Efficient Inner-product Algorithm for Stabilizer States

Efficient Inner-product Algorithm for Stabilizer States H´ector J. Garc´ıa∗ [email protected]

Igor L. Markov∗ [email protected]

Andrew W. Cross† [email protected]

∗ University of Michigan – EECS Department 2260 Hayward Street, Ann Arbor, MI, 48109-2121 † IBM T. J. Watson Research Center Yorktown Heights, NY 10598

arXiv:1210.6646v3 [cs.ET] 7 Aug 2013

Abstract Large-scale quantum computation is likely to require massive quantum error correction (QEC). QEC codes and circuits are described via the stabilizer formalism, which represents stabilizer states by keeping track of the operators that preserve them. Such states are obtained by stabilizer circuits (consisting of CNOT, Hadamard and Phase only) and can be represented compactly on conventional computers using Ω(n2 ) bits, where n is the number of qubits [10]. Although techniques for the efficient simulation of stabilizer circuits have been studied extensively [1], [9], [10], techniques for efficient manipulation of stabilizer states are not currently available. To this end, we leverage the theoretical insights from [1] and [16] to design new algorithms for: (i) obtaining canonical generators for stabilizer states, (ii) obtaining canonical stabilizer circuits, and (iii) computing the inner product between stabilizer states. Our inner-product algorithm takes O(n3 ) time in general, but observes quadratic behavior for many practical instances relevant to QECC (e.g., GHZ states). We prove that each n-qubit stabilizer state has exactly 4(2n − 1) nearest-neighbor stabilizer states, and verify this claim experimentally using our algorithms. We design techniques for representing arbitrary quantum states using stabilizer frames and generalize our algorithms to compute the inner product between two such frames.

I. I NTRODUCTION Gottesman [9] and Knill showed that for certain types of non-trivial quantum circuits known as stabilizer circuits, efficient simulation on classical computers is possible. Stabilizer circuits are exclusively composed of stabilizer gates – Controlled-NOT, Hadamard and Phase gates (Figure 1a) – followed by measurements in the computational basis. Such circuits are applied to a computational basis state (usually |00...0i) and produce output states called stabilizer states. The case of purely unitary stabilizer circuits (without measurement gates) is considered often, e.g., by consolidating measurements at the end. Stabilizer circuits can be simulated in poly-time by keeping track of a set Pauli operators that stabilize1 the quantum state. Such stabilizer operators uniquely represent a stabilizer state up to an unobservable global phase. Equation 1 shows that the number of n-qubit 2 stabilizer states grows as 2n /2 , therefore, describing a generic stabilizer state requires at least n2 /2 bits. Despite their compact representation, stabilizer states can exhibit multi-qubit entanglement and are often encountered in many quantum information applications such as Bell states, GHZ states, error-correcting codes and one-way quantum computation. To better understand the role stabilizer states play in such applications, researchers have designed techniques to quantify the amount of entanglement [8], [18], [12] in such states and studied relevant properties such as purification schemes [7], Bell inequalities [11] and equivalence classes [17]. Efficient algorithms for the manipulation of stabilizer states (e.g., computing the angle between them), can help lead to additional insights related to linear-algebraic and geometric properties of stabilizer states. In this work, we describe in detail algorithms for the efficient computation of the inner product between stabilizer states. We adopt the approach outlined in [1], which requires the synthesis of a unitary stabilizer circuit that maps a stabilizer state to a computational basis state. The work in [1] shows that, for any unitary stabilizer circuit, there exists an equivalent blockstructured canonical circuit that applies a block of Hadamard (H) gates only, followed by a block of CNOT (C) only, then a block of Phase (P ) gates only, and so on in the 7-block sequence H-C-P -C-P -C-H. Using an alternate representation for stabilizer states, the work in [16] proves the existence of a (H-C-P -CZ)-canonical circuit, where the CZ block consists of Controlled-Z (CPHASE) gates. However, no algorithms are known to synthesize such smaller 4-block circuits given an arbitrary stabilizer state. In contrast, we describe an algorithm for synthesizing (H-C-CZ-P -H)-canonical circuits given any input stabilizer state. We prove that any n-qubit stabilizer state |ψi has exactly 4(2n − 1) nearest-neighbors – stabilizer states |ϕi such that |hψ|ϕi| attains the largest possible value 6= 1. Furthermore, we design techniques for representing arbitrary quantum states using stabilizer frames and generalize our algorithms to compute the inner product between two such frames. This paper is structured as follows. Section II reviews the stabilizer formalism and relevant algorithms for manipulating stabilizer-based representations of quantum states. Section III describes our circuit-synthesis and inner-product algorithms. In Section IV, we evaluate the performance of our algorithms. Our findings related to geometric properties of stabilizer states are 1 An

operator U is said to stabilize a state iff U |ψi = |ψi.

1

1 0 CN OT =  0 

H=

√1 2

 1 1

 1 −1

 P =

1 0

 0 i

0 Fig. 1.

0 1 0 0

0 0 0 1

 0 0 1 0

X=

 0 1

 1 0

Y =

 0 i

−i 0

 Z=

 1 0

 0 −1

Fig. 1. (b) The Pauli matrices.

(a) Unitary stabilizer gates Hadamard (H), Phase (P) and Controlled-NOT (CNOT).

described in Section V. In Section VI, we discuss stabilizer frames and how they can be used to represent arbitrary states and extend our algorithms to compute the inner product between frames. Section VII closes with concluding remarks. II. BACKGROUND AND P REVIOUS W ORK Gottesman [10] developed a description for quantum states involving the Heisenberg representation often used by physicists to describe atomic phenomena. In this model, one describes quantum states by keeping track of their symmetries rather than explicitly maintaining complex vectors. The symmetries are operators for which these states are 1-eigenvectors. Algebraically, symmetries form group structures, which can be specified compactly by group generators. It turns out that this approach, also known as the stabilizer formalism, can be used to represent an important class of quantum states. A. The stabilizer formalism A unitary operator U stabilizes a state |ψi if |ψi is a 1–eigenvector of U , i.e., U |ψi = |ψi [9], [14]. We are interested in operators U derived from the Pauli matrices shown in Figure 1b The following table lists the one-qubit states stabilized by the Pauli matrices. √ √ −X : (|0i − |1i)/ √2 X : (|0i + |1i)/ √2 Y : (|0i + i |1i)/ 2 −Y : (|0i − i |1i)/ 2 Z : |0i −Z : |1i √ Observe that I stabilizes all states and −I does not stabilize any state. As an example, the entangled state (|00i + |11i)/ 2 is stabilized by the Pauli operators X ⊗ X, −Y ⊗ Y , Z ⊗ Z and I ⊗ I. As shown in Table I, it turns out that the Pauli matrices along with I and the multiplicative factors ±1, ±i, form a closed group under matrix multiplication [14]. Formally, the Pauli group Gn on n qubits consists of the n-fold tensor product of Pauli matrices, P = ik P1 ⊗ · · · ⊗ Pn such that Pj ∈ {I, X, Y, Z} and k ∈ {0, 1, 2, 3}. For brevity, the tensor-product symbol is often omitted so that P is denoted by a string of I, X, Y and Z characters or Pauli literals and a separate integer value k for the phase ik . This string-integer pair representation allows us to compute the product of Pauli operators without explicitly computing the tensor products,2 e.g., (−IIXI)(iIY II) = −iIY XI. Since | Gn |= 4n+1 , Gn can have at most log2 | Gn |= log2 4n+1 = 2(n + 1) irredundant generators [14]. The key idea behind the stabilizer formalism is to represent an n-qubit quantum state |ψi by its stabilizer group S(|ψi) – the subgroup of Gn that stabilizes |ψi. As the following theorem shows, if |S(|ψi)| = 2n , the group uniquely specifies |ψi. Theorem 1. For an n-qubit pure state |ψi and k ≤ n, S(|ψi) ∼ = Zk2 . If k = n, |ψi is specified uniquely by S(|ψi) and is called a stabilizer state. Proof: (i) To prove that S(|ψi) is commutative, let P, Q ∈ S(|ψi) such that P Q |ψi = |ψi. If P and Q anticommute, −QP |ψi = −Q(P |ψi) = −Q |ψi = − |ψi = 6 |ψi. Thus, P and Q cannot both be elements of S(|ψi). (ii) To prove that every element of S(|ψi) is of degree 2, let P ∈ S(|ψi) such that P |ψi = |ψi. Observe that P 2 = il I for l ∈ {0, 1, 2, 3}. Since P 2 |ψi = P (P |ψi) = P |ψi = |ψi, we obtain il = 1 and P 2 = I. (iii) From group theory, a finite Abelian group with a2 = a for every element must be ∼ = Zk2 . (iv) We now prove that k ≤ n. First note that each independent generator P ∈ S(|ψi) imposes the linear constraint P |ψi = |ψi on the 2n -dimensional vector space. The subspace of vectors that satisfy such a constraint has dimension 2n−1 , or half the space. Let gen(|ψi) be the set of generators for S(|ψi). We add independent generators to gen(|ψi) one by one and impose their

TABLE I M ULTIPLICATION TABLE FOR PAULI MATRICES . S HADED CELLS INDICATE ANTICOMMUTING PRODUCTS .

I X Y Z

I I X Y Z

X X I −iZ iY

2 This holds true due to the identity: (A ⊗ B)(C ⊗ D) = (AC ⊗ BD).

2

Y Y iZ I −iX

Z Z −iY iX I

linear constraints, to limit |ψi to the shared 1-eigenvector. Thus the size of gen(|ψi) is at most n. In the case |gen(|ψi)| = n, the n independent generators reduce the subspace of possible states to dimension one. Thus, |ψi is uniquely specified. The proof of Theorem 1 shows that S(|ψi) is specified by only log2 2n = n irredundant stabilizer generators. Therefore, an arbitrary n-qubit stabilizer state can be represented by a stabilizer matrix M whose rows represent a set of generators g1 , . . . , gn for S(|ψi). (Hence we use the terms generator set and stabilizer matrix interchangeably.) Since each gi is a string of n Pauli literals, the size of the matrix is n × n. The phases of each gi are stored separately using a vector of n integers. Therefore, the storage cost for M is Θ(n2 ), which is an exponential improvement over the O(2n ) cost often encountered in vector-based representations. Theorem 1 suggests that Pauli literals can be represented using only two bits, e.g., 00 = I, 01 = Z, 10 = X and 11 = Y . Therefore, a stabilizer matrix can be encoded using an n × 2n binary matrix or tableau. The advantage of this approach is that 2 this literal-to-bits mapping induces an isomorphism Z2n 2 → Gn because vector addition in Z2 is equivalent to multiplication of Pauli operators up to a global phase. The tableau implementation of the stabilizer formalism is covered in [1], [14]. Proposition 2. The number of n-qubit pure stabilizer states is given by N (n) = 2n

n−1 Y

(2n−k + 1) = 2(.5+o(1))n

2

(1)

k=0

The proof of Proposition 2 can be found in [1]. An alternate interpretation of Equation 1 is given by the simple recurrence relation N (n) = 2(2n + 1)N (n − 1) with base case N (1) = 6. For example, for n = 2 the number of stabilizer states is 60, and for n = 3 it is 1080. This recurrence relation stems from the fact that there are 2n + 1 ways of combining the generators of N (n − 1) with additional Pauli matrices to form valid n-qubit generators. The factor of 2 accounts for the increase in the number of possible sign configurations. Table II and Appendix A list all two-qubit and three-qubit stabilizer states, respectively. Observation 3. Consider a stabilizer state |ψi represented by a set of generators of its stabilizer group S(|ψi). Recall from the proof of Theorem 1 that, since S(|ψi) ∼ = Zn2 , each generator imposes a linear constraint on |ψi. Therefore, the set of generators can be viewed as a system of linear equations whose solution yields the 2n basis amplitudes that make up |ψi. Thus, one needs to perform Gaussian elimination to obtain the basis amplitudes from a generator set. Canonical stabilizer matrices. Although stabilizer states are uniquely determined √ by their stabilizer group, the set of generators may be selected in different ways. For example, the state |ψi = (|00i + |11i)/ 2 is uniquely specified by any of the following stabilizer matrices: XX XX -Y Y M1 = M2 = M3 = ZZ -Y Y ZZ One obtains M2 from M1 by left-multiplying the second row by the first. Similarly, one can also obtain M3 from M1 or M2 via row multiplication. Observe that, multiplying any row by itself yields II, which stabilizes |ψi. However, II cannot be

E NTANGLED

S EPARABLE

TABLE II S IXTY TWO - QUBIT STABILIZER STATES AND THEIR CORRESPONDING PAULI GENERATORS . S HORTHAND NOTATION REPRESENTS A STABILIZER STATE AS α0 , α1 , α2 , α3 WHERE αi ARE THE NORMALIZED AMPLITUDES OF THE BASIS STATES . T HE BASIS STATES ARE EMPHASIZED IN BOLD . T HE FIRST COLUMN LISTS STATES WHOSE GENERATORS DO NOT INCLUDE AN UPFRONT MINUS SIGN , AND OTHER COLUMNS INTRODUCE THE SIGNS . A SIGN CHANGE CREATES AN ORTHOGONAL VECTOR . T HEREFORE , EACH ROW OF THE TABLE GIVES AN ORTHOGONAL BASIS . T HE CELLS IN DARK GREY INDICATE STABILIZER STATES WITH FOUR NON - ZERO BASIS AMPLITUDES , I . E ., αi 6= 0 ∀ i. T HE ∠ COLUMN INDICATES THE ANGLE BETWEEN THAT STATE AND |00i, WHICH HAS 12 NEAREST- NEIGHBOR STATES ( LIGHT GRAY ) AND 15 ORTHOGONAL STATES (⊥). S TATE G EN ’ TORS 1, 1, 1, 1 IX, XI 1, 1, i, i IX, YI 1, 1, 0, 0 IX, ZI 1, i, 1, i IY, XI 1, i, i, −1 IY, YI 1, i, 0, 0 IY, ZI 1, 0, 1, 0 IZ, XI 1, 0, i, 0 IZ, YI 1, 0, 0, 0 IZ, ZI 0, 1, 1, 0 XX, YY 1, 0, 0, i XY, YX 1, 1, 1, −1 XZ, ZX 1, i, 1, −i XZ, ZY 1, 1, i, −i YZ, ZX 1, i, i, 1 YZ, ZY

∠ π/3 π/3 π/4 π/3 π/3 π/4 π/4 π/4 0 ⊥ π/4 π/3 π/3 π/3 π/3

S TATE 1, −1, 1, −1 1, −1, i, −i 1, −1, 0, 0 1, −i, 1, −i 1, −i, i, 1 1, −i, 0, 0 0, 1, 0, 1 0, 1, 0, i 0, 1, 0, 0 1, 0, 0, −1 0, 1, i, 0 1, 1, −1, 1 1, i, −1, i 1, 1, −i, i 1, i, −i, −1

G EN ’ TORS -IX, XI -IX, YI -IX, ZI -IY, XI -IY, YI -IY, ZI -IZ, XI -IZ, YI -IZ, ZI -XX, YY -XY, YX -XZ, ZX -XZ, ZY -YZ, ZX -YZ, ZY

∠ π/3 π/3 π/4 π/3 π/3 π/4 ⊥ ⊥ ⊥ π/4 ⊥ π/3 π/3 π/3 π/3

S TATE 1, 1, −1, −1 1, 1, −i, −i 0, 0, 1, 1 1, i, −1, −i 1, i, −i, 1 0, 0, 1, i 1, 0, −1, 0 1, 0, −i, 0 0, 0, 1, 0 1, 0, 0, 1 0, 1, −i, 0 1, −1, 1, 1 1, −i, 1, i 1, −1, i, i 1, −i, i, −1

3

G EN ’ TORS IX, -XI IX, -YI IX, -ZI IY, -XI IY, -YI IY, -ZI IZ, -XI IZ, -YI IZ, -ZI XX, -YY XY, -YX XZ, -ZX XZ, -ZY YZ, -ZX YZ, -ZY

∠ π/3 π/3 ⊥ π/3 π/3 ⊥ π/4 π/4 ⊥ π/4 ⊥ π/3 π/3 π/3 π/3

S TATE 1, −1, −1, 1 1, −1, −i, i 0, 0, 1, −1 1, −i, −1, i 1, −i, −i, −1 0, 0, 1, −i 0, 1, 0, −1 0, 1, 0, −i 0, 0, 0, 1 0, 1, −1, 0 1, 0, 0, −i 1, −1, −1, −1 1, −i, −1, −i 1, −1, −i, −i 1, −i, −i, 1

G EN ’ TORS -IX, -XI -IX, -YI -IX, -ZI -IY, -XI -IY, -YI -IY, -ZI -IZ, -XI -IZ, -YI -IZ, -ZI -XX, -YY -XY, -YX -XZ, -ZX -XZ, -ZY -YZ, -ZX -YZ, -ZY

