integer, is well-known as Pell’s− equation. Recall that x y 1 x y − − 1 = 1 x yx y : x2 + y2 x y + xy =0, x 1 . R 3 X − y − 1 x y x − − − ≤ ≤ − − (17) Now treating x2 + y2 x y + xy = 0 as a polynomial in indeterminate y, we obtain − − (1 x) (1 x)(3x + 1) y = − ± − . p 2 Then y Q if and only if (1 x)(3x+1) is zero or perfect square of a nonzero rational number. It is zero∈ if x 1, 1/3 . If−x = 1 then y = 0 and hence the corresponding orthogonal matrix ∈{ − }
9 in Q is the identity matrix, otherwise if x = 1/3 then y = 2/3 which gives the Grover X − matrix as an element of Q. If it is nonzero then X p2 (1 x)(3x +1)= − q2
q 4q2 3p2 for some nonzero rational number p/q. This gives x = ± − . p 3q Now we need to find values of p, q for which x Q. Note that 4q2 3p2 =0, since otherwise p/q =2/√3 which is not rational. Thus 4q2 3p∈2 has to be square− of a nonzero6 integer, say m. Hence 4q2 3p2 = m2 which is a Pell’s equation,− which finally taks the form − X2 3Y 2 =1 − where X =2q/m and Y = p/m. Now we apply Hilbert’s Theorem 90 (see Section 2). Set E = Q(√3) and F = Q. Besides N(x)= X2 3Y 2 = 1 where x = X + √3Y Q(√3) as given above, and σ(X + √3Y ) = X √3Y where− σ : Q(√3) Q(√3) is the generator∈ of Gal(Q(√3)/Q). Then due to Hilbert’s− Theorem 90 there must exist→ y = a + √3b Q(√3) such that ∈ a + √3b a + √3b a2 +3b2 2ab X + √3Y = = = + √3 − . σ(a + √3b) a √3b a2 3b2 a2 3b2 − − − Thus r2 +3 2r X = and Y = − r2 3 r2 3 − − for some r = a/b Q. Consequently we obtain ∈ m(r2 + 3) 2mr q = and p = − 2(r2 3) r2 3 − − which satisfy the equation 4q2 3p2 = m2, for some nonzero integer m. Then we have the following− proposition.
Proposition 3.7. The set Q, Q Q, Q Q, Q, Q = Q Q Q Q are matrix groups of orthogonalX rationalX ∪ matrices, Y X ∪ ZwhereX ∪W PO X ∪ Y ∪ Z ∪W
1 2(r2 3) 1 r2 3 6r Q = A R : x = − , y = − ± , r Q , X ∈ X 3 ± 3(r2 + 3) 3 ∓ 3(r2 + 3) ∈ 1 2(r2 3) 1 r2 3 6r Q = A R : x = − , y = − ± , r Q Y ∈ Y −3 ± 3(r2 + 3) −3 ∓ 3(r2 + 3) ∈ Q = P A : A Q Z { 4 ∈ X } Q = P A : A Q . W { 4 ∈ Y } Proof: From the above discussion it follows that
x y 1 x y p − − q m (1 x) q = 1 x yx y : x = ± ,y = − ± Q 3q 2 X − y − 1 x y x − − 2mr m(r2 + 3) p = − , q = , r Q,m Z . r2 3 2(r2 3) ∈ ∈ − − Then the desired result for Q can be obtained by a straightforward calculation. For Q, note X 2 2 Y p2 that the parameters x, y satisfy x + y + xy + x + y = 0. This implies (1 + x)(1 3x)= q2 (1+x) p − q m − ± q and using above arguments it follows that x = − 3±q and y = 2 . Thus the desired results follows for other cases using Proposition 3.5.