∠ π/3 π/3 ⊥ π/3 π/3 ⊥ ⊥ ⊥ ⊥ ⊥ π/4 π/3 π/3 π/3 π/3

Fig. 2. Canonical (row-reduced echelon) form for stabilizer matrices. The X-block contains a minimal set of rows with X/Y literals. The rows with Z literals only appear in the Z-block. Each block is arranged so that the leading non-I literal of each row is strictly to the right of the leading non-I literal in the row above. The number of Pauli (non-I) literals in each block is minimal.

used as a stabilizer generator because it is redundant and carries no information about the structure of |ψi. This also holds true in general for M of any size. Any stabilizer matrix can be rearranged by applying sequences of elementary row operations in order to obtain a particular matrix structure. Such operations do not modify the stabilizer state. The elementary row operations that can be performed on a stabilizer matrix are transposition, which swaps two rows of the matrix, and multiplication, which left-multiplies one row with another. Such operations allow one to rearrange the stabilizer matrix in a series of steps that resemble Gauss-Jordan elimination.3 Given an n × n stabilizer matrix, row transpositions are performed in constant time4 while row multiplications require Θ(n) time. Algorithm 1 rearranges a stabilizer matrix into a row-reduced echelon form that contains: (i) a minimum set of generators with X and Y literals appearing at the top, and (ii) generators containing a minimum set of Z literals only appearing at the bottom of the matrix. This particular stabilizer-matrix structure, shown in Figure 2, defines a canonical representation for stabilizer states [6], [10]. The algorithm iteratively determines which row operations to apply based on the Pauli (non-I) literals contained in the first row and column of an increasingly smaller submatrix of the full stabilizer matrix. Initially, the submatrix considered is the full stabilizer matrix. After the proper row operations are applied, the dimensions of the submatrix decrease by one until the size of the submatrix reaches one. The algorithm performs this process twice, once to position the rows with X(Y ) literals at the top, and then again to position the remaining rows containing Z literals only at the bottom. Let i ∈ {1, . . . , n} and j ∈ {1, . . . , n} be the index of the first row and first column, respectively, of submatrix A. The steps to construct the upper-triangular portion of the row-echelon form shown in Figure 2 are as follows. 1. Let k be a row in A whose j th literal is X(Y ). Swap rows k and i such that k is the first row of A. Decrease the height of A by one (i.e., increase i). 2. For each row m ∈ {0, . . . , n}, m 6= i that has an X(Y ) in column j, use row multiplication to set the j th literal in row m to I or Z. 3. Decrease the width of A by one (i.e., increase j). To bring the matrix to its lower-triangular form, one executes the same process with the following difference: (i) step 1 looks for rows that have a Z literal (instead of X or Y ) in column j, and (ii) step 2 looks for rows that have Z or Y literals (instead of X or Y ) in column j. Observe that Algorithm 1 ensures that the columns in M have at most two distinct types of non-I literals. Since Algorithm 1 inspects all n2 entries in the matrix and performs a constant number of row multiplications each time, the runtime of the algorithm is O(n3 ). An alternative row-echelon form for stabilizer generators along with relevant algorithms to obtain them were introduced in [2]. However, their matrix structure is not canonical as it does not guarantee a minimum set of generators with X/Y literals. Stabilizer circuit simulation. The computational basis states are stabilizer states that can be represented using the following stabilizer-matrix structure. Definition 4. A stabilizer matrix is in basis form if it has the following structure.   ± Z I ··· I  ±   I Z ··· I  ..  .. .. . . ..  . .  .  . . ±

I

I

···

Z

In this matrix form, the ± sign of each row along with its corresponding Zj -literal designates whether the state of the j th qubit is |0i (+) or |1i (−). Suppose we want to simulate circuit C. Stabilizer-based simulation first initializes M to specify some basis state |ψi. To simulate the action of each gate U ∈ C, we conjugate each row gi of M by U .5 We require that 3 Since Gaussian elimination essentially inverts the n × 2n matrix, this could be sped up to O(n2.376 ) time by using fast matrix inversion algorithms. However, O(n3 )- time Gaussian elimination seems more practical. 4 Storing pointers to rows facilitates O(1)-time row transpositions – one simply swaps relevant pointers. 5 Since g |ψi = |ψi, the resulting state U |ψi is stabilized by U g U † because (U g U † )U |ψi = U g |ψi = U |ψi. i i i i

4

U gi U † maps to another string of Pauli literals so that the resulting stabilizer matrix M0 is well-formed. It turns out that the Hadamard, Phase and CNOT gates (Figure 1a) have such mappings, i.e., these gates conjugate the Pauli group onto itself [10], [14]. Table III lists the mapping for each of these gates. √ For example, suppose we simulate a CNOT operation on |ψi = (|00i + |11i)/ 2 using M, XI CN OT 0 −−−−→ M = IZ √ CN OT One can verify that the rows of M0 stabilize |ψi −−−−→ (|00i + |10i)/ 2 as required. Since Hadamard, Phase and CNOT gates are directly simulated using stabilizers, these gates are commonly called stabilizer gates. They are also called Clifford gates because they generate the Clifford group of unitary operators. We use these names interchangeably. Any circuit composed exclusively of stabilizer gates is called a unitary stabilizer circuit. Table III shows that at most two columns of M are updated when one simulates a stabilizer gate. Thus, such gates are simulated in Θ(n) time. M=

XX ZZ

Theorem 5. An n-qubit stabilizer state |ψi can be obtained by applying a stabilizer circuit to the |0i

⊗n

basis state.

Proof: The work in [1] represents the generators using a tableau, and then shows how to construct a unitary stabilizer circuit from the tableau. We refer the reader to [1, Theorem 8] for details of the proof. Corollary 6. An n-qubit stabilizer state |ψi can be transformed by stabilizer gates into the |0i

⊗n

basis state. ⊗n

Proof: Since every stabilizer state can be produced by applying some unitary stabilizer circuit C to the |0i state, it suffices to reverse C to perform the inverse transformation. To reverse a stabilizer circuit, reverse the order of gates and replace every P gate with P P P . The stabilizer formalism also admits one-qubit measurements in the computational basis [10]. However, the updates to M for such gates are not as efficient as for stabilizer gates. Note that any qubit j in a stabilizer state is either in a |0i (|1i) state or in an unbiased6 superposition of both. The former case is called a deterministic outcome and the latter a random outcome. We can tell these cases apart in Θ(n) time by searching for X or Y literals in the j th column of M. If such literals are found, the qubit must be in a superposition and the outcome is random with equal probability (p(0) = p(1) = .5); otherwise the outcome is deterministic (p(0) = 1 or p(1) = 1). Algorithm 1 Canonical form reduction for stabilizer matrices Input: Stabilizer matrix M for S(|ψi) with rows R1 , . . . , Rn Output: M is reduced to row-echelon form ⇒ ROWSWAP(M, i, j) swaps rows Ri and Rj of M ⇒ ROWMULT(M, i, j) left-multiplies rows Ri and Rj , returns updated Ri 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: 18: 19: 20: 21: 22: 23: 24: 25:

i←1 for j ∈ {1, . . . , n} do k ← index of row Rk∈{i,...,n} with j th literal set to X(Y ) if k exists then ROWSWAP(M, i, k) for m ∈ {0, . . . , n} do if j th literal of Rm is X or Y and m 6= i then Rm = ROWMULT(M, Ri , Rm ) end if end for i←i+1 end if end for for j ∈ {1, . . . , n} do k ← index of row Rk∈{i,...,n} with j th literal set to Z if k exists then ROWSWAP(M, i, k) for m ∈ {0, . . . , n} do if j th literal of Rm is Z or Y and m 6= i then Rm = ROWMULT(M, Ri , Rm ) end if end for i←i+1 end if end for

. Setup X block

. Gauss-Jordan elimination step

. Setup Z block

. Gauss-Jordan elimination step

2 2 6 An arbitrary state |ψi with computational basis decomposition Pn k=0 λk |ki is said to be unbiased if for all λi 6= 0 and λj 6= 0, |λi | = |λj | . Otherwise, the state is biased. One can verify that none of the stabilizer gates produce biased states.

5

TABLE III C ONJUGATION OF THE PAULI - GROUP ELEMENTS BY THE STABILIZER GATES [14]. F OR THE CNOT CASE , SUBSCRIPT 1 INDICATES THE CONTROL AND 2 THE TARGET. G ATE H

P

I NPUT X Y Z X Y Z

O UTPUT Z -Y X Y -X Z

G ATE

CN OT

I NPUT I1 X2 X1 I2 I1 Y2 Y1 I 2 I1 Z2 Z1 I2

O UTPUT I1 X2 X1 X2 Z1 Y2 Y1 X2 Z1 Z2 Z1 I2

Random case: one flips an unbiased coin to decide the outcome and then updates M to make it consistent with the outcome obtained. This requires at most n row multiplications leading to O(n2 ) runtime [1], [14]. Deterministic case: no updates to M are necessary but we need to figure out whether the state of the qubit is |0i or |1i, i.e., whether the qubit is stabilized by Z or -Z, respectively. One approach is to apply Algorithm 1 to put M in row-echelon form. This removes redundant literals from M in order to identify the row containing a Z in its j th position and I everywhere else. The ± phase of this row decides the outcome of the measurement. Since this approach is a form of Gaussian elimination, it takes O(n3 ) time in practice. Aaronson and Gottesman [1] improved the runtime of deterministic measurements by doubling the size of M to include n destabilizer generators in addition to the n stabilizer generators. Such destabilizer generators help identify exactly which row multiplications to compute in order to decide the measurement outcome. This approach avoids Gaussian elimination and thus deterministic measurements are computed in O(n2 ) time. III. I NNER - PRODUCT AND CIRCUIT- SYNTHESIS ALGORITHMS Given hψ|ϕi = reiα , we normalize the global phase of |ψi to ensure, without loss of generality, that hψ|ϕi ∈ R+ . Theorem 7. Let S(|ψi) and S(|ϕi) be the stabilizer groups for |ψi and |ϕi, respectively. If there exist P ∈ S(|ψi) and Q ∈ S(|ϕi) such that P = -Q, then |ψi ⊥ |ϕi. Proof: Since |ψi is a 1-eigenvector of P and |ϕi is a (−1)-eigenvector of P , they must be orthogonal. Theorem 8. [1] Let |ψi, |ϕi be non-orthogonal stabilizer states. Let s be the minimum, over all sets of generators {P1 , . . . , Pn } for S(|ψi) and {Q1 , . . . , Qn } for S(|ϕi), of the number of different i values for which Pi 6= Qi . Then, |hψ|ϕi| = 2−s/2 . Proof: Since hψ|ϕi is not affected by unitary transformations U , we choose a stabilizer circuit such that U |ψi = |bi, where |bi is a basis state. For this state, select the stabilizer generators M of the form I . . . IZI . . . I. Perform Gaussian elimination on M to minimize the incidence of Pi 6= Qi . Consider two cases. If U |ϕi = 6 |bi and its generators contain only I/Z literals, then U |ϕi ⊥ U |ψi, which contradicts the assumption that |ψi and |ϕi are non-orthogonal. Otherwise, each generator of U |ϕi √ containing X/Y literals contributes a factor of 1/ 2 to the inner product. Synthesizing canonical circuits. A crucial step in the proof of Theorem 8 is the computation of a stabilizer circuit that brings an n-qubit stabilizer state |ψi to a computational basis state |bi. Consider a stabilizer matrix M that uniquely identifies |ψi. M is reduced to basis form (Definition 4) by applying a series of elementary row and column operations. Recall that row operations (transposition and multiplication) do not modify the state, but column (Clifford) operations do. Thus, the column operations involved in the reduction process constitute a unitary stabilizer circuit C such that C |ψi = |bi, where |bi is a basis state. Algorithm 2 reduces an input stabilizer matrix M to basis form and returns the circuit C that performs such a mapping. Definition 9. Given a finite sequence of quantum gates, a circuit template describes a segmentation of the circuit into blocks where each block uses only one gate type. The blocks must correspond to the sequence and be concatenated in that order. For example, a circuit satisfying the H-C-P template starts with a block of Hadamard (H) gates, followed by a block of CNOT (C) gates, followed by a block of Phase (P ) gates. Definition 10. A circuit with a template structure consisting entirely of CNOT, Hadamard and Phase blocks is called a canonical stabilizer circuit. Canonical forms are useful for synthesizing stabilizer circuits that minimize the number of gates and qubits required to produce a particular computation. This is particularly important in the context of quantum fault-tolerant architectures that are based on stabilizer codes. Given any stabilizer matrix, Algorithm 2 synthesizes a 5-block canonical circuit with template HC-CZ-P -H (Figure 3-a), where the CZ block consists of Controlled-Z (CPHASE) gates. Such gates are stabilizer gates since CPHASEi,j =Hj CNOTi,j Hj (Figure 3-b). In our implementation, such gates are simulated directly on the stabilizer. The work 6

|ψi

|b1 i

      

• |b2 i H

     

CNOT

CPHASE

.. .

P

H .. .

|b3 i

• ≡

Z

H

H

|bn i

(a)

(b)

Fig. 3. (a) Template structure for the basis-normalization circuit synthesized by Algorithm 2. The input is an arbitrary stabilizer state |ψi while the output is a basis state |b1 , . . . , bn i, where b1 , . . . , bn ∈ {0, 1}n . (b) Controlled-Z gates used in the CPHASE block. CPHASE gates can be implemented directly or using the equivalence shown here.

in [1] establishes a longer 7-block7 H-C-P -C-P -C-H canonical-circuit template. The existence of a H-C-P -CZ template is proven in [16] but no algorithms are known for obtaining such 4-block canonical circuits given an arbitrary stabilizer state. We now describe the main steps in Algorithm 2. For simplicity, the updates to the phase array under row and column operations will be left out of our discussion as such updates do not affect the overall execution of the algorithm. 1. Reduce M to canonical form. 2. Use row transposition to diagonalize M. For j ∈ {1, . . . , n}, if the diagonal literal Mj,j = Z and there are other Pauli (non-I) literals in the row (qubit is entangled), conjugate M by Hj . Elements below the diagonal are Z/I literals. 3. For each above-diagonal element Mj,k = X/Y , conjugate by CNOTj,k . Elements above the diagonal are now I/Z literals. 4. For each above-diagonal element Mj,k = Z, conjugate by CPHASEj,k . Elements above the diagonal are now I literals. 5. For each diagonal literal Mj,j = Y , conjugate by Pj . 6. For each diagonal literal Mj,j = X, conjugate by Hj . 7. Use row multiplication to eliminate trailing Z literals below the diagonal and arrive at basis form. Proposition 11. For an n × n stabilizer matrix M, the number of gates in the circuit C returned by Algorithm 2 is O(n2 ). Proof: The number of gates in C is dominated by the CNOT and CPHASE blocks, which have O(n2 ) gates each. This agrees with previous results regarding the number of gates needed for an n-qubit stabilizer circuit in the worst case [4], [5]. Observe that, for each gate added to C, the corresponding column operation is applied to M. Therefore, since column operations run in Θ(n) time, it follows from Proposition 11 that the runtime of Algorithm 2 is O(n3 ). Canonical stabilizer circuits that follow the 7-block template structure from [1] can be optimized to obtain a tighter bound on the number of gates. As in our approach, such circuits are dominated by the size of the CNOT blocks, which contain O(n2 ) gates. The work in [15] shows that that any CNOT circuit has an equivalent CNOT circuit with only O(n2 / log n) gates. Thus, one simply applies such techniques to each of the CNOT blocks in the canonical circuit. It is an open problem whether one can apply the techniques from [15] directly to CPHASE blocks, which would facilitate similar optimizations to our proposed 5-block canonical form. Inner-product algorithm. Let |ψi and |φi be two stabilizer states represented by stabilizer matrices Mψ and Mφ , respectively. Our approach for computing the inner product between these two states is shown in Algorithm 3. Following the proof of Theorem 8, Algorithm 2 is applied to Mψ in order to reduce it to basis form. The stabilizer circuit generated by Algorithm 2 is then applied to Mφ in order to preserve the inner product. Then, we minimize the number of X and Y literals in Mφ by applying Algorithm 1. Finally, each generator in Mφ that anticommutes with Mψ √ (since Mψ is in basis form, we only need to check φ which generators in M have X or Y literals) contributes a factor of 1/ 2 to the inner product. If a generator in Mφ , say Qi , commutes with Mψ , then we check orthogonality by determining whether Qi is in the stabilizer group generated by Mψ . This is accomplished by multiplying the appropriate generators in Mψ such that we create Pauli operator R, which has the same literals as Qi , and check whether R has an opposite sign to Qi . If this is the case, then, by Theorem 7, the states are orthogonal. Clearly, the most time-consuming step of Algorithm 3 is the call to Algorithm 2, therefore, the overall runtime is O(n3 ). However, as we show in Section IV, the performance of our algorithm depends strongly on the stabilizer matrices considered and exhibits quadratic behaviour for certain stabilizer states. IV. E MPIRICAL VALIDATION We implemented our algorithms in C++ and designed a benchmark set to validate the performance of our inner-product algorithm. Recall that the runtime of Algorithm 2 is dominated by the two nested for-loops (lines 20-35). The number of times 7 Theorem 8 in [1] actually describes an 11-step canonical procedure. However, the last four steps pertain to reducing destabilizer rows, which we do not consider in our approach.

7

Algorithm 2 Synthesis of basis normalization circuit Input: Stabilizer matrix M for S(|ψi) with rows R1 , . . . , Rn Output: (i) Unitary stabilizer circuit C such that C |ψi equals basis state |bi, and (ii) reduce M to basis form ⇒ GAUSS(M) reduces M to canonical form (Figure 2) ⇒ ROWSWAP(M, i, j) swaps rows Ri and Rj of M ⇒ ROWMULT(M, i, j) left-multiplies rows Ri and Rj , returns updated Ri ⇒ CONJ(M, αj ) conjugates j th column of M by Clifford sequence α 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: 18: 19: 20: 21: 22: 23: 24: 25: 26: 27: 28: 29: 30: 31: 32: 33: 34: 35: 36: 37: 38: 39: 40: 41: 42: 43: 44: 45: 46: 47: 48: 49: 50: 51: 52: 53: 54: 55:

GAUSS(M) . Set M to canonical form C←∅ i←1 for j ∈ {1, . . . , n} do . Apply block of Hadamard gates k ← index of row Rk∈{i,...,n} with j th literal set to X or Y if k exists then ROWSWAP(M, i, k) else k2 ← index of last row Rk2 ∈{i,...,n} with j th literal set to Z if k2 exists then ROWSWAP(M, i, k2 ) if Ri has X, Y or Z literals in columns {j + 1, . . . , n} then CONJ(M, Hj ) C ← C ∪ Hj end if end if end if i←i+1 end for for j ∈ {1, . . . , n} do . Apply block of CNOT gates for k ∈ {j + 1, . . . , n} do if kth literal of row Rj is set to X or Y then CONJ(M, CNOTj,k ) C ← C ∪ CNOTj,k end if end for end for for j ∈ {1, . . . , n} do . Apply a block of Controlled-Z gates (Figure 3b) for k ∈ {j + 1, . . . , n} do if kth literal of row Rj is set to Z then CONJ(M, CPHASEj,k ) C ← C ∪ CPHASEj,k end if end for end for for j ∈ {1, . . . , n} do . Apply block of Phase gates if j th literal of row Rj is set to Y then CONJ(M, Pj ) C ← C ∪ Pj end if end for for j ∈ {1, . . . , n} do . Apply block of Hadamard gates if j th literal of row Rj is set to X then CONJ(M, Hj ) C ← C ∪ Hj end if end for for j ∈ {1, . . . , n} do . Eliminate trailing Z literals to ensure basis form (Definition 4) for k ∈ {j + 1, . . . , n} do if j th literal of row Rk is set to Z then Rk = ROWMULT(M, Rj , Rk ) end if end for end for return C

8

Algorithm 3 Inner product for stabilizer states Input: Stabilizer matrices (i) Mψ for |ψi with rows P1 , . . . , Pn , and (ii) Mφ for |φi with rows Q1 , . . . , Qn Output: Inner product between |ψi and |φi ⇒ BASISNORMCIRC(M) reduces M to basis form, i.e, C |ψi = |bi, where |bi is a basis state, and returns C ⇒ CONJ(M, C) conjugates M by Clifford circuit C ⇒ GAUSS(M) reduces M to canonical form (Figure 2) ⇒ LEFTMULT(P, Q) left-multiplies Pauli operators P and Q, and returns the updated Q 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: 18:

C ← BASISNORMCIRC(Mψ ) CONJ(Mφ , C) GAUSS(Mφ ) k←0 for each row Qi ∈ Mφ do if Qi contains X or Y literals then k ←k+1 else R ← I ⊗n for each Z literal in Qi found at position j do R ← LEFTMULT(Pj , R) end for if R = −Qi then return 0 end if end if end for return 2−k/2

. Apply Algorithm 2 to Mψ . Compute C |φi . Set Mφ to canonical form

. Check orthogonality, i.e., Qi ∈ / S(|bi).

. By Theorem 7

. By Theorem 8

these loops execute depends on the amount of entanglement in the input stabilizer state. In turn, the number of entangled qubits depends on the the number of CNOT gates in the circuit C used to generate the stabilizer state C |0⊗n i (Theorem 5). By a simple heuristic argument [1], one generates highly entangled stabilizer states as long as the number of CNOT gates in C is proportional to n lg n. Therefore, we generated random n-qubit stabilizer circuits for n ∈ {20, 40, . . . , 500} as follows: fix a parameter β > 0; then choose βdn log2 ne unitary gates (CNOT, Phase or Hadamard) each with probability 1/3. Then, each random C is applied to the |00 . . . 0i basis state to generate random stabilizer matrices (states). The use of randomly generated benchmarks is justified for our experiments because (i) our algorithms are not explicitly sensitive to circuit topology and (ii) random stabilizer circuits are considered representative [13]. For each n, we applied Algorithm 3 to pairs of random stabilizer matrices and measured the number of seconds needed to compute the inner product. The entire procedure was repeated for increasing degrees of entanglement by ranging β from 0.6 to 1.2 in increments of 0.1. Our results are shown in Figure 4-a. The runtime of Algorithm 3 appears to grow quadratically in n when β = 0.6. However, when the number of unitary gates is doubled (β = 1.2), the runtime exhibits cubic growth. Therefore, Figure 4-a shows that the performance of Algorithm 3 is highly dependent on the degree of entanglement in the input stabilizer states. Figure 4-b shows the average size of the basis-normalization circuit returned by the calls to Algorithm 2. As expected (Proposition 11), the size of the circuit grows

90K

70 60 50 40

β = .6 β = .7 β = .8 β = .9 β= 1.0 β= 1.1 β = 1.2

80K Average circuit size (number of gates)

Average runtime (secs)

80

30 20 10

70K 60K 50K 40K

β = .6 β = .7 β = .8 β = .9 β= 1.0 β= 1.1 β = 1.2

30K 20K 10K

0 50 100 150 200 250 300 350 400 450 500

50 100 150 200 250 300 350 400 450 500

Number of qubits (a)

(b)

Fig. 4. Average runtime for Algorithm 3 to compute the inner product between two random n-qubit stabilizer states.  The stabilizer matrices that represent the input states are generated by applying βn log2 n unitary gates to 0⊗n .

9

2.0

150

100

β = .6 β = .7 β = .8 β = .9 β= 1.0 β= 1.1 β = 1.2

〈00...0|φ〉

Average runtime (secs)

Average runtime (millisecs)

200

50

β = .6 β = .7 β = .8 β = .9 β= 1.0 β= 1.1 β = 1.2 |φ=0...0〉

1.5

1.0

〈GHZ|φ〉

0.5

0

0

50 100 150 200 250 300 350 400 450 500

50 100 150 200 250 300 350 400 450 500

Number of qubits (a)

(b)

 Fig. 5. Average runtime for Algorithm 3 to compute the inner product between (a) 0⊗n and random stabilizer state |φi and (b) the n-qubit GHZ state and random stabilizer state |φi.

quadratically in n. Figure 5 shows the average runtime for Algorithm 3 to compute the inner product between: (i) the allzeros basis state and random n-qubit stabilizer states, and (ii) the n-qubit GHZ state8 and random stabilizer states. GHZ states are maximally entangled states that have been realized experimentally using several quantum technologies and are often encountered in practical applications such as error-correcting codes and fault-tolerant architectures. Figure 5 shows that, for such practical instances, Algorithm 3 can compute the inner product in roughly O(n2 ) time (e.g. hGHZ|0i). However, without apriori information about the input stabilizer matrices, one can only say that the performance of Algorithm 3 will be somewhere between quadratic and cubic in n. V. N EAREST- NEIGHBOR STABILIZER STATES We used Algorithm 3 to compute the inner product between |00i and all two-qubit stabilizer states. Our results are shown in Table II. We leveraged these results to formulate the following properties related to the geometry of stabilizer states. Definition 12. Given an arbitrary state |ψi with ||ψ|| = 1, a stabilizer state |ϕi is a nearest stabilizer state to |ψi if |hψ|ϕi| attains the largest possible value 6= 1. Proposition 13. Consider two orthogonal stabilizer states |αi and |βi whose unbiased superposition |ψi is also a stabilizer state. Then |ψi is a nearest stabilizer state to |αi and |βi. Proof: Since stabilizer states are unbiased, |hψ|αi| = |hψ|βi| = Thus, |ψi is a nearest stabilizer state to |αi and |βi.

√1 . 2

By Theorem 8, this is the largest possible value 6= 1.

Lemma 14. For any two stabilizer states, the numbers of nearest-neighbor stabilizer states are equal. Proof: By Corollary 6, any stabilizer state can be mapped to another stabilizer state by a stabilizer circuit. Since the operators effected by these circuits are unitary, inner products are preserved. Lemma 15. Let |ψi and |ϕi be orthogonal stabilizer states such that |ϕi = P |ψi where P is an element of the Pauli group. √ Then |ψi+|ϕi is a stabilizer state. 2 Proof: Suppose |ψi = hgk ik=1,2,...,n is generated by elements gk of the n-qubit Pauli group. Let  0 if [P, gk ] = 0 f (k) = 1 otherwise and write |ϕi = h(−1)f (k) gk i. Conjugating each generator gk by P we see that |ϕi is stabilized by h(−1)f (k) gk i. Let Zk (respectively Xk ) denote the Pauli operator Z (X) acting on the k th qubit. By Corollary 6, there exists an element L of the ⊗n n-qubit Clifford group such that L|ψi = |0i and L|ϕi = (LP L† )L|ψi = it |f (1)f (2) . . . f (n)i. The second equality follows from the fact that LP L† is an element of the Pauli group and can therefore be written as it X(v)Z(u) for some t ∈ {0, 1, 2, 3} and u, v ∈ Zk2 . Therefore, ⊗n |ψi + |ϕi L† (|0i + it |f (1)f (2) . . . f (n)i) √ √ = 2 2 8 An

n-qubit GHZ state is an equal superposition of the all-zeros and all-ones states, i.e.,

10

|0⊗n i√+|1⊗n i 2

.

The state in parenthesis on the right-hand side is the product of an all-zeros state and a GHZ state. Therefore, the sum is stabilized by S 0 = L† hSzero , Sghz iL where Szero = hZi , i ∈ {k|f (k) = 0}i and Sghz is supported on {k|f (k) = 1} and equals h(−1)t/2 XX . . . X, ∀i Zi Zi+1 i if t = 0 mod 2 or h(−1)(t−1)/2 Y Y . . . Y, ∀i Zi Zi+1 i if t = 1 mod 2. Theorem 16. For any n-qubit stabilizer state |ψi, there are 4(2n − 1) nearest-neighbor stabilizer states, and these states can be produced as described in Lemma 15. √ ⊗n Proof: The all-zeros basis amplitude of any stabilizer state |ψi that is a nearest neighbor to |0i must be ∝ 1/ 2. ⊗n ⊗n ⊗n |0i √ Therefore, |ψi is an unbiased superposition of |0i and one of the other 2n − 1 basis states, i.e., |ψi = |0i +P , where 2 ⊗n

⊗n

⊗n

t

|ϕi , where |ϕi is a basis state P ∈ Gn such that P |0i 6= α |0i . As in the proof of Lemma 15, we have |ψi = |0i √+i 2 n and t ∈ {0, 1, 2, 3}. Thus, there are 4 possible unbiased superpositions, and a total of 4(2 − 1) nearest stabilizer states. Since ⊗n |0i is a stabilizer state, all stabilizer states have the same number of nearest stabilizer states by Lemma 14. Table II shows that |00i has 12 nearest-neighbor states. We computed inner products between all-pairs of 2-qubit stabilizer states and confirmed that each had exactly 12 nearest neighbors. We used the same procedure to verify that all 3-qubit stabilizer states have 28 nearest neighbors. We verified the correctness of our algorithm by comparing against inner product computations based on explicit basis amplitudes.

VI. S TABILIZER FRAMES Given an n-qubit stabilizer state |ψi, there exists an orthonormal basis including |ψi and consisting entirely of stabilizer states. Using Theorem 7, one can generate such a basis from the stabilizer representation of |ψi. Observe that, one can create a state |ϕi that is orthogonal to |ψi by changing the signs of an arbitrary non-empty subset of generators of S(|ψi), i.e., by permuting the phase vector of the stabilizer matrix for |ψi. Moreover, selecting two different subsets will produce two mutually orthogonal states. Thus, one can produce 2n − 1 additional orthogonal stabilizer states. Such states, together with |ψi, form an orthonormal basis. This is illustrated by Table II were each row constitutes an orthonormal basis. Definition 17. A stabilizer frame F is a set of k ≤ 2n stabilizer states that forms an orthonormal basis {|ψ1 i , . . . , |ψk i} and spans a subspace of the n-qubit Hilbert space. F is represented by a pair consisting of (i) a stabilizer matrix M and (ii) a set of k distinct phase vectors σj (M), j ∈ {1, . . . , k}. The size of the frame, which we denote by |F|, is equal to k. Stabilizer frames are useful for representing arbitrary quantum states and for simulating the action of stabilizer circuits on such states. Let α = (α1 , . .P . , αk ) ∈ Ck be the decomposition of the arbitrary n-qubit state |φi onto the basis {|ψ1 i , . . . , |ψk i} k defined by F, i.e., |φi = i=1 αk |ψi i. Furthermore, let U be a stabilizer gate. To simulate U |φi, one simply rotates the basis defined by F to get the new basis {U |ψ1 i , . . . , U |ψk i}. This is accomplished with the following two-step process: (i) update the stabilizer matrix M associated with F as per Section II-A; (ii) iterate over the phase vectors in F and update each accordingly (Table III). The second step is linear in the number of phase vectors as only a constant number of elements in each vector needs to be updated. Also, α may need to be updated, which requires the computation of the global phase of each U |ψi i. Since the stabilizer does not maintain global phases directly, each αi is updated as follows: 1. Use Gaussian elimination to obtain a basis state |bi from M (Observation 3) and store its non-zero amplitude β. If U is the Hadamard gate, it may be necessary to sample a sum of two non-zero (one real, one imaginary) basis amplitudes. 2. Compute U β |bi = β 0 |b0 i directly using the state-vector representation. 3. Obtain |b0 i from U MU † and store its non-zero amplitude γ. 4. Compute the global-phase factor generated as αi = (αi · β 0 )/γ. Observe that, all the above processes take time polynomial in k, therefore, if k = poly(n), U |φi can be simulated efficiently on a classical computer via frame-based simulation. Inner product between frames. We now discuss how to use our algorithms to compute the inner product between arbitrary quantum states. Let |φi and |ϕi be quantum states represented by the pairs < F φ , α = (α1 , . . . , αk ) > and < F ϕ , β = (β1 , . . . , βl ) >, respectively. The following steps compute |hφ|ϕi|. 1. Apply Algorithm 2 to Mφ (the stabilizer matrix associated with F φ ) to obtain basis-normalization circuit C. 2. Rotate frames F φ and F ϕ by C as outlined in our previous discussion. 3. Reduce Mφ to canonical form (Algorithm 1) and record the row operations applied. Apply the same row operations to each phase vector σiφ , i ∈ {1, . . . , k} in F φ . Repeat this step for Mϕ and the phase vectors in F ϕ . 4. Let Mφi denote that the leading-phases of the rows in Mφ are set equal to σiφ . Similarly, Mϕ j denotes that the phases ϕ φ ϕ ϕ of M are equal to σj . Furthermore, let δ(Mi , Mj ) be the function that returns 0 if the orthogonality check from Algorithm 3 (lines 9–15) returns 0, and 1 otherwise. The inner product is computed as, k l 1 XX ∗ |hφ|ϕi| = s/2 |αi βj | · δ(Miφ , Mjϕ ) 2 i=1 j=1 11