10 4 Periodicity of lively quantum walks on cycles
In this section we generalize the lively quantum walk on cycles by considering the coin operator as an orthogonal matrix which is a linear sum of permutation matrices. The lively quantum walk on cycles is defined as follows. The state space of the walk is C3 Cn where n is the ⊗ number of nodes in the cycle Cn, and denotes the tensor product as usual. The evolution operator is given by ⊗ U = S(n,a)(C I), (18) (n,a) ⊗ (n,a) n where C is the coin operator, S is the shift operator, and a 2 is called the liveliness parameter [30]. If a = 0 the corresponding walk is called the lazy≤⌊ walk.⌋ The shift operator is defined by n 1 (n,a) − (n,a) S = Sx (19) x=0 X where
S(n,a) = 0 0 x 1(mod n) x x | i h | ⊗ | − i h | + 1 1 x + 1(mod n) x | i h | ⊗ | i h | + 2 2 x + a(mod n) x . (20) | i h | ⊗ | i h | (n,a) Thus Sx acts as shifting the position of the walking particle from the position at the vertex x of to one of the positions x 1, x 1 and x + a depending on the coin state. The position state is described by x , x −0, 1,...,n− 1 which form a set of orthonormal standard basis n | i ∈{ − } vectors of C . Obviously, the vertex set of Cn is given by 0, 1,...,n 1 . The periodicity of this walk is investigated in [30] where{ the coin operator− } C is the Grover 2 Grov matrix G = 3 J I . Recently, periodicity of three state lazy quantum walk on cycles is considered in [17−]. In∈ this section we investigate the periodicity of the lively quantum walk on n cycle Cn when C R and the pair (n,a) satisfies the condition divides l, for some ∈ PO gcd(n,a) l 0, 1,...,n 1 . We follow a similar approach to derive the periodicity of lively quantum walks∈{ where the− coin} operator is an orthogonal matrix which is a linear sum of permutation matrices, as described in Section 3. 3 n If Ψ0 C C denote the initial state of the lively quantum walk then the discrete time evolution∈ of the⊗ walk is given by t Ψ(t)= U(n,a)Ψ0 (21) 0 where t N denotes the time, and U(n,a) = Ψ(0) = Ψ0. The walk is said to be periodic t ∈ if U(n,a) = I for some t N. The smallest t = T for which the walk is periodic, is called the period of the walk. If∈ no such t exists then the walk is called not periodic. One of the fundamental problems in the area of quantum walks is to decide if a walk is periodic and then find the period. The periodicity of a walk can be investigated through the eigenvalues of the t corresponding evolution operator of the walk. It can be easily shown that U(n,a) = I for some t t if and only if λ = 1 for any eigenvalue λ of U(n,a) [17]. In this paper we consider the lively quantum walk defined by the evolution operator U(n,a) given by the equation (18), where the coin operator C R and the shift operator is given ∈ PO by equation (19). The following proposition shows that eigenvalues of U(n,a) can be accessed 3 3 from the eigenvalues of a unitary matrix Uk = DkC C × where Dk is a unitary diagonal matrix. ∈