where s is the number of rows in Mϕ that contain X or Y literals. Prior work on representation of arbitrary states using the stabilizer formalism can be found in [1]. The authors propose an approach that represents a quantum state as a sum of density-matrix terms. Our frame-based technique offers more compact storage (|F| ≤ 2n whereas a density matrix may have 4n non-zero entries) but requires more sophisticated book-keeping. VII. C ONCLUSION The stabilizer formalism facilitates compact representation of stabilizer states and efficient simulation of stabilizer circuits. Stabilizer states appear in many different quantum-information applications, and their efficient manipulation via geometric and linear-algebraic operations may lead to additional insights. To this end, we study algorithms to efficiently compute the inner product between stabilizer states. A crucial step of this computation is the synthesis of a canonical circuit that transforms a stabilizer state into a computational basis state. We designed an algorithm to synthesize such circuits using a 5-block template structure and showed that these circuits contain O(n2 ) stabilizer gates. We analysed the performance of our inner-product algorithm and showed that, although its runtime is O(n3 ), there are practical instances in which it runs in linear or quadratic time. Furthermore, we proved that an n-qubit stabilizer state has exactly 4(2n − 1) nearest-neighbor states and verified this result experimentally. Finally, we designed techniques for representing arbitrary quantum states using stabilizer frames and generalize our algorithms to compute the inner product between two such frames. R EFERENCES [1] S. Aaronson and D. Gottesman, “Improved simulation of stabilizer circuits,” Phys. Rev. A, vol. 70, no. 052328 (2004). [2] K. M. R. Audenaert and M. B. Plenio, “Entanglement on mixed stabiliser states: normal norms and reduction procedures,” New J. Phys., vol. 7, no. 170 (2005). [3] A. Calderbank, E. Rains, P. Shor, and N. Sloane, “Quantum error correction via codes over GF(4),” IEEE Trans. Inf. Theory, vol. 44, pp. 1369–1387 (1998). [4] R. Cleve and D. Gottesman, “Efficient computations of encodings for quantum error correction,” Phys. Rev. A, vol. 56, no. 1 (1997). [5] J. Dehaene and B. De Moor, “Clifford group, stabilizer states, and linear and quadratic operations over GF(2),” Phys. Rev. A, vol. 68, no. 042318 (2003). [6] I. Djordjevic, “Quantum information processing and quantum error correction: an engineering approach, ” Academic press (2012). [7] W. Dur, H. Aschauer and H.J. Briegel, “Multiparticle entanglement purification for graph states,” Phys. Rev. Lett., vol. 91, no. 10 (2003). [8] D. Fattal, T. S. Cubitto, Y. Yamamoto, S. Bravyi and I. L. Chuang, “Entanglement in the stabilizer formalism,” arXiv:0406168 (2004). [9] D. Gottesman, “Stabilizer codes and quantum error correction,” Caltech Ph.D. thesis (1997). [10] D. Gottesman, “The Heisenberg representation of quantum computers,” arXiv:9807006v1 (1998). [11] O. Guehne, G. Toth, P. Hyllus and H.J. Briegel,“Bell inequalities for graph states,” Phys. Rev. Lett., vol. 95, no. 120405 (2005). [12] M. Hein, J. Eisert, and H.J. Briegel,“Multi-party entanglement in graph states,” Phys. Rev. A, vol. 69, no. 6 (2004). [13] E. Knill, et. al., “Randomized benchmarking of quantum gates,” Phys. Rev. A, vol. 77, no. 1 (2007). [14] M. A. Nielsen and I. L. Chuang, “Quantum Computation and Quantum Information,” Cambridge University Press (2000). [15] K. N. Patel, I. L. Markov and J. P. Hayes, “Optimal synthesis of linear reversible circuits,” Quant. Inf. Comp., vol. 8, no. 3 (2008). [16] M. Van den Nest, “Classical simulation of quantum computation, the Gottesman-Knill theorem, and slightly beyond,” Quant. Inf. Comp., vol. 10, pp. 0258–0271 (2010). [17] M. Van den Nest, J. Dehaene and B. De Moor, “On local unitary versus local Clifford equivalence of stabilizer states,” Phys. Rev. A, vol. 71, no. 062323 (2005). [18] H. Wunderlich and M. B. Plenio, “Quantitative verification of fidelities and entanglement from incomplete measurement data”, J. Mod. Opt., vol. 56, pp. 2100–2105 (2009).

12

A PPENDIX A T HE 1080 THREE - QUBIT STABILIZER STATES Shorthand notation represents a stabilizer state as α0 , α1 , α2 , α3 where αi are the normalized amplitudes of the basis states. Basis states are emphasized in bold. The ∠ column indicates the angle between that state and |000i, which has 28 nearestneighbor states and 315 orthogonal states (⊥). S TATE

G EN ’ TORS

11111111 IIX IXI XII 1111iiii IIX IXI YII 11110000 IIX IXI ZII 11ii11ii IIX IYI XII 11iiii-IIX IYI YII 11ii0000 IIX IYI ZII 11001100 IIX IZI XII 1100ii00 IIX IZI YII 11000000 IIX IZI ZII 00111100 IIX XXI YYI 11iiii11 IIX XXI YZI 110000ii IIX XYI YXI 11--iiii IIX XYI YZI 11¯ i¯ i11ii IIX XZI YXI 111111-IIX XZI YYI 1i1i1i1i IIY IXI XII 1i1ii-iIIY IXI YII 1i1i0000 IIY IXI ZII 1ii-1iiIIY IYI XII 1ii-i--¯ i IIY IYI YII 1ii-0000 IIY IYI ZII 1i001i00 IIY IZI XII 1i00i-00 IIY IZI YII 1i000000 IIY IZI ZII 001i1i00 IIY XXI YYI 1ii-i-1i IIY XXI YZI 1i0000iIIY XYI YXI 1i-¯ ii-iIIY XYI YZI 1i¯ i11iiIIY XZI YXI 1i1i1i-¯ i IIY XZI YYI 10101010 IIZ IXI XII 1010i0i0 IIZ IXI YII 10100000 IIZ IXI ZII 10i010i0 IIZ IYI XII 10i0i0-0 IIZ IYI YII 10i00000 IIZ IYI ZII 10001000 IIZ IZI XII 1000i000 IIZ IZI YII 10000000 IIZ IZI ZII 00101000 IIZ XXI YYI 10i0i010 IIZ XXI YZI 100000i0 IIZ XYI YXI 10-0i0i0 IIZ XYI YZI 10¯ i010i0 IIZ XZI YXI 101010-0 IIZ XZI YYI 01011010 IXI XIX YIY 1i1ii1i1 IXI XIX YIZ 10100i0i IXI XIY YIX 1-1-iiii IXI XIY YIZ 1¯ i1¯ i1i1i IXI XIZ YIX 11111-1IXI XIZ YIY 01100110 IXX IYY XII 01100ii0 IXX IYY YII 01100000 IXX IYY ZII 1ii11ii1 IXX IYZ XII 1ii1i--i IXX IYZ YII 1ii10000 IXX IYZ ZII 1¯ i¯ i1¯ i11¯ i IXX XIX YYY 01101001 IXX XIX YYZ 1ii1-ii- IXX XYY YIX 1--1---- IXX XYY YYZ 10010ii0 IXX XYZ YIX 1111¯ iii¯ i IXX XYZ YYY 100i100i IXY IYX XII 100ii00IXY IYX YII 100i0000 IXY IYX ZII 1-ii1-ii IXY IYZ XII 1-iii¯ i-IXY IYZ YII 1-ii0000 IXY IYZ ZII 1-iiii1- IXY XIY YYX 0i10100i IXY XIY YYZ 11¯ ii1-ii IXY XYX YIY 1i1i1¯ i-i IXY XYX YYZ 100i0-i0 IXY XYZ YIY 1¯ i-ii-i- IXY XYZ YYX 1¯ i1i1¯ i1i IXZ IYX XII 1¯ i1ii1iIXZ IYX YII 1¯ i1i0000 IXZ IYX ZII 111-111IXZ IYY XII 111-iii¯ i IXZ IYY YII 111-0000 IXZ IYY ZII 1-11111- IXZ XIZ YYX 1¯ i1i1i1¯ i IXZ XIZ YYY 111-i¯ iii IXZ XYX YIZ 0¯ i0i1010 IXZ XYX YYY 1i1¯ ii1i- IXZ XYY YIZ 1010010- IXZ XYY YYX 010i10i0 IYI XIX YIY 1ii-i1-i IYI XIX YIZ 10i00i0IYI XIY YIX 1-i¯ iii-IYI XIY YIZ 1¯ ii11iiIYI XIZ YIX 11ii1-i¯ i IYI XIZ YIY 1i-ii1i- IYX XIX YXY 100i01i0 IYX XIX YXZ 1¯ i1i1i-i IYX XXY YIX 1-¯ ii11ii IYX XXY YXZ 0¯ i10100i IYX XXZ YIX 11iii¯ i1- IYX XXZ YXY 111-¯ iiii IYY XIY YXX 100-0ii0 IYY XIY YXZ 1------1 IYY XXX YIY 1ii--ii1 IYY XXX YXZ 0110100- IYY XXZ YIY 1¯ i¯ i-¯ i11i IYY XXZ YXX 11i¯ i1-ii IYZ XIZ YXX 1ii11¯ ii- IYZ XIZ YXY 1-iiii-1 IYZ XXX YIZ 0i0110i0 IYZ XXX YXY 1¯ ii-i--i IYZ XXY YIZ 10i00-0i IYZ XXY YXX 01001000 IZI XIX YIY 1i00i100 IZI XIX YIZ 10000i00 IZI XIY YIX 1-00ii00 IZI XIY YIZ 1¯ i001i00 IZI XIZ YIX 11001-00 IZI XIZ YIY 11-1111- IZX XIX YXY 11¯ ii11i¯ i IZX XIX YXZ 111-ii¯ ii IZX XXY YIX 00¯ ii1100 IZX XXY YXZ 11i¯ iii1- IZX XXZ YIX 1100001- IZX XXZ YXY 1i1¯ i1i-i IZY XIY YXX 1i¯ i-1ii1 IZY XIY YXZ 1i-ii-i1 IZY XXX YIY 1i0000i1 IZY XXX YXZ 1ii1i-1¯ i IZY XXZ YIY 001¯ i1i00 IZY XXZ YXX 100¯ i100i IZZ XIZ YXX 1001100- IZZ XIZ YXY 100ii001 IZZ XXX YIZ 00011000 IZZ XXX YXY 100-i00i IZZ XXY YIZ 1000000i IZZ XXY YXX

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1111---IIX IXI -XII 1111¯ i¯ i¯ i¯ i IIX IXI -YII 00001111 IIX IXI -ZII 11ii--¯ i¯ i IIX IYI -XII 11ii¯ i¯ i11 IIX IYI -YII 000011ii IIX IYI -ZII 1100--00 IIX IZI -XII 1100¯ i¯ i00 IIX IZI -YII 00001100 IIX IZI -ZII 11--11-IIX XII -IXI 11¯ i¯ i11¯ i¯ i IIX XII -IYI 00110011 IIX XII -IZI 11000011 IIX XXI -YYI 11¯ i¯ i¯ i¯ i11 IIX XXI -YZI 00ii1100 IIX XYI -YXI 1111¯ i¯ iii IIX XYI -YZI 11ii11¯ i¯ i IIX XZI -YXI 11--1111 IIX XZI -YYI 11--ii¯ i¯ i IIX YII -IXI 11¯ i¯ iii11 IIX YII -IYI 001100ii IIX YII -IZI 00¯ i¯ i1100 IIX YXI -XYI 11ii--ii IIX YXI -XZI 110000-- IIX YYI -XXI 11-----IIX YYI -XZI 11¯ i¯ iii-IIX YZI -XXI 1111ii¯ i¯ i IIX YZI -XYI 11--0000 IIX ZII -IXI 11¯ i¯ i0000 IIX ZII -IYI 00110000 IIX ZII -IZI 1i1i-¯ i-¯ i IIY IXI -XII 1i1i¯ i1¯ i1 IIY IXI -YII 00001i1i IIY IXI -ZII 1ii--¯ i¯ i1 IIY IYI -XII 1ii-¯ i11i IIY IYI -YII 00001iiIIY IYI -ZII 1i00-¯ i00 IIY IZI -XII 1i00¯ i100 IIY IZI -YII 00001i00 IIY IZI -ZII 1i-¯ i1i-¯ i IIY XII -IXI 1i¯ i11i¯ i1 IIY XII -IYI 001i001i IIY XII -IZI 1i00001i IIY XXI -YYI 1i¯ i1¯ i11i IIY XXI -YZI 00i-1i00 IIY XYI -YXI 1i1i¯ i1iIIY XYI -YZI 1ii-1i¯ i1 IIY XZI -YXI 1i-¯ i1i1i IIY XZI -YYI 1i-¯ ii-¯ i1 IIY YII -IXI 1i¯ i1i-1i IIY YII -IYI 001i00iIIY YII -IZI 00¯ i11i00 IIY YXI -XYI 1ii--¯ iiIIY YXI -XZI 1i0000-¯ i IIY YYI -XXI 1i-¯ i-¯ i-¯ i IIY YYI -XZI 1i¯ i1i--¯ i IIY YZI -XXI 1i1ii-¯ i1 IIY YZI -XYI 1i-¯ i0000 IIY ZII -IXI 1i¯ i10000 IIY ZII -IYI 001i0000 IIY ZII -IZI 1010-0-0 IIZ IXI -XII 1010¯ i0¯ i0 IIZ IXI -YII 00001010 IIZ IXI -ZII 10i0-0¯ i0 IIZ IYI -XII 10i0¯ i010 IIZ IYI -YII 000010i0 IIZ IYI -ZII 1000-000 IIZ IZI -XII 1000¯ i000 IIZ IZI -YII 00001000 IIZ IZI -ZII 10-010-0 IIZ XII -IXI 10¯ i010¯ i0 IIZ XII -IYI 00100010 IIZ XII -IZI 10000010 IIZ XXI -YYI 10¯ i0¯ i010 IIZ XXI -YZI 00i01000 IIZ XYI -YXI 1010¯ i0i0 IIZ XYI -YZI 10i010¯ i0 IIZ XZI -YXI 10-01010 IIZ XZI -YYI 10-0i0¯ i0 IIZ YII -IXI 10¯ i0i010 IIZ YII -IYI 001000i0 IIZ YII -IZI 00¯ i01000 IIZ YXI -XYI 10i0-0i0 IIZ YXI -XZI 100000-0 IIZ YYI -XXI 10-0-0-0 IIZ YYI -XZI 10¯ i0i0-0 IIZ YZI -XXI 1010i0¯ i0 IIZ YZI -XYI 10-00000 IIZ ZII -IXI 10¯ i00000 IIZ ZII -IYI 00100000 IIZ ZII -IZI 1-1-1-1IXI XII -IIX 1¯ i1¯ i1¯ i1¯ i IXI XII -IIY 01010101 IXI XII -IIZ 10100101 IXI XIX -YIY 1¯ i1¯ i¯ i1¯ i1 IXI XIX -YIZ 0i0i1010 IXI XIY -YIX 1111¯ ii¯ ii IXI XIY -YIZ 1i1i1¯ i1¯ i IXI XIZ -YIX 1-1-1111 IXI XIZ -YIY 1-1-i¯ ii¯ i IXI YII -IIX 1¯ i1¯ ii1i1 IXI YII -IIY 01010i0i IXI YII -IIZ 0¯ i0¯ i1010 IXI YIX -XIY 1i1i-i-i IXI YIX -XIZ 10100-0- IXI YIY -XIX 1-1----IXI YIY -XIZ 1¯ i1¯ ii-iIXI YIZ -XIX 1111i¯ ii¯ i IXI YIZ -XIY 1-1-0000 IXI ZII -IIX 1¯ i1¯ i0000 IXI ZII -IIY 01010000 IXI ZII -IIZ 01100--0 IXX IYY -XII 01100¯ i¯ i0 IXX IYY -YII 00000110 IXX IYY -ZII 1ii1-¯ i¯ iIXX IYZ -XII 1ii1¯ i11¯ i IXX IYZ -YII 00001ii1 IXX IYZ -ZII 10011001 IXX XII -IYY 1¯ i¯ i11¯ i¯ i1 IXX XII -IYZ 1ii1i11i IXX XIX -YYY 10010110 IXX XIX -YYZ 1¯ i¯ i1-¯ i¯ i- IXX XYY -YIX 1111-11- IXX XYY -YYZ 0ii01001 IXX XYZ -YIX 1--1iiii IXX XYZ -YYY 1001i00i IXX YII -IYY 1¯ i¯ i1i11i IXX YII -IYZ 1¯ i¯ i11ii1 IXX YIX -XYY 0¯ i¯ i01001 IXX YIX -XYZ 1ii1¯ i--¯ i IXX YYY -XIX 1--1¯ i¯ i¯ i¯ i IXX YYY -XYZ 10010--0 IXX YYZ -XIX 11111--1 IXX YYZ -XYY 10010000 IXX ZII -IYY 1¯ i¯ i10000 IXX ZII -IYZ