(n,a) Proposition 4.1. Let C R. Then the lively quantum walk operator U = S (C ∈ PO (n,a) ⊗ In) has eigenvalues λk,j with a corresponding eigenvector ψk,j = νk,j φk where λk,j is an | i | i⊗| ni 1 ikx n eigenvalue of Uk = DkC corresponding to an eigenvector νk,j , φk = x=0− e x C , 2πl ik| iik | iika | i ∈ k = ,l 0, 1, ..., n 1 , j 0, 1, 2 , and Dk = diag(e ,e− ,e− ). n ∈{ − } ∈{ } P
11 (n,a) Proof: First, we consider the image of j φk under S for j =0, 1, 2. For j =0 | i × | i n 1 (n,a) − (n,a) S ( 0 φk ) = S ( 0 φk ) | i ⊗ | i x | i ⊗ | i x=0 X ik ik(n 2) ik(n 1) = 0 n 1 + 0 e + ... + n 3 e − + n 2 e − | i⊗ | − i | i | − i | − i ik+i(n 1)k ik ik+ik(n 3) ik+ik(n 2) = 0 n 1 e − + 0 e + ... + n 3 e − + n 2 e − | i⊗ | − i | i | − i | − i ik ik = 0 e φk = e ( 0 φk ). | i⊗ | i | i ⊗ | i Similarly,
(n,a) ik (n,a) ika S ( 1 φk )= e− ( 1 φk ) and S ( 2 φk )= e− ( 2 φk ). | i ⊗ | i | i ⊗ | i | i ⊗ | i | i ⊗ | i
Let νk,j be an eigenvector of Uk corresponding to the eigenvalue λk,j , j =0, 1, 2. Suppose | i 2 C νk,j = αj j for some αj C. Then | i j=0 | i ∈ P (n,a) U ψk,j = S (C νk,j φk ) (n,a) | i | i × | i 2 (n,a) = S αj j φk | i ⊗ | i j=0 X ik ik ika = α e 0 + α e− 1 + α e− 2 φk 0 | i 1 | i 2 | i ⊗ | i = DkC νk,j φk | i ⊗ | i = λk,j ψk,j . | i This completes the proof. Here we mention that the above proposition is proved in [Lemma 1, [30]] for the case when C = G, the Grover matrix. However, there is a typo in the statement of Lemma 1 about φk . Below, we investigate periodicity of lively quantum walks when the coin operator | i belongs to subsets of R. Since periodicity of the walk is decided by the eigenvalues of U PO (n,a) that are indeed eigenvalues of Uk, we derive eigenvalues of Uk = DkC, C R, and hence determine the periodicity of the corresponding walk. First, we recall the following∈ PO results for lazy quantum walks on cycles [17]. Theorem 4.2. Consider the lazy quantum walk corresponding to the coin operator C = Z Z [ζlcm(n,t)] Z[ζt] [ζlcm(n,t)] t [cij ]3 3. If c11 or c22 or c33 , then U(n,0) = I for any t N. × 6∈ n 6∈ n 6∈ n 6 ∈ t Lemma 4.3. If U(n,0) = I for some positive integer t then λ1(l)+ λ2(l)+ λ3(l) Z[ζt] where 2πl ∈ l 0, 1,...,n 1 and λj (l) are eigenvalues of Uk, k = , j =1, 2, 3. ∈{ − } n A straightforward generalization of the above theorem provides a necessary condition for finite periodicity for lively (a = 0) quantum walks on cycles can be stated as follows. 6 Lemma 4.4. Let C = [cij ]3 3 be a coin operator of order 3 in a lively quantum walk with × Z Z [ζlcm(n,t)] [ζlcm(n,t)] liveliness ′a′ where n la for l 0, 1, ..., n 1 . If c11 n or c22 n or Z | ∈ { − } 6∈ 6∈ c [ζt] , then U t = I for any t N. 33 6∈ n (n,a) 6 ∈ ika 2πl Proof: The proof follows from the fact that e = 1 when n divides la, where k = n and l 0, 1, ..., n 1 and Theorem 4.2. ∈{ − }
4.1 C R ∈ X In this section we derive periodicity of the lively quantum walks defined by the walk operator (n,a) U = S (C I) when C R. (n,a) ⊗ ∈ X
12 Lemma 4.5. Eigenvalues of Uk = DkC, C R are given by 1 and ∈ X x 1 (x +2x cos k + 1)(x +2x cos k 3) λk± = − + x cos k − 2 ± p 2 1 n 2πl where x 1, and divides l 0, 1,...,n 1 , k = n . Moreover, λk± = λ2±π k. −3 ≤ ≤ gcd(n,a) ∈{ − } − Proof: Note that eikx eiky eik(1 x y) ik ik ika ik ik −ik − Uk = diag(e ,e− ,e− )C = e− (1 x y) e− x e− y −ika − ika ika e− y e− (1 x y) e− x − − 2 2 1 where x + y x y + xy = 0, x 1. Then the characteristic polynomial of Uk is − − −3 ≤ ≤ given by 3 ika 2 ika ika χ(λ)= λ x(e− +2cos k)λ + x(e− (2 cos k)+1)λ e− . − − ika n Further, e = 1 since gcd(n,a) divides l implies n divides la. Consequently,
χ(λ) = (λ 1) λ2 λ(x +2x cos k 1)+1 . − − − Then the roots of χ(λ)=0 are
x (x +2x cos k + 1)(x +2x cos k 3) 1 1, + x cos k − . 2 ± p 2 − 2 Hence the desired result follows. Now we show that the Grover coin operator is special among all coin operators in R I in terms of its eigenvalues. X \{ }
2 Theorem 4.6. All the eigenvalues of a matrix C R are real if and only if C = G = 3 J I, the Grover matrix or C is the identity matrix. ∈ X −
Proof: Using Lemma 4.5 the eigenvalues of Uk for C R are ∈ X x (x +2x cos k + 1)(x +2x cos k 3) 1 1, + x cos k − , 2 ± p 2 − 2 where 1 x 1, k = 2πl and l =0, 1, 2. Now k = 0 when l =0. Then U = diag(1, 1, 1)C. − 3 ≤ ≤ 3 0 Thus U0 = C, and hence the eigenvalues of C are
3x 1 (3x + 1)(3x 3) − − . 2 ± p 2 1 Obviously (3x + 1)(3x 3) R if and only if x 3 , 1 . Thus either C = I or C = G. The following theorem− describes∈ the periodicity∈ {− of lively} quantum walks when the coin operator belongs to R. The proof is similar to the proof of [Theorem 2.2, [17]]. X Theorem 4.7. The period of a lively quantum walk on Cn defined by the walk operator (n,a) U = S (C I), C R = θ is given by (n,a) ⊗ ∈ X X
2l ml lcm clpl :0 (2 cos θ 2) (2 cos θ + 1) 2π i (2 cos θ + 1) 2π i − + e n + e− n Z[ζn]. 3 3 3 ∈ Consequently, 2π i 2π i 2π i e n (2 cos θ 2) + (2cos θ + 1)e n + (2cos θ + 1)e− n 3Z[ζn], − ∈ which implies 2π i 4π i (2 cos θ 2)e n + (2cos θ + 1)e n + (2cos θ + 1) 3Z[ζn]. − ∈ φ(n) 1 i Further, any τ Z[ζn] can be written as j=0 − ζn where φ to be the Euler totient ∈ 2πi×0 2πi×1 2πi×2 function. If φ(n) > 2, the coefficients of e n ,eP n ,e n do not belong to 3Z except for cos θ 1, 1/2 that is for permutation matrices where θ 2π/3, 0 . For φ(n) 2, that is, for∈{n −2, 4,}6 , the coin operator C are not permutation∈ {± matrices.} As above, the≤ proof follows. Indeed,∈ { for} n =2, (2 cos θ 2)ζ1 + (2cos θ + 1)(ζ2 + ζ0) = (2cos θ + 4)ζ0 3Z[ζ ]. − 2 2 2 2 6∈ 2 Similarly for n =4 (2 cos θ 2)ζ1 + (2cos θ + 1)(ζ2 + ζ0) = (2cos θ 2)ζ1 3Z[ζ ], − 4 4 4 − 4 6∈ 4 and (2 cos θ 2)ζ1 + (2cos θ + 1)(ζ2 + ζ0) = (4cos θ 1)ζ1 3Z[ζ ] − 6 6 6 − 6 6∈ 6 14 for n =6. Hence the necessary condition for finite periodicity is not satisfied. Thus for n =3 the walk has no finite period. This completes the proof. 6 π Now we restrict our attention to two specific coin operators in R = θ. Indeed for θ = 2 3π X X and 2 , the corresponding coin operators are given by 1 1+√3 1 √3 1 1 √3 1+√3 3 3 −3 3 −3 3 ∆ = 1 √3 1 1+√3 and ∆ = 1+√3 1 1 √3 1 −3 3 3 2 3 3 −3 1+√3 1 √3 1 1 √3 1+√3 1 3 −3 3 −3 3 3 2 respectively. Moreover, ∆i = G, the Grover matrix, i = 1, 2. Then we have the following corollary. Corollary 4.8. The period of lively quantum walks on cycles Cn corresponding to the coin operator ∆i,i =1, 2 are given by 12, if n =3 , otherwise. ∞ 4.2 C R ∈ Z Let C . Then the characteristic polynomial of Uk = SC, is given by ∈ ZR 3 2 ik ik ik ik χ(λ)=1+ λ λ (e− + y ye− + i2x sin k) λ(e + y ye i2x sin