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100i-00¯ i IXY IYX -XII 100i¯ i001 IXY IYX -YII 0000100i IXY IYX -ZII 1-ii-1¯ i¯ i IXY IYZ -XII 1-ii¯ ii11 IXY IYZ -YII 00001-ii IXY IYZ -ZII 0i100i10 IXY XII -IYX 11¯ ii11¯ ii IXY XII -IYZ 11¯ ii¯ ii11 IXY XIY -YYX 100i0i10 IXY XIY -YYZ 1-ii11¯ ii IXY XYX -YIY 1¯ i-i1i1i IXY XYX -YYZ 0-i0100i IXY XYZ -YIY 1i1i¯ i-i1 IXY XYZ -YYX 0i100-i0 IXY YII -IYX 11¯ iiii1- IXY YII -IYZ 1-ii--i¯ i IXY YIY -XYX 01¯ i0100i IXY YIY -XYZ 11¯ iii¯ i-- IXY YYX -XIY 1i1ii1¯ i- IXY YYX -XYZ 100i0¯ i-0 IXY YYZ -XIY 1¯ i-i-¯ i-¯ i IXY YYZ -XYX 0i100000 IXY ZII -IYX 11¯ ii0000 IXY ZII -IYZ 1¯ i1i-i-¯ i IXZ IYX -XII 1¯ i1i¯ i-¯ i1 IXZ IYX -YII 00001¯ i1i IXZ IYX -ZII 111----1 IXZ IYY -XII 111-¯ i¯ i¯ ii IXZ IYY -YII 0000111- IXZ IYY -ZII 1i1¯ i1i1¯ i IXZ XII -IYX 1-111-11 IXZ XII -IYY 111-1-11 IXZ XIZ -YYX 1i1¯ i1¯ i1i IXZ XIZ -YYY 1-11¯ i¯ i¯ ii IXZ XYX -YIZ 10100¯ i0i IXZ XYX -YYY 1¯ i1i¯ i1¯ i- IXZ XYY -YIZ 010-1010 IXZ XYY -YYX 1i1¯ ii-i1 IXZ YII -IYX 1-11i¯ iii IXZ YII -IYY 1-11iii¯ i IXZ YIZ -XYX 1¯ i1ii-i1 IXZ YIZ -XYY 111--1-- IXZ YYX -XIZ 0-011010 IXZ YYX -XYY 1i1¯ i-i-¯ i IXZ YYY -XIZ 10100i0¯ i IXZ YYY -XYX 1i1¯ i0000 IXZ ZII -IYX 1-110000 IXZ ZII -IYY 1-i¯ i1-i¯ i IYI XII -IIX 1¯ ii11¯ ii1 IYI XII -IIY 010i010i IYI XII -IIZ 10i0010i IYI XIX -YIY 1¯ ii1¯ i11i IYI XIX -YIZ 0i0-10i0 IYI XIY -YIX 11ii¯ ii1- IYI XIY -YIZ 1ii-1¯ ii1 IYI XIZ -YIX 1-i¯ i11ii IYI XIZ -YIY 1-i¯ ii¯ i-1 IYI YII -IIX 1¯ ii1i1-i IYI YII -IIY 010i0i0IYI YII -IIZ 0¯ i0110i0 IYI YIX -XIY 1ii--i¯ iIYI YIX -XIZ 10i00-0¯ i IYI YIY -XIX 1-i¯ i--¯ i¯ i IYI YIY -XIZ 1¯ ii1i--¯ i IYI YIZ -XIX 11iii¯ i-1 IYI YIZ -XIY 1-i¯ i0000 IYI ZII -IIX 1¯ ii10000 IYI ZII -IIY 010i0000 IYI ZII -IIZ 0¯ i100¯ i10 IYX XII -IXY 1i-i1i-i IYX XII -IXZ 1¯ i1i¯ i1i1 IYX XIX -YXY 01i0100i IYX XIX -YXZ 1i-i1¯ i1i IYX XXY -YIX 11ii1-¯ ii IYX XXY -YXZ 100i0¯ i10 IYX XXZ -YIX 1-¯ ii¯ i¯ i11 IYX XXZ -YXY 0¯ i1001i0 IYX YII -IXY 1i-ii-¯ iIYX YII -IXZ 1i-i-i-¯ i IYX YIX -XXY 100i0i-0 IYX YIX -XXZ 1¯ i1ii-¯ i- IYX YXY -XIX 1-¯ iiii-- IYX YXY -XXZ 0-¯ i0100i IYX YXZ -XIX 11ii-1i¯ i IYX YXZ -XXY 0¯ i100000 IYX ZII -IXY 1i-i0000 IYX ZII -IXZ 100-100- IYY XII -IXX 1---1--IYY XII -IXZ 1---iii¯ i IYY XIY -YXX 0ii0100- IYY XIY -YXZ 111--111 IYY XXX -YIY 1¯ i¯ i--¯ i¯ i1 IYY XXX -YXZ 100-0110 IYY XXZ -YIY 1ii-i11¯ i IYY XXZ -YXX 100-i00¯ i IYY YII -IXX 1---i¯ i¯ i¯ i IYY YII -IXZ 111-1--- IYY YIY -XXX 100-0--0 IYY YIY -XXZ 1---¯ i¯ i¯ ii IYY YXX -XIY 1ii-¯ i--i IYY YXX -XXZ 0¯ i¯ i0100- IYY YXZ -XIY 1¯ i¯ i-1ii- IYY YXZ -XXX 100-0000 IYY ZII -IXX 1---0000 IYY ZII -IXZ 1¯ ii-1¯ iiIYZ XII -IXX 11i¯ i11i¯ i IYZ XII -IXY 1-ii11i¯ i IYZ XIZ -YXX 1¯ ii-1ii1 IYZ XIZ -YXY 11i¯ i¯ ii11 IYZ XXX -YIZ 10i00i01 IYZ XXX -YXY 1ii1¯ i-1i IYZ XXY -YIZ 0-0i10i0 IYZ XXY -YXX 1¯ ii-i1-¯ i IYZ YII -IXX 11i¯ iii-1 IYZ YII -IXY 11i¯ ii¯ i-- IYZ YIZ -XXX 1ii1i1-¯ i IYZ YIZ -XXY 1-ii--¯ ii IYZ YXX -XIZ 010¯ i10i0 IYZ YXX -XXY 1¯ ii--¯ i¯ i- IYZ YXY -XIZ 10i00¯ i0- IYZ YXY -XXX 1¯ ii-0000 IYZ ZII -IXX 11i¯ i0000 IYZ ZII -IXY 1-001-00 IZI XII -IIX 1¯ i001¯ i00 IZI XII -IIY 01000100 IZI XII -IIZ 10000100 IZI XIX -YIY 1¯ i00¯ i100 IZI XIX -YIZ 0i001000 IZI XIY -YIX 1100¯ ii00 IZI XIY -YIZ 1i001¯ i00 IZI XIZ -YIX 1-001100 IZI XIZ -YIY 1-00i¯ i00 IZI YII -IIX 1¯ i00i100 IZI YII -IIY 01000i00 IZI YII -IIZ

S TATE G EN ’ TORS 0¯ i001000 IZI YIX -XIY 1i00-i00 IZI YIX -XIZ 10000-00 IZI YIY -XIX 1-00--00 IZI YIY -XIZ 1¯ i00i-00 IZI YIZ -XIX 1100i¯ i00 IZI YIZ -XIY 1-000000 IZI ZII -IIX 1¯ i000000 IZI ZII -IIY 01000000 IZI ZII -IIZ 111-11-1 IZX XIX -YXY 11i¯ i11¯ ii IZX XIX -YXZ 11-1¯ i¯ i¯ ii IZX XXY -YIX 110000¯ ii IZX XXY -YXZ 11¯ ii¯ i¯ i1- IZX XXZ -YIX 001-1100 IZX XXZ -YXY 11-1iii¯ i IZX YIX -XXY 11¯ iiii-1 IZX YIX -XXZ 111---1- IZX YXY -XIX 00-11100 IZX YXY -XXZ 11i¯ i--i¯ i IZX YXZ -XIX 110000i¯ i IZX YXZ -XXY 1i-i1i1¯ i IZY XIY -YXX 1ii11i¯ i- IZY XIY -YXZ 1i1¯ i¯ i1i1 IZY XXX -YIY 00i11i00 IZY XXX -YXZ 1i¯ i-¯ i11¯ i IZY XXZ -YIY 1i00001¯ i IZY XXZ -YXX 1i1¯ ii-¯ i- IZY YIY -XXX 1i¯ i-i--i IZY YIY -XXZ 1i-i-¯ i-i IZY YXX -XIY 1i0000-i IZY YXX -XXZ 1ii1-¯ ii1 IZY YXZ -XIY 00¯ i-1i00 IZY YXZ -XXX 100i100¯ i IZZ XIZ -YXX 100-1001 IZZ XIZ -YXY 100¯ i¯ i001 IZZ XXX -YIZ 10000001 IZZ XXX -YXY 1001¯ i00i IZZ XXY -YIZ 000i1000 IZZ XXY -YXX 100¯ ii00- IZZ YIZ -XXX 1001i00¯ i IZZ YIZ -XXY 100i-00i IZZ YXX -XIZ 000¯ i1000 IZZ YXX -XXY 100--00- IZZ YXY -XIZ 1000000- IZZ YXY -XXX 010-10-0 XIX YIY -IXI 010¯ i10¯ i0 XIX YIY -IYI 00010010 XIX YIY -IZI 1i-¯ ii1¯ iXIX YIZ -IXI 1i¯ i1i11¯ i XIX YIZ -IYI 001i00i1 XIX YIZ -IZI 1¯ i-¯ i¯ i1¯ i- XIX YXY -IYX 1----1-XIX YXY -IZX 01¯ i0100¯ i XIX YXZ -IYX 1-ii-1ii XIX YXZ -IZX 1i¯ i-i1-¯ i XIX YYY -IXX 100-01-0 XIX YYZ -IXX 10-00i0¯ i XIY YIX -IXI 10¯ i00i01 XIY YIX -IYI 0010000i XIY YIX -IZI 1--1ii¯ i¯ i XIY YIZ -IXI 1-¯ iiii11 XIY YIZ -IYI 001-00ii XIY YIZ -IZI 1-11ii¯ ii XIY YXX -IYY 1¯ i1i-i1i XIY YXX -IZY 0i¯ i01001 XIY YXZ -IYY 1¯ ii--iiXIY YXZ -IZY 11i¯ i¯ ii-- XIY YYX -IXY 100¯ i0i-0 XIY YYZ -IXY 1¯ i-i1i-¯ i XIZ YIX -IXI 1¯ i¯ i-1i¯ i1 XIZ YIX -IYI 001¯ i001i XIZ YIX -IZI 11--1--1 XIZ YIY -IXI 11¯ i¯ i1-¯ ii XIZ YIY -IYI 0011001- XIZ YIY -IZI 1-¯ i¯ i11¯ ii XIZ YXX -IYZ 0¯ i100i10 XIZ YXX -IZZ 1¯ i¯ i11i¯ i- XIZ YXY -IYZ 01100-10 XIZ YXY -IZZ 11-11--- XIZ YYX -IXZ 1i-i1¯ i-¯ i XIZ YYY -IXZ 001-1-00 XXI YYI -IIX 001¯ i1¯ i00 XXI YYI -IIY 00010100 XXI YYI -IIZ 1-i¯ ii¯ i1XXI YZI -IIX 1¯ ii1i11¯ i XXI YZI -IIY 010i0i01 XXI YZI -IIZ 11-11-11 XXX YIY -IYY 1¯ i-¯ i¯ i-¯ i1 XXX YIY -IZY 11¯ iii¯ i11 XXX YIZ -IYZ 0i1001i0 XXX YIZ -IZZ 0¯ i0110¯ i0 XXX YXY -IYZ 01000010 XXX YXY -IZZ 1i¯ i11¯ ii1 XXX YXZ -IYY 00¯ i11¯ i00 XXX YXZ -IZY 1i1¯ i-i1i XXY YIX -IYX 1-11¯ iiii XXY YIX -IZX 1i¯ i-i11i XXY YIZ -IYZ 0-100ii0 XXY YIZ -IZZ 10¯ i0010i XXY YXX -IYZ 00100i00 XXY YXX -IZZ 1-i¯ i--ii XXY YXZ -IYX 1-0000ii XXY YXZ -IZX 100¯ i0i10 XXZ YIX -IYX 1-¯ i¯ i¯ ii11 XXZ YIX -IZX 10010-10 XXZ YIY -IYY 1¯ i¯ i1¯ i-1i XXZ YIY -IZY 1¯ ii1i-1i XXZ YXX -IYY 1¯ i00001i XXZ YXX -IZY 11¯ i¯ i¯ ii1- XXZ YXY -IYX 00111-00 XXZ YXY -IZX 1-0000i¯ i XYI YXI -IIX 1¯ i0000i1 XYI YXI -IIY 0100000i XYI YXI -IIZ 1--1i¯ ii¯ i XYI YZI -IIX 1¯ i-ii1i1 XYI YZI -IIY 010-0i0i XYI YZI -IIZ 1-¯ i¯ i--¯ ii XYX YIY -IXY 1---ii¯ ii XYX YIZ -IXZ 0i0i10-0 XYX YYY -IXZ 1i-¯ i-i-i XYX YYZ -IXY 1¯ ii-1i¯ i- XYY YIX -IXX 1¯ i-¯ ii-¯ iXYY YIZ -IXZ 10-00-0- XYY YYX -IXZ 1-1-11-- XYY YYZ -IXX 0¯ ii0100- XYZ YIX -IXX 01i0100¯ i XYZ YIY -IXY 1¯ i1¯ i¯ i1i- XYZ YYX -IXY 11--i¯ ii¯ i XYZ YYY -IXX 1-¯ ii1-i¯ i XZI YXI -IIX 1¯ i¯ i-1¯ ii1 XZI YXI -IIY 010¯ i010i XZI YXI -IIZ 1-1-1--1 XZI YYI -IIX 1¯ i1¯ i1¯ i-i XZI YYI -IIY 0101010- XZI YYI -IIZ

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11----11 IIX -IXI -XII 11--¯ i¯ iii IIX -IXI -YII 000011-IIX -IXI -ZII 11¯ i¯ i--ii IIX -IYI -XII 11¯ i¯ i¯ i¯ i-IIX -IYI -YII 000011¯ i¯ i IIX -IYI -ZII 001100-IIX -IZI -XII 001100¯ i¯ i IIX -IZI -YII 00000011 IIX -IZI -ZII 00--1100 IIX -XXI -YYI 11ii¯ i¯ i-IIX -XXI -YZI 110000¯ i¯ i IIX -XYI -YXI 11--¯ i¯ i¯ i¯ i IIX -XYI -YZI 11¯ i¯ i--¯ i¯ i IIX -XZI -YXI 1111--11 IIX -XZI -YYI 1i-¯ i-¯ i1i IIY -IXI -XII 1i-¯ i¯ i1iIIY -IXI -YII 00001i-¯ i IIY -IXI -ZII 1i¯ i1-¯ iiIIY -IYI -XII 1i¯ i1¯ i1-¯ i IIY -IYI -YII 00001i¯ i1 IIY -IYI -ZII 001i00-¯ i IIY -IZI -XII 001i00¯ i1 IIY -IZI -YII 0000001i IIY -IZI -ZII 00-¯ i1i00 IIY -XXI -YYI 1ii-¯ i1-¯ i IIY -XXI -YZI 1i0000¯ i1 IIY -XYI -YXI 1i-¯ i¯ i1¯ i1 IIY -XYI -YZI 1i¯ i1-¯ i¯ i1 IIY -XZI -YXI 1i1i-¯ i1i IIY -XZI -YYI 10-0-010 IIZ -IXI -XII 10-0¯ i0i0 IIZ -IXI -YII 000010-0 IIZ -IXI -ZII 10¯ i0-0i0 IIZ -IYI -XII 10¯ i0¯ i0-0 IIZ -IYI -YII 000010¯ i0 IIZ -IYI -ZII 001000-0 IIZ -IZI -XII 001000¯ i0 IIZ -IZI -YII 00000010 IIZ -IZI -ZII 00-01000 IIZ -XXI -YYI 10i0¯ i0-0 IIZ -XXI -YZI 100000¯ i0 IIZ -XYI -YXI 10-0¯ i0¯ i0 IIZ -XYI -YZI 10¯ i0-0¯ i0 IIZ -XZI -YXI 1010-010 IIZ -XZI -YYI 1-1--1-1 IXI -IIX -XII 1-1-¯ ii¯ ii IXI -IIX -YII 00001-1IXI -IIX -ZII 1¯ i1¯ i-i-i IXI -IIY -XII 1¯ i1¯ i¯ i-¯ iIXI -IIY -YII 00001¯ i1¯ i IXI -IIY -ZII 01010-0IXI -IIZ -XII 01010¯ i0¯ i IXI -IIZ -YII 00000101 IXI -IIZ -ZII 0-0-1010 IXI -XIX -YIY 1i1i¯ i-¯ iIXI -XIX -YIZ 10100¯ i0¯ i IXI -XIY -YIX 1-1-¯ i¯ i¯ i¯ i IXI -XIY -YIZ 1¯ i1¯ i-¯ i-¯ i IXI -XIZ -YIX 1111-1-1 IXI -XIZ -YIY 1001-00- IXX -IYY -XII 1001¯ i00¯ i IXX -IYY -YII 00001001 IXX -IYY -ZII 1¯ i¯ i1-iiIXX -IYZ -XII 1¯ i¯ i1¯ i--¯ i IXX -IYZ -YII 00001¯ i¯ i1 IXX -IYZ -ZII 1¯ i¯ i1i--i IXX -XIX -YYY 0--01001 IXX -XIX -YYZ 1ii11¯ i¯ i1 IXX -XYY -YIX 1--11111 IXX -XYY -YYZ 10010¯ i¯ i0 IXX -XYZ -YIX 1111i¯ i¯ ii IXX -XYZ -YYY 0i100¯ i-0 IXY -IYX -XII 0i1001¯ i0 IXY -IYX -YII 00000i10 IXY -IYX -ZII 11¯ ii--i¯ i IXY -IYZ -XII 11¯ ii¯ i¯ i-1 IXY -IYZ -YII 000011¯ ii IXY -IYZ -ZII 1-ii¯ i¯ i-1 IXY -XIY -YYX 0¯ i-0100i IXY -XIY -YYZ 11¯ ii-1¯ i¯ i IXY -XYX -YIY 1i1i-i1¯ i IXY -XYX -YYZ 100i01¯ i0 IXY -XYZ -YIY 1¯ i-i¯ i1¯ i1 IXY -XYZ -YYX 1i1¯ i-¯ i-i IXZ -IYX -XII 1i1¯ i¯ i1¯ iIXZ -IYX -YII 00001i1¯ i IXZ -IYX -ZII 1-11-1-IXZ -IYY -XII 1-11¯ ii¯ i¯ i IXZ -IYY -YII 00001-11 IXZ -IYY -ZII 1-11---1 IXZ -XIZ -YYX 1¯ i1i-¯ i-i IXZ -XIZ -YYY 111-¯ ii¯ i¯ i IXZ -XYX -YIZ 0i0¯ i1010 IXZ -XYX -YYY 1i1¯ i¯ i-¯ i1 IXZ -XYY -YIZ 10100-01 IXZ -XYY -YYX 1-i¯ i-1¯ ii IYI -IIX -XII 1-i¯ i¯ ii1IYI -IIX -YII 00001-i¯ i IYI -IIX -ZII 1¯ ii1-i¯ iIYI -IIY -XII 1¯ ii1¯ i-1¯ i IYI -IIY -YII 00001¯ ii1 IYI -IIY -ZII 010i0-0¯ i IYI -IIZ -XII 010i0¯ i01 IYI -IIZ -YII 0000010i IYI -IIZ -ZII 0-0¯ i10i0 IYI -XIX -YIY 1ii-¯ i-1¯ i IYI -XIX -YIZ 10i00¯ i01 IYI -XIY -YIX 1-i¯ i¯ i¯ i11 IYI -XIY -YIZ 1¯ ii1-¯ i¯ i1 IYI -XIZ -YIX 11ii-1¯ ii IYI -XIZ -YIY 0¯ i100i-0 IYX -IXY -XII 0¯ i100-¯ i0 IYX -IXY -YII 00000¯ i10 IYX -IXY -ZII 1i-i-¯ i1¯ i IYX -IXZ -XII 1i-i¯ i1i1 IYX -IXZ -YII 00001i-i IYX -IXZ -ZII 1i-i¯ i-¯ i1 IYX -XIX -YXY 100i0-¯ i0 IYX -XIX -YXZ 1¯ i1i-¯ i1¯ i IYX -XXY -YIX 1-¯ ii--¯ i¯ i IYX -XXY -YXZ 0i-0100i IYX -XXZ -YIX 11ii¯ ii-1 IYX -XXZ -YXY 100--001 IYY -IXX -XII 100-¯ i00i IYY -IXX -YII 0000100- IYY -IXX -ZII 1----111 IYY -IXZ -XII 1---¯ iiii IYY -IXZ -YII 00001--IYY -IXZ -ZII 111-i¯ i¯ i¯ i IYY -XIY -YXX 100-0¯ i¯ i0 IYY -XIY -YXZ 1---111- IYY -XXX -YIY 1ii-1¯ i¯ i- IYY -XXX -YXZ 0--0100- IYY -XXZ -YIY 1¯ i¯ i-i--¯ i IYY -XXZ -YXX