k). − − − − − Obviously, the product of the eigenvalues is 1. Then we have the following theorem. − Theorem 4.9. The period of the lively quantum walk on a cycle Cn with walk operator (n,a) U = S (C I), C R = θ is given by (n,a) ⊗ ∈ Z Z l ml lcm 2, 2pl :0 15 Now let n =3. Then the characteristic equation is 1 + λ3 λ λ2 = 0 when l =0, that is, − − k =0. Thus the eigenvalues of Uk are 1, 1 and hence the period of the corresponding walk is 2π θ π θ π 2π ± i(θ 3 ) i( 2 3 ) i( 2 3 ) 2. For l =1, and hence k = 3 the eigenvalues of Uk are e− − ,e − , e − , which 2pπi i( pπ ) i( pπ ) − are of the form e− q ,e q , e q , considering the one parameter representation θ of R, θ π pπ − Z Z and setting 2 = 3 q . Then assuming q,p as even or odd the desired result follows, that is, − π 2π θ π θ π 4 i(θ 3 ) i( 2 3 ) i( 2 + 3 ) the period is 2q. For l =2, that is, k = 3 , the eigenvalues are e − ,e− − , e− 2pπi i( pπ ) i( pπ ) − which are of the form e q ,e− q , e− q for some nonnegative rational number p/q. Consequently the desired result follows− as above, and hence the period of the corresponding walk is lcm(2, 2q) when n =3. t Now, let n =3. If possible let U(n,a) = I for some positive integer t. Then by Lemma 4.3, 3 6 3 i=1 λi(l) Z[ζt], so that i=1 λi(l) Q[ζn]. Then from the characteristic polynomial of ik ∈ ik ik ik ∈ 1+2 cos θ Uk, e− + y ye− + xe xe− Q[ζn]. Setting x = 3 and the corresponding y Pfrom equation− (15), we obtainP− ∈ ik (2 cos θ + 1) ik ik (1 cos θ + √3 sin θ) ik e− + (e e− )+ − (1 e− ) Q[ζn] 3 − 3 − ∈ 2π for l 0, 1, 2, ..., n 1 . Whenever l = 1, k = and using definition of Z[ζn] we have ∈{ − } n ik (2 cos θ + 1) (1 cos θ + √3 sin θ) ik 2cos θ +1 (1 cos θ + √3 sin θ) e− 1 − +e + − Z[ζn], − 3 − 3 3 3 ∈ ! which implies ik 1 cos θ √3 sin θ ik 2cos θ +1 (1 cos θ + √3 sin θ) e− − + e + − Z[ζn]. 3 − 3 3 3 ∈ ! Therefore, ik ik ik e e− (1 cos θ √3 sin θ)+ e (1+2cos θ)+(1 cos θ + √3 sin θ) 3Z[ζn], − − − ∈ which further implies ik 2ik (1 cos θ √3 sin θ)+ e (1 cos θ + √3 sin θ)+ e (1+2cos θ) 3Z[ζn]. − − − ∈ 0 2πi 4πi When Euler’s totient function φ(n) > 2, the coefficients of e ,e n ,e n 3Z[ζn] if and only if θ = 0, 2π that is, the coin is a permutation matrix. If φ(n) ∈2, that is, for ± 3 ≤ n 2, 4, 6 , we consider the following cases. For non-permutation matrices in R: ∈{ } Z 1. n =2: (1+2cos θ) 3Z[ζ ] 6∈ 2 2. n =4: ( 3cos θ √3 sin θ)+(1 cos θ + √3 sin θ)ζ1 3Z[ζ ] − − − 4 6∈ 4 3. n =6: (1 cos θ √3 sin θ)ζ1 +(1+2cos θ)ζ2 3Z[ζ ]. − − 6 6 6∈ 6 The above equations are satisfied only for all permutation matrices in R. However the conditions not satisfied otherwise, when l = 1. This completes the proof. Z 4.3 C R ∈Y The characteristic polynomial of Uk = DkC, C R is given by ∈ Y χ(λ)= λ3 +1 λ2(x +2x cos k) λ(x +2x cos k). − − Then the following theorem describes the periodicity of the corresponding quantum walk. 16 Theorem 4.10. The period of the lively quantum walk on a cycle Cn with walk operator (n,a) U = S (C I), C R = θ is given by (n,a) ⊗ ∈ Y Y 2l ml lcm 2, 2pl :0 4.4 C R ∈ W The characteristic polynomial of Uk = DkC, C R is given by ∈W 3 2 ik ik ik ik ik ik ik ik χ(λ)= λ + λ (e− y + ye− + xe + xe− ) λ(e y + ye