13

S TATE

G EN ’ TORS

S TATE

G EN ’ TORS

S TATE

G EN ’ TORS

1¯ ii--i¯ i1 IYZ -IXX -XII 10-00i0i XYX -IXZ -YYY 1--1-11-IIX -IXI -XII 1¯ ii-¯ i-1i IYZ -IXX -YII 1i¯ i-1¯ ii- XYY -IXX -YIX 1--1¯ iii¯ i -IIX -IXI -YII 00001¯ iiIYZ -IXX -ZII 11--1-1- XYY -IXX -YYZ 00001--1 -IIX -IXI -ZII 11i¯ i--¯ ii IYZ -IXY -XII 1i-i¯ i-iXYY -IXZ -YIZ 1-¯ ii-1i¯ i -IIX -IYI -XII 11i¯ i¯ i¯ i1IYZ -IXY -YII 0-0-10-0 XYY -IXZ -YYX 1-¯ ii¯ ii-1 -IIX -IYI -YII 000011i¯ i IYZ -IXY -ZII 100-0¯ ii0 XYZ -IXX -YIX 00001-¯ ii -IIX -IYI -ZII 11i¯ i-1¯ i¯ i IYZ -XIZ -YXX 1-1-¯ i¯ iii XYZ -IXX -YYY 001-00-1 -IIX -IZI -XII 1ii1-i¯ i1 IYZ -XIZ -YXY 100¯ i01i0 XYZ -IXY -YIY 001-00¯ ii -IIX -IZI -YII 1-ii¯ i¯ i1IYZ -XXX -YIZ 1i-¯ ii1i1 XYZ -IXY -YYX 0000001-IIX -IZI -ZII 0¯ i0-10i0 IYZ -XXX -YXY 1-i¯ i1-¯ ii XZI -IIX -YXI 00-11-00 -IIX -XXI -YYI 1¯ ii-¯ i11¯ i IYZ -XXY -YIZ 1--11-1XZI -IIX -YYI 1-i¯ i¯ ii-1 -IIX -XXI -YZI 10i0010¯ i IYZ -XXY -YXX 1¯ ii11¯ i¯ iXZI -IIY -YXI 1-0000¯ ii -IIX -XYI -YXI 1-00-100 IZI -IIX -XII 1¯ i-i1¯ i1¯ i XZI -IIY -YYI 1--1¯ ii¯ ii -IIX -XYI -YZI 1-00¯ ii00 IZI -IIX -YII 010i010¯ i XZI -IIZ -YXI 1-¯ ii-1¯ ii -IIX -XZI -YXI 00001-00 IZI -IIX -ZII 010-0101 XZI -IIZ -YYI 1-1--11-IIX -XZI -YYI 1¯ i00-i00 IZI -IIY -XII 1--1i¯ i¯ ii YII -IIX -IXI 1¯ i-i-i1¯ i -IIY -IXI -XII 1¯ i00¯ i-00 IZI -IIY -YII 1-¯ iii¯ i1YII -IIX -IYI 1¯ i-i¯ i-i1 -IIY -IXI -YII 00001¯ i00 IZI -IIY -ZII 001-00i¯ i YII -IIX -IZI 00001¯ i-i -IIY -IXI -ZII 01000-00 IZI -IIZ -XII 1¯ i-ii1¯ iYII -IIY -IXI 1¯ i¯ i--ii1 -IIY -IYI -XII 01000¯ i00 IZI -IIZ -YII 1¯ i¯ i-i11¯ i YII -IIY -IYI 1¯ i¯ i-¯ i--i -IIY -IYI -YII 00000100 IZI -IIZ -ZII 001¯ i00i1 YII -IIY -IZI 00001¯ i¯ i-IIY -IYI -ZII 0-001000 IZI -XIX -YIY 010-0i0¯ i YII -IIZ -IXI 001¯ i00-i -IIY -IZI -XII 1i00¯ i-00 IZI -XIX -YIZ 010¯ i0i01 YII -IIZ -IYI 001¯ i00¯ i-IIY -IZI -YII 10000¯ i00 IZI -XIY -YIX 0001000i YII -IIZ -IZI 0000001¯ i -IIY -IZI -ZII 1-00¯ i¯ i00 IZI -XIY -YIZ 0-100¯ ii0 YII -IXX -IYY 00-i1¯ i00 -IIY -XXI -YYI 1¯ i00-¯ i00 IZI -XIZ -YIX 1i¯ i-i-1¯ i YII -IXX -IYZ 1¯ ii1¯ i--i -IIY -XXI -YZI 1100-100 IZI -XIZ -YIY 100¯ ii001 YII -IXY -IYX 1¯ i0000¯ i-IIY -XYI -YXI 11-1---1 IZX -XIX -YXY 1-¯ i¯ ii¯ i11 YII -IXY -IYZ 1¯ i-i¯ i-¯ i-IIY -XYI -YZI 11¯ ii--¯ ii IZX -XIX -YXZ 1¯ i-¯ ii1¯ i1 YII -IXZ -IYX 1¯ i¯ i--i¯ i-IIY -XZI -YXI 111-¯ i¯ ii¯ i IZX -XXY -YIX 11-1ii¯ ii YII -IXZ -IYY 1¯ i1¯ i-i1¯ i -IIY -XZI -YYI 00i¯ i1100 IZX -XXY -YXZ 0¯ i0i10-0 YIX -IXI -XIY 010-0-01 -IIZ -IXI -XII 11i¯ i¯ i¯ i-1 IZX -XXZ -YIX 1i-¯ i-i1¯ i YIX -IXI -XIZ 010-0¯ i0i -IIZ -IXI -YII 110000-1 IZX -XXZ -YXY 1i¯ i--i¯ i1 YIX -IXX -XYY 0000010-IIZ -IXI -ZII 1i1¯ i-¯ i1¯ i IZY -XIY -YXX 100-0i¯ i0 YIX -IXX -XYZ 010¯ i0-0i -IIZ -IYI -XII 1i¯ i--¯ i¯ iIZY -XIY -YXZ 0¯ i0-10¯ i0 YIX -IYI -XIY 010¯ i0¯ i0-IIZ -IYI -YII 1i-i¯ i1¯ i- IZY -XXX -YIY 1i¯ i1-ii1 YIX -IYI -XIZ 0000010¯ i -IIZ -IYI -ZII 1i0000¯ i- IZY -XXX -YXZ 1¯ i-¯ i1i1¯ i YIX -IYX -XXY 0001000-IIZ -IZI -XII 1ii1¯ i1-i IZY -XXZ -YIY 0¯ i-0100¯ i YIX -IYX -XXZ 0001000¯ i -IIZ -IZI -YII 00-i1i00 IZY -XXZ -YXX 000¯ i0010 YIX -IZI -XIY 00000001 -IIZ -IZI -ZII 100¯ i-00¯ i IZZ -XIZ -YXX 001i00-i YIX -IZI -XIZ 000-0100 -IIZ -XXI -YYI 1001-001 IZZ -XIZ -YXY 1---¯ ii¯ i¯ i YIX -IZX -XXY 010i0¯ i0-IIZ -XXI -YZI 100i¯ i00- IZZ -XXX -YIZ 1-ii¯ ii-YIX -IZX -XXZ 0100000¯ i -IIZ -XYI -YXI 000-1000 IZZ -XXX -YXY 10-00-01 YIY -IXI -XIX 010-0¯ i0¯ i -IIZ -XYI -YZI 100-¯ i00¯ i IZZ -XXY -YIZ 1--1--11 YIY -IXI -XIZ 010¯ i0-0¯ i -IIZ -XZI -YXI 1000000¯ i IZZ -XXY -YXX 11i¯ i1-¯ i¯ i YIY -IXY -XYX 01010-01 -IIZ -XZI -YYI 1--11--1 XII -IIX -IXI 100¯ i0-¯ i0 YIY -IXY -XYZ 0-0110-0 -IXI -XIX -YIY 1-¯ ii1-¯ ii XII -IIX -IYI 10¯ i00-0i YIY -IYI -XIX 1i-¯ i¯ i-i1 -IXI -XIX -YIZ 001-001XII -IIX -IZI 1-¯ ii--ii YIY -IYI -XIZ 10-00¯ i0i -IXI -XIY -YIX 1¯ i-i1¯ i-i XII -IIY -IXI 1-11--1- YIY -IYY -XXX 1--1¯ i¯ iii -IXI -XIY -YIZ 1¯ i¯ i-1¯ i¯ iXII -IIY -IYI 01-01001 YIY -IYY -XXZ 1¯ i-i-¯ i1i -IXI -XIZ -YIX 001¯ i001¯ i XII -IIY -IZI 0010000- YIY -IZI -XIX 11---11-IXI -XIZ -YIY 010-010XII -IIZ -IXI 001-00-YIY -IZI -XIZ 0-1001-0 -IXX -IYY -XII 010¯ i010¯ i XII -IIZ -IYI 1¯ i1i¯ i-i- YIY -IZY -XXX 0-100i¯ i0 -IXX -IYY -YII 00010001 XII -IIZ -IZI 1¯ ii-¯ i--¯ i YIY -IZY -XXZ 00000-10 -IXX -IYY -ZII 0-100-10 XII -IXX -IYY 1¯ i-ii-¯ i1 YIZ -IXI -XIX 1i¯ i--¯ ii1 -IXX -IYZ -XII 1i¯ i-1i¯ iXII -IXX -IYZ 11--i¯ i¯ ii YIZ -IXI -XIY 1i¯ i-¯ i1-i -IXX -IYZ -YII 100¯ i100¯ i XII -IXY -IYX 11-1i¯ i¯ i¯ i YIZ -IXZ -XYX 00001i¯ i-IXX -IYZ -ZII 1-¯ i¯ i1-¯ i¯ i XII -IXY -IYZ 1i-ii1¯ i1 YIZ -IXZ -XYY 1i¯ i-¯ i-1i -IXX -XIX -YYY 1¯ i-¯ i1¯ i-¯ i XII -IXZ -IYX 1¯ i¯ i-i-1i YIZ -IYI -XIX 100-0-10 -IXX -XIX -YYZ 11-111-1 XII -IXZ -IYY 11¯ i¯ ii¯ i1YIZ -IYI -XIY 1¯ ii--¯ ii1 -IXX -XYY -YIX 10-0010- XIX -IXI -YIY 1-¯ i¯ iii1YIZ -IYZ -XXX 1-1---11 -IXX -XYY -YYZ 1¯ i-i¯ i1iXIX -IXI -YIZ 1¯ i¯ i1i-1¯ i YIZ -IYZ -XXY 0i¯ i0100- -IXX -XYZ -YIX 1¯ ii-¯ i1-i XIX -IXX -YYY 001¯ i00iYIZ -IZI -XIX 11--¯ ii¯ ii -IXX -XYZ -YYY 01-0100- XIX -IXX -YYZ 001100i¯ i YIZ -IZI -XIY 100¯ i-00i -IXY -IYX -XII 10¯ i0010¯ i XIX -IYI -YIY 0¯ i100-i0 YIZ -IZZ -XXX 100¯ i¯ i00-IXY -IYX -YII 1¯ i¯ i-¯ i1-¯ i XIX -IYI -YIZ 01100¯ ii0 YIZ -IZZ -XXY 0000100¯ i -IXY -IYX -ZII 1i1¯ ii1¯ i1 XIX -IYX -YXY 00¯ ii1-00 YXI -IIX -XYI 1-¯ i¯ i-1ii -IXY -IYZ -XII 100¯ i01¯ i0 XIX -IYX -YXZ 1-i¯ i-1i¯ i YXI -IIX -XZI 1-¯ i¯ i¯ ii--IXY -IYZ -YII 00100001 XIX -IZI -YIY 00¯ i-1¯ i00 YXI -IIY -XYI 00001-¯ i¯ i -IXY -IYZ -ZII 001¯ i00¯ i1 XIX -IZI -YIZ 1¯ ii1-ii1 YXI -IIY -XZI 11i¯ ii¯ i11 -IXY -XIY -YYX 1-11-111 XIX -IZX -YXY 000¯ i0100 YXI -IIZ -XYI 100¯ i0¯ i10 -IXY -XIY -YYZ 1-¯ i¯ i-1¯ i¯ i XIX -IZX -YXZ 010i0-0i YXI -IIZ -XZI 1-¯ i¯ i11i¯ i -IXY -XYX -YIY 0i0¯ i10-0 XIY -IXI -YIX 11-1i¯ iii YXX -IYY -XIY 1i-¯ i1¯ i1¯ i -IXY -XYX -YYZ 11--¯ iii¯ i XIY -IXI -YIZ 1i¯ i1i1-i YXX -IYY -XXZ 0-¯ i0100¯ i -IXY -XYZ -YIY 1-¯ i¯ iii-1 XIY -IXY -YYX 11¯ ii-1ii YXX -IYZ -XIZ 1¯ i1¯ ii-¯ i1 -IXY -XYZ -YYX 0i-0100¯ i XIY -IXY -YYZ 0-0¯ i10¯ i0 YXX -IYZ -XXY 1¯ i-¯ i-i1i -IXZ -IYX -XII 0i0110¯ i0 XIY -IYI -YIX 1¯ i-¯ i1¯ i1i YXX -IZY -XIY 1¯ i-¯ i¯ i-i-IXZ -IYX -YII 11¯ i¯ i¯ ii-1 XIY -IYI -YIZ 00-¯ i1¯ i00 YXX -IZY -XXZ 00001¯ i-¯ i -IXZ -IYX -ZII 11-1¯ ii¯ i¯ i XIY -IYY -YXX 0i100i-0 YXX -IZZ -XIZ 11-1--1-IXZ -IYY -XII 10010i¯ i0 XIY -IYY -YXZ 0¯ i000010 YXX -IZZ -XXY 11-1¯ i¯ ii¯ i -IXZ -IYY -YII 000i0010 XIY -IZI -YIX 1i1¯ i¯ i-i- YXY -IYX -XIX 000011-1 -IXZ -IYY -ZII 001100¯ ii XIY -IZI -YIZ 1-i¯ i¯ i¯ i-- YXY -IYX -XXZ 11-1-111 -IXZ -XIZ -YYX 1¯ i-¯ i-i-¯ i XIY -IZY -YXX 1i¯ i--iiYXY -IYZ -XIZ 1i-i-i1i -IXZ -XIZ -YYY 1¯ i¯ i1-i¯ i1 XIY -IZY -YXZ 10¯ i00i0- YXY -IYZ -XXX 1---¯ i¯ ii¯ i -IXZ -XYX -YIZ 1i-¯ i1¯ i-i XIZ -IXI -YIX 1-111--- YXY -IZX -XIX 0¯ i0¯ i10-0 -IXZ -XYX -YYY 1--111-XIZ -IXI -YIY 1-0000-- YXY -IZX -XXZ 1¯ i-¯ i¯ i1i1 -IXZ -XYY -YIZ 1---11-1 XIZ -IXZ -YYX 0-100--0 YXY -IZZ -XIZ 10-00101 -IXZ -XYY -YYX 1¯ i-¯ i1i-i XIZ -IXZ -YYY 00100-00 YXY -IZZ -XXX 0-0i10¯ i0 -IYI -XIX -YIY 1i¯ i11¯ i¯ iXIZ -IYI -YIX 100¯ i0-i0 YXZ -IYX -XIX 1i¯ i1¯ i--i -IYI -XIX -YIZ 1-¯ ii11¯ i¯ i XIZ -IYI -YIY 11¯ i¯ i1-i¯ i YXZ -IYX -XXY 10¯ i00¯ i0-IYI -XIY -YIX 11¯ ii1-¯ i¯ i XIZ -IYZ -YXX 10010¯ ii0 YXZ -IYY -XIY 1-¯ ii¯ i¯ i--IYI -XIY -YIZ 1i¯ i-1¯ i¯ i1 XIZ -IYZ -YXY 1¯ ii1-¯ ii- YXZ -IYY -XXX 1¯ i¯ i--¯ ii-IYI -XIZ -YIX 001i001¯ i XIZ -IZI -YIX 1-¯ i¯ i1-ii YXZ -IZX -XIX 11¯ i¯ i-1i¯ i -IYI -XIZ -YIY 001-0011 XIZ -IZI -YIY 00¯ i¯ i1-00 YXZ -IZX -XXY 1¯ i-¯ ii-i1 -IYX -XIX -YXY 0i100¯ i10 XIZ -IZZ -YXX 1¯ i¯ i11¯ ii- YXZ -IZY -XIY 0-i0100¯ i -IYX -XIX -YXZ 0-100110 XIZ -IZZ -YXY 1¯ i0000i- YXZ -IZY -XXX 1i1¯ i1¯ i-¯ i -IYX -XXY -YIX 1-00001- XXI -IIX -YYI 1-0000-1 YYI -IIX -XXI 1-i¯ i11¯ i¯ i -IYX -XXY -YXZ 1-¯ ii¯ ii1XXI -IIX -YZI 1--1-1-1 YYI -IIX -XZI 100¯ i0¯ i-0 -IYX -XXZ -YIX 1¯ i00001¯ i XXI -IIY -YYI 1¯ i0000-i YYI -IIY -XXI 11¯ i¯ ii¯ i-1 -IYX -XXZ -YXY 1¯ i¯ i-¯ i-1¯ i XXI -IIY -YZI 1¯ i-i-i-i YYI -IIY -XZI 1-11¯ i¯ ii¯ i -IYY -XIY -YXX 01000001 XXI -IIZ -YYI 0100000- YYI -IIZ -XXI 0¯ ii01001 -IYY -XIY -YXZ 010¯ i0¯ i01 XXI -IIZ -YZI 010-0-0YYI -IIZ -XZI 11-1-1-- -IYY -XXX -YIY 1-1111-1 XXX -IYY -YIY 1-¯ i¯ i¯ i¯ i1- YYX -IXY -XIY 1i¯ i1-i¯ i- -IYY -XXX -YXZ 1¯ ii11i¯ i1 XXX -IYY -YXZ 1i-¯ i¯ i-¯ i- YYX -IXY -XYZ 100101-0 -IYY -XXZ -YIY 1-¯ i¯ i¯ i¯ i-1 XXX -IYZ -YIZ 1-----1YYX -IXZ -XIZ 1¯ ii1¯ i1-¯ i -IYY -XXZ -YXX 10¯ i00¯ i01 XXX -IYZ -YXY 010110-0 YYX -IXZ -XYY 1-¯ i¯ i--i¯ i -IYZ -XIZ -YXX 1¯ i1ii1¯ i1 XXX -IZY -YIY 1¯ ii-i-1¯ i YYY -IXX -XIX 1¯ i¯ i1-¯ ii1 -IYZ -XIZ -YXY 1¯ i0000¯ i1 XXX -IZY -YXZ 1-1-ii¯ i¯ i YYY -IXX -XYZ 11¯ ii¯ ii--IYZ -XXX -YIZ 0¯ i1001¯ i0 XXX -IZZ -YIZ 1¯ i-¯ i-¯ i1¯ i YYY -IXZ -XIZ 0i0-10¯ i0 -IYZ -XXX -YXY 00100100 XXX -IZZ -YXY 10-00¯ i0¯ i YYY -IXZ -XYX 1i¯ i-¯ i--¯ i -IYZ -XXY -YIZ 1¯ i-¯ i-¯ i-i XXY -IYX -YIX 0-10100- YYZ -IXX -XIX 10¯ i00-0¯ i -IYZ -XXY -YXX 11¯ i¯ i-1¯ ii XXY -IYX -YXZ 11---1-1 YYZ -IXX -XYY 000-0010 -IZI -XIX -YIY 1¯ i¯ i1¯ i1-i XXY -IYZ -YIZ 0¯ i10100¯ i YYZ -IXY -XIY 001i00¯ i-IZI -XIX -YIZ 010i10¯ i0 XXY -IYZ -YXX 1¯ i1¯ i1i-¯ i YYZ -IXY -XYX 0010000¯ i -IZI -XIY -YIX 1---i¯ iii XXY -IZX -YIX 1-¯ iii¯ i-1 YZI -IIX -XXI 001-00¯ i¯ i -IZI -XIY -YIZ 00ii1-00 XXY -IZX -YXZ 1-1-i¯ i¯ ii YZI -IIX -XYI 001¯ i00-¯ i -IZI -XIZ -YIX 01100i¯ i0 XXY -IZZ -YIZ 1¯ i¯ i-i1-i YZI -IIY -XXI 001100-1 -IZI -XIZ -YIY 0i000010 XXY -IZZ -YXX 1¯ i1¯ ii1¯ iYZI -IIY -XYI 1---1-11 -IZX -XIX -YXY 0i10100¯ i XXZ -IYX -YIX 010¯ i0i0YZI -IIZ -XXI 1-ii1-¯ i¯ i -IZX -XIX -YXZ 1-i¯ iii11 XXZ -IYX -YXY 01010i0¯ i YZI -IIZ -XYI 1-11i¯ i¯ i¯ i -IZX -XXY -YIX 0-101001 XXZ -IYY -YIY 1--10000 ZII -IIX -IXI 1-0000¯ i¯ i -IZX -XXY -YXZ 1i¯ i1¯ i-1¯ i XXZ -IYY -YXX 1-¯ ii0000 ZII -IIX -IYI 1-¯ i¯ ii¯ i--IZX -XXZ -YIX 1-iii¯ i11 XXZ -IZX -YIX 001-0000 ZII -IIX -IZI 00--1-00 -IZX -XXZ -YXY 1-000011 XXZ -IZX -YXY 1¯ i-i0000 ZII -IIY -IXI 1¯ i1i1¯ i-¯ i -IZY -XIY -YXX 1¯ ii-i11i XXZ -IZY -YIY 1¯ i¯ i-0000 ZII -IIY -IYI 1¯ ii-1¯ i¯ i1 -IZY -XIY -YXZ 001i1¯ i00 XXZ -IZY -YXX 001¯ i0000 ZII -IIY -IZI 1¯ i-¯ ii1i- -IZY -XXX -YIY 00i¯ i1-00 XYI -IIX -YXI 010-0000 ZII -IIZ -IXI 00i-1¯ i00 -IZY -XXX -YXZ 1-1-¯ iii¯ i XYI -IIX -YZI 010¯ i0000 ZII -IIZ -IYI 1¯ i¯ i1i1-¯ i -IZY -XXZ -YIY 00i11¯ i00 XYI -IIY -YXI 00010000 ZII -IIZ -IZI 1¯ i0000-¯ i -IZY -XXZ -YXX 1¯ i1¯ i¯ i-i1 XYI -IIY -YZI 0-100000 ZII -IXX -IYY 0¯ i100¯ i-0 -IZZ -XIZ -YXX 000i0100 XYI -IIZ -YXI 1i¯ i-0000 ZII -IXX -IYZ 011001-0 -IZZ -XIZ -YXY 01010¯ i0i XYI -IIZ -YZI 100¯ i0000 ZII -IXY -IYX 0i100-¯ i0 -IZZ -XXX -YIZ 11i¯ i-1ii XYX -IXY -YIY 1-¯ i¯ i0000 ZII -IXY -IYZ 0-000010 -IZZ -XXX -YXY 1¯ i1¯ i-¯ i1i XYX -IXY -YYZ 1¯ i-¯ i0000 ZII -IXZ -IYX 0-100¯ i¯ i0 -IZZ -XXY -YIZ 11-1¯ iiii XYX -IXZ -YIZ 11-10000 ZII -IXZ -IYY 00100¯ i00 -IZZ -XXY -YXX

S TATE

∠

S TATE

∠

S TATE

∠

11111111 1111iiii 11110000 11ii11ii 11iiii-11ii0000 11001100 1100ii00 11000000 00111100 11iiii11 110000ii 11--iiii 11¯ i¯ i11ii 111111-1i1i1i1i 1i1ii-i1i1i0000 1ii-1ii1ii-i--¯ i 1ii-0000 1i001i00 1i00i-00 1i000000 001i1i00 1ii-i-1i 1i0000i1i-¯ ii-i1i¯ i11ii1i1i1i-¯ i 10101010 1010i0i0 10100000 10i010i0 10i0i0-0 10i00000 10001000 1000i000 10000000 00101000 10i0i010 100000i0 10-0i0i0 10¯ i010i0 101010-0 01011010 1i1ii1i1 10100i0i 1-1-iiii 1¯ i1¯ i1i1i 11111-101100110 01100ii0 01100000 1ii11ii1 1ii1i--i 1ii10000 1¯ i¯ i1¯ i11¯ i 01101001 1ii1-ii1--1---10010ii0 1111¯ iii¯ i 100i100i 100ii00100i0000 1-ii1-ii 1-iii¯ i-1-ii0000 1-iiii10i10100i 11¯ ii1-ii 1i1i1¯ i-i 100i0-i0 1¯ i-ii-i1¯ i1i1¯ i1i 1¯ i1ii1i1¯ i1i0000 111-111111-iii¯ i 111-0000 1-111111¯ i1i1i1¯ i 111-i¯ iii 0¯ i0i1010 1i1¯ ii1i1010010010i10i0 1ii-i1-i 10i00i01-i¯ iii-1¯ ii11ii11ii1-i¯ i 1i-ii1i100i01i0 1¯ i1i1i-i 1-¯ ii11ii 0¯ i10100i 11iii¯ i1111-¯ iiii 100-0ii0 1------1 1ii--ii1 01101001¯ i¯ i-¯ i11i 11i¯ i1-ii 1ii11¯ ii1-iiii-1 0i0110i0 1¯ ii-i--i 10i00-0i 01001000 1i00i100 10000i00 1-00ii00 1¯ i001i00 11001-00 11-111111¯ ii11i¯ i 111-ii¯ ii 00¯ ii1100 11i¯ iii111000011i1¯ i1i-i 1i¯ i-1ii1 1i-ii-i1 1i0000i1 1ii1i-1¯ i 001¯ i1i00 100¯ i100i 1001100100ii001 00011000 100-i00i 1000000i

π/2.59 π/2.59 π/3 π/2.59 π/2.59 π/3 π/3 π/3 π/4 ⊥ π/2.59 π/3 π/2.59 π/2.59 π/2.59 π/2.59 π/2.59 π/3 π/2.59 π/2.59 π/3 π/3 π/3 π/4 ⊥ π/2.59 π/3 π/2.59 π/2.59 π/2.59 π/3 π/3 π/4 π/3 π/3 π/4 π/4 π/4 0 ⊥ π/3 π/4 π/3 π/3 π/3 ⊥ π/2.59 π/3 π/2.59 π/2.59 π/2.59 ⊥ ⊥ ⊥ π/2.59 π/2.59 π/3 π/2.59 ⊥ π/2.59 π/2.59 π/3 π/2.59 π/3 π/3 π/4 π/2.59 π/2.59 π/3 π/2.59 ⊥ π/2.59 π/2.59 π/3 π/2.59 π/2.59 π/2.59 π/3 π/2.59 π/2.59 π/3 π/2.59 π/2.59 π/2.59 ⊥ π/2.59 π/3 ⊥ π/2.59 π/3 π/2.59 π/2.59 π/2.59 π/2.59 π/3 π/2.59 π/2.59 ⊥ π/2.59 π/2.59 π/3 π/2.59 π/2.59 ⊥ π/2.59 π/2.59 π/2.59 π/2.59 ⊥ π/2.59 π/3 ⊥ π/3 π/4 π/3 π/3 π/3 π/2.59 π/2.59 π/2.59 ⊥ π/2.59 π/3 π/2.59 π/2.59 π/2.59 π/3 π/2.59 ⊥ π/3 π/3 π/3 ⊥ π/3 π/4

1111---1111¯ i¯ i¯ i¯ i 00001111 11ii--¯ i¯ i 11ii¯ i¯ i11 000011ii 1100--00 1100¯ i¯ i00 00001100 11--11-11¯ i¯ i11¯ i¯ i 00110011 11000011 11¯ i¯ i¯ i¯ i11 00ii1100 1111¯ i¯ iii 11ii11¯ i¯ i 11--1111 11--ii¯ i¯ i 11¯ i¯ iii11 001100ii 00¯ i¯ i1100 11ii--ii 110000-11-----11¯ i¯ iii-1111ii¯ i¯ i 11--0000 11¯ i¯ i0000 00110000 1i1i-¯ i-¯ i 1i1i¯ i1¯ i1 00001i1i 1ii--¯ i¯ i1 1ii-¯ i11i 00001ii1i00-¯ i00 1i00¯ i100 00001i00 1i-¯ i1i-¯ i 1i¯ i11i¯ i1 001i001i 1i00001i 1i¯ i1¯ i11i 00i-1i00 1i1i¯ i1i1ii-1i¯ i1 1i-¯ i1i1i 1i-¯ ii-¯ i1 1i¯ i1i-1i 001i00i00¯ i11i00 1ii--¯ ii1i0000-¯ i 1i-¯ i-¯ i-¯ i 1i¯ i1i--¯ i 1i1ii-¯ i1 1i-¯ i0000 1i¯ i10000 001i0000 1010-0-0 1010¯ i0¯ i0 00001010 10i0-0¯ i0 10i0¯ i010 000010i0 1000-000 1000¯ i000 00001000 10-010-0 10¯ i010¯ i0 00100010 10000010 10¯ i0¯ i010 00i01000 1010¯ i0i0 10i010¯ i0 10-01010 10-0i0¯ i0 10¯ i0i010 001000i0 00¯ i01000 10i0-0i0 100000-0 10-0-0-0 10¯ i0i0-0 1010i0¯ i0 10-00000 10¯ i00000 00100000 1-1-1-11¯ i1¯ i1¯ i1¯ i 01010101 10100101 1¯ i1¯ i¯ i1¯ i1 0i0i1010 1111¯ ii¯ ii 1i1i1¯ i1¯ i 1-1-1111 1-1-i¯ ii¯ i 1¯ i1¯ ii1i1 01010i0i 0¯ i0¯ i1010 1i1i-i-i 10100-01-1----1¯ i1¯ ii-i1111i¯ ii¯ i 1-1-0000 1¯ i1¯ i0000 01010000 01100--0 01100¯ i¯ i0 00000110 1ii1-¯ i¯ i1ii1¯ i11¯ i 00001ii1 10011001 1¯ i¯ i11¯ i¯ i1 1ii1i11i 10010110 1¯ i¯ i1-¯ i¯ i1111-110ii01001 1--1iiii 1001i00i 1¯ i¯ i1i11i 1¯ i¯ i11ii1 0¯ i¯ i01001 1ii1¯ i--¯ i 1--1¯ i¯ i¯ i¯ i 10010--0 11111--1 10010000 1¯ i¯ i10000