xe− + xe ) 1. − − − − − − Then we have the following theorem. Theorem 4.11. The period of the lively quantum walk on a cycle Cn with walk operator (n,a) U = S (C I), C R = θ is given by (n,a) ⊗ ∈W W l ml lcm 2, 2pl :0 4.5 C Q ∈ PO Now we consider coins which belong to Q = Q Q Q Q, the group of all rational PO X ∪ Y ∪ Z ∪W matrices which are linear sum of permutation matrices of order 3 3. Obviously, Grov Q, 2 × ⊂ PO Q and the Grover coin G = J I Grov. In [17], it has been shown that period P ⊂PO 3 − ∈ of lively quantum walks on cycles Cn with Grover coin is finite, which is 6, if and only if n =3. We have already seen that period is finite when the coin operator C or C , ∈P − ∈P see Theorem 4.7 - 4.11. We mention that Q R and hence the periodicity can be PO ⊂ PO calculated for a particular coin C Q. However, the following result is remarkable since it proves the importance of rational∈ coins PO which are of Grover-type matrices. We recall Niven’s theorem which states that if x and sin x are both rational, then sin x 0, 1 , 1 [28]. π ∈{ ± 2 ± } Theorem 4.12. Period of lively quantum walks on cycles Cn defined by the walk operator (n,a) U(n,a) = S (C I), C Q is finite if and only if C or C Grov. In particular, the period of the quantum⊗ walk∈ PO corresponding to the coin operators− C∈or P ∪ C in Grov are given by − 2 2 6, if n =3 and, C or C 3 J P2, 3 J P3 12, if n =3 and, C or− C∈= 2 J − P − 3 6 4, if n =3 and, C or −C = 2 J −P . − 3 − 5 Proof: First consider C Q. Then the eigenvalues of Uk that are not 1 are given as ∈ X x (x +2x cos k + 1)(x +2x cos k 3) 1 + x cos k − 2 ± p 2 − 2 17 iπθ x 1 which must be of the form e± and cos πθ = 2 + x cos k 2 , for some θ R. Finite peri- t − ∈ odicity of the corresponding walk implies U(n,a) = I for some positive integer t which further iπtθ implies e± = 1. Consequently, θ must be rational. By Niven’s Theorem, it can be con- 1 cluded that when θ and cos πθ are rational, cos πθ 0, 1, 2 . Using the description of 1 2 ∈ { ± ± } θ, when k = 0, x 3 , 3 , 1, 0 . Thus the matrices in R are the permutation matrices or XGrover Matrix or the∈ {± matrices 2}J P and 2 J P . However,X if x = 1 , the corresponding 3 − 2 3 − 3 3 matrix C is no longer rational. Hence, only non-permutation matrices in Q that have finite 2 2 2 X period in quantum lively walk on Cn are J I, J P and J P and n = 3. Similarly, 3 − 3 − 2 3 − 3 considering the coin operators belonging to R, R and R, the desired result follows. The periods follows from Theorem 4.7-4.11. Y Z W Conclusion. We have derived period of three state lively quantum walk on cycle when the coin operator of the walk is a real orthogonal matrix which can be expressed as linear sum of permutation matrices. This is achieved by first classifying all orthogonal matrices of order 3 3 that can be expressed as linear sum of permutation matrices. We have also established that× the set of such matrices is same as the group of orthogonal permutative matrices. We mention that Grover matrix has been used in literature as coin operator to define many quantum mechanical algorithms like quantum walks. 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