π/2.59 π/2.59 ⊥ π/2.59 π/2.59 ⊥ π/3 π/3 ⊥ π/2.59 π/2.59 ⊥ π/3 π/2.59 ⊥ π/2.59 π/2.59 π/2.59 π/2.59 π/2.59 ⊥ ⊥ π/2.59 π/3 π/2.59 π/2.59 π/2.59 π/3 π/3 ⊥ π/2.59 π/2.59 ⊥ π/2.59 π/2.59 ⊥ π/3 π/3 ⊥ π/2.59 π/2.59 ⊥ π/3 π/2.59 ⊥ π/2.59 π/2.59 π/2.59 π/2.59 π/2.59 ⊥ ⊥ π/2.59 π/3 π/2.59 π/2.59 π/2.59 π/3 π/3 ⊥ π/3 π/3 ⊥ π/3 π/3 ⊥ π/4 π/4 ⊥ π/3 π/3 ⊥ π/4 π/3 ⊥ π/3 π/3 π/3 π/3 π/3 ⊥ ⊥ π/3 π/4 π/3 π/3 π/3 π/4 π/4 ⊥ π/2.59 π/2.59 ⊥ π/3 π/2.59 ⊥ π/2.59 π/2.59 π/2.59 π/2.59 π/2.59 ⊥ ⊥ π/2.59 π/3 π/2.59 π/2.59 π/2.59 π/3 π/3 ⊥ ⊥ ⊥ ⊥ π/2.59 π/2.59 ⊥ π/3 π/2.59 π/2.59 π/3 π/2.59 π/2.59 ⊥ π/2.59 π/3 π/2.59 π/2.59 ⊥ π/2.59 π/2.59 π/3 π/2.59 π/4 π/3

100i-00¯ i 100i¯ i001 0000100i 1-ii-1¯ i¯ i 1-ii¯ ii11 00001-ii 0i100i10 11¯ ii11¯ ii 11¯ ii¯ ii11 100i0i10 1-ii11¯ ii 1¯ i-i1i1i 0-i0100i 1i1i¯ i-i1 0i100-i0 11¯ iiii11-ii--i¯ i 01¯ i0100i 11¯ iii¯ i-1i1ii1¯ i100i0¯ i-0 1¯ i-i-¯ i-¯ i 0i100000 11¯ ii0000 1¯ i1i-i-¯ i 1¯ i1i¯ i-¯ i1 00001¯ i1i 111----1 111-¯ i¯ i¯ ii 00001111i1¯ i1i1¯ i 1-111-11 111-1-11 1i1¯ i1¯ i1i 1-11¯ i¯ i¯ ii 10100¯ i0i 1¯ i1i¯ i1¯ i010-1010 1i1¯ ii-i1 1-11i¯ iii 1-11iii¯ i 1¯ i1ii-i1 111--1-0-011010 1i1¯ i-i-¯ i 10100i0¯ i 1i1¯ i0000 1-110000 1-i¯ i1-i¯ i 1¯ ii11¯ ii1 010i010i 10i0010i 1¯ ii1¯ i11i 0i0-10i0 11ii¯ ii11ii-1¯ ii1 1-i¯ i11ii 1-i¯ ii¯ i-1 1¯ ii1i1-i 010i0i00¯ i0110i0 1ii--i¯ i10i00-0¯ i 1-i¯ i--¯ i¯ i 1¯ ii1i--¯ i 11iii¯ i-1 1-i¯ i0000 1¯ ii10000 010i0000 0¯ i100¯ i10 1i-i1i-i 1¯ i1i¯ i1i1 01i0100i 1i-i1¯ i1i 11ii1-¯ ii 100i0¯ i10 1-¯ ii¯ i¯ i11 0¯ i1001i0 1i-ii-¯ i1i-i-i-¯ i 100i0i-0 1¯ i1ii-¯ i1-¯ iiii-0-¯ i0100i 11ii-1i¯ i 0¯ i100000 1i-i0000 100-1001---1--1---iii¯ i 0ii0100111--111 1¯ i¯ i--¯ i¯ i1 100-0110 1ii-i11¯ i 100-i00¯ i 1---i¯ i¯ i¯ i 111-1--100-0--0 1---¯ i¯ i¯ ii 1ii-¯ i--i 0¯ i¯ i01001¯ i¯ i-1ii100-0000 1---0000 1¯ ii-1¯ ii11i¯ i11i¯ i 1-ii11i¯ i 1¯ ii-1ii1 11i¯ i¯ ii11 10i00i01 1ii1¯ i-1i 0-0i10i0 1¯ ii-i1-¯ i 11i¯ iii-1 11i¯ ii¯ i-1ii1i1-¯ i 1-ii--¯ ii 010¯ i10i0 1¯ ii--¯ i¯ i10i00¯ i01¯ ii-0000 11i¯ i0000 1-001-00 1¯ i001¯ i00 01000100 10000100 1¯ i00¯ i100 0i001000 1100¯ ii00 1i001¯ i00 1-001100 1-00i¯ i00 1¯ i00i100 01000i00

π/3 π/3 ⊥ π/2.59 π/2.59 ⊥ ⊥ π/2.59 π/2.59 π/3 π/2.59 π/2.59 ⊥ π/2.59 ⊥ π/2.59 π/2.59 ⊥ π/2.59 π/2.59 π/3 π/2.59 ⊥ π/3 π/2.59 π/2.59 ⊥ π/2.59 π/2.59 ⊥ π/2.59 π/2.59 π/2.59 π/2.59 π/2.59 π/3 π/2.59 ⊥ π/2.59 π/2.59 π/2.59 π/2.59 π/2.59 ⊥ π/2.59 π/3 π/3 π/3 π/2.59 π/2.59 ⊥ π/3 π/2.59 ⊥ π/2.59 π/2.59 π/2.59 π/2.59 π/2.59 ⊥ ⊥ π/2.59 π/3 π/2.59 π/2.59 π/2.59 π/3 π/3 ⊥ ⊥ π/2.59 π/2.59 ⊥ π/2.59 π/2.59 π/3 π/2.59 ⊥ π/2.59 π/2.59 π/3 π/2.59 π/2.59 ⊥ π/2.59 ⊥ π/3 π/3 π/2.59 π/2.59 ⊥ π/2.59 π/2.59 π/3 π/2.59 π/3 π/2.59 π/2.59 π/3 π/2.59 π/2.59 ⊥ π/2.59 π/4 π/3 π/2.59 π/2.59 π/2.59 π/2.59 π/2.59 π/3 π/2.59 ⊥ π/2.59 π/2.59 π/2.59 π/2.59 π/2.59 ⊥ π/2.59 π/3 π/3 π/3 π/3 π/3 ⊥ π/4 π/3 ⊥ π/3 π/3 π/3 π/3 π/3 ⊥

S TATE 0¯ i001000 1i00-i00 10000-00 1-00--00 1¯ i00i-00 1100i¯ i00 1-000000 1¯ i000000 01000000 111-11-1 11i¯ i11¯ ii 11-1¯ i¯ i¯ ii 110000¯ ii 11¯ ii¯ i¯ i1001-1100 11-1iii¯ i 11¯ iiii-1 111---100-11100 11i¯ i--i¯ i 110000i¯ i 1i-i1i1¯ i 1ii11i¯ i1i1¯ i¯ i1i1 00i11i00 1i¯ i-¯ i11¯ i 1i00001¯ i 1i1¯ ii-¯ i1i¯ i-i--i 1i-i-¯ i-i 1i0000-i 1ii1-¯ ii1 00¯ i-1i00 100i100¯ i 100-1001 100¯ i¯ i001 10000001 1001¯ i00i 000i1000 100¯ ii001001i00¯ i 100i-00i 000¯ i1000 100--001000000010-10-0 010¯ i10¯ i0 00010010 1i-¯ ii1¯ i1i¯ i1i11¯ i 001i00i1 1¯ i-¯ i¯ i1¯ i1----1-01¯ i0100¯ i 1-ii-1ii 1i¯ i-i1-¯ i 100-01-0 10-00i0¯ i 10¯ i00i01 0010000i 1--1ii¯ i¯ i 1-¯ iiii11 001-00ii 1-11ii¯ ii 1¯ i1i-i1i 0i¯ i01001 1¯ ii--ii11i¯ i¯ ii-100¯ i0i-0 1¯ i-i1i-¯ i 1¯ i¯ i-1i¯ i1 001¯ i001i 11--1--1 11¯ i¯ i1-¯ ii 00110011-¯ i¯ i11¯ ii 0¯ i100i10 1¯ i¯ i11i¯ i01100-10 11-11--1i-i1¯ i-¯ i 001-1-00 001¯ i1¯ i00 00010100 1-i¯ ii¯ i11¯ ii1i11¯ i 010i0i01 11-11-11 1¯ i-¯ i¯ i-¯ i1 11¯ iii¯ i11 0i1001i0 0¯ i0110¯ i0 01000010 1i¯ i11¯ ii1 00¯ i11¯ i00 1i1¯ i-i1i 1-11¯ iiii 1i¯ i-i11i 0-100ii0 10¯ i0010i 00100i00 1-i¯ i--ii 1-0000ii 100¯ i0i10 1-¯ i¯ i¯ ii11 10010-10 1¯ i¯ i1¯ i-1i 1¯ ii1i-1i 1¯ i00001i 11¯ i¯ i¯ ii100111-00 1-0000i¯ i 1¯ i0000i1 0100000i 1--1i¯ ii¯ i 1¯ i-ii1i1 010-0i0i 1-¯ i¯ i--¯ ii 1---ii¯ ii 0i0i10-0 1i-¯ i-i-i 1¯ ii-1i¯ i1¯ i-¯ ii-¯ i10-00-01-1-11-0¯ ii010001i0100¯ i 1¯ i1¯ i¯ i1i11--i¯ ii¯ i 1-¯ ii1-i¯ i 1¯ i¯ i-1¯ ii1 010¯ i010i 1-1-1--1 1¯ i1¯ i1¯ i-i 0101010-

∠

S TATE

∠

S TATE

∠

S TATE

∠

S TATE

∠

⊥ π/3 π/4 π/3 π/3 π/3 π/4 π/4 ⊥ π/2.59 π/2.59 π/2.59 π/3 π/2.59 ⊥ π/2.59 π/2.59 π/2.59 ⊥ π/2.59 π/3 π/2.59 π/2.59 π/2.59 ⊥ π/2.59 π/3 π/2.59 π/2.59 π/2.59 π/3 π/2.59 ⊥ π/3 π/3 π/3 π/4 π/3 ⊥ π/3 π/3 π/3 ⊥ π/3 π/4 ⊥ ⊥ ⊥ π/2.59 π/2.59 ⊥ π/2.59 π/2.59 ⊥ π/2.59 π/2.59 π/3 π/3 π/3 ⊥ π/2.59 π/2.59 ⊥ π/2.59 π/2.59 ⊥ π/2.59 π/2.59 π/3 π/2.59 π/2.59 ⊥ π/2.59 π/2.59 ⊥ π/2.59 ⊥ π/2.59 ⊥ π/2.59 π/2.59 ⊥ ⊥ ⊥ π/2.59 π/2.59 ⊥ π/2.59 π/2.59 π/2.59 ⊥ ⊥ ⊥ π/2.59 ⊥ π/2.59 π/2.59 π/2.59 ⊥ π/3 ⊥ π/2.59 π/3 π/3 π/2.59 π/3 π/2.59 π/2.59 π/3 π/2.59 ⊥ π/3 π/3 ⊥ π/2.59 π/2.59 ⊥ π/2.59 π/2.59 ⊥ π/2.59 π/2.59 π/2.59 π/3 π/2.59 ⊥ ⊥ π/2.59 π/2.59 π/2.59 π/2.59 ⊥ π/2.59 π/2.59 ⊥

11----11 11--¯ i¯ iii 000011-11¯ i¯ i--ii 11¯ i¯ i¯ i¯ i-000011¯ i¯ i 001100-001100¯ i¯ i 00000011 00--1100 11ii¯ i¯ i-110000¯ i¯ i 11--¯ i¯ i¯ i¯ i 11¯ i¯ i--¯ i¯ i 1111--11 1i-¯ i-¯ i1i 1i-¯ i¯ i1i00001i-¯ i 1i¯ i1-¯ ii1i¯ i1¯ i1-¯ i 00001i¯ i1 001i00-¯ i 001i00¯ i1 0000001i 00-¯ i1i00 1ii-¯ i1-¯ i 1i0000¯ i1 1i-¯ i¯ i1¯ i1 1i¯ i1-¯ i¯ i1 1i1i-¯ i1i 10-0-010 10-0¯ i0i0 000010-0 10¯ i0-0i0 10¯ i0¯ i0-0 000010¯ i0 001000-0 001000¯ i0 00000010 00-01000 10i0¯ i0-0 100000¯ i0 10-0¯ i0¯ i0 10¯ i0-0¯ i0 1010-010 1-1--1-1 1-1-¯ ii¯ ii 00001-11¯ i1¯ i-i-i 1¯ i1¯ i¯ i-¯ i00001¯ i1¯ i 01010-001010¯ i0¯ i 00000101 0-0-1010 1i1i¯ i-¯ i10100¯ i0¯ i 1-1-¯ i¯ i¯ i¯ i 1¯ i1¯ i-¯ i-¯ i 1111-1-1 1001-001001¯ i00¯ i 00001001 1¯ i¯ i1-ii1¯ i¯ i1¯ i--¯ i 00001¯ i¯ i1 1¯ i¯ i1i--i 0--01001 1ii11¯ i¯ i1 1--11111 10010¯ i¯ i0 1111i¯ i¯ ii 0i100¯ i-0 0i1001¯ i0 00000i10 11¯ ii--i¯ i 11¯ ii¯ i¯ i-1 000011¯ ii 1-ii¯ i¯ i-1 0¯ i-0100i 11¯ ii-1¯ i¯ i 1i1i-i1¯ i 100i01¯ i0 1¯ i-i¯ i1¯ i1 1i1¯ i-¯ i-i 1i1¯ i¯ i1¯ i00001i1¯ i 1-11-1-1-11¯ ii¯ i¯ i 00001-11 1-11---1 1¯ i1i-¯ i-i 111-¯ ii¯ i¯ i 0i0¯ i1010 1i1¯ i¯ i-¯ i1 10100-01 1-i¯ i-1¯ ii 1-i¯ i¯ ii100001-i¯ i 1¯ ii1-i¯ i1¯ ii1¯ i-1¯ i 00001¯ ii1 010i0-0¯ i 010i0¯ i01 0000010i 0-0¯ i10i0 1ii-¯ i-1¯ i 10i00¯ i01 1-i¯ i¯ i¯ i11 1¯ ii1-¯ i¯ i1 11ii-1¯ ii 0¯ i100i-0 0¯ i100-¯ i0 00000¯ i10 1i-i-¯ i1¯ i 1i-i¯ i1i1 00001i-i 1i-i¯ i-¯ i1 100i0-¯ i0 1¯ i1i-¯ i1¯ i 1-¯ ii--¯ i¯ i 0i-0100i 11ii¯ ii-1 100--001 100-¯ i00i 00001001----111 1---¯ iiii 00001--111-i¯ i¯ i¯ i 100-0¯ i¯ i0 1---1111ii-1¯ i¯ i0--01001¯ i¯ i-i--¯ i

π/2.59 π/2.59 ⊥ π/2.59 π/2.59 ⊥ ⊥ ⊥ ⊥ ⊥ π/2.59 π/3 π/2.59 π/2.59 π/2.59 π/2.59 π/2.59 ⊥ π/2.59 π/2.59 ⊥ ⊥ ⊥ ⊥ ⊥ π/2.59 π/3 π/2.59 π/2.59 π/2.59 π/3 π/3 ⊥ π/3 π/3 ⊥ ⊥ ⊥ ⊥ ⊥ π/3 π/4 π/3 π/3 π/3 π/2.59 π/2.59 ⊥ π/2.59 π/2.59 ⊥ ⊥ ⊥ ⊥ ⊥ π/2.59 π/3 π/2.59 π/2.59 π/2.59 π/3 π/3 ⊥ π/2.59 π/2.59 ⊥ π/2.59 ⊥ π/2.59 π/2.59 π/3 π/2.59 ⊥ ⊥ ⊥ π/2.59 π/2.59 ⊥ π/2.59 ⊥ π/2.59 π/2.59 π/3 π/2.59 π/2.59 π/2.59 ⊥ π/2.59 π/2.59 ⊥ π/2.59 π/2.59 π/2.59 ⊥ π/2.59 π/3 π/2.59 π/2.59 ⊥ π/2.59 π/2.59 ⊥ ⊥ ⊥ ⊥ ⊥ π/2.59 π/3 π/2.59 π/2.59 π/2.59 ⊥ ⊥ ⊥ π/2.59 π/2.59 ⊥ π/2.59 π/3 π/2.59 π/2.59 ⊥ π/2.59 π/3 π/3 ⊥ π/2.59 π/2.59 ⊥ π/2.59 π/3 π/2.59 π/2.59 ⊥ π/2.59

1¯ ii--i¯ i1 1¯ ii-¯ i-1i 00001¯ ii11i¯ i--¯ ii 11i¯ i¯ i¯ i1000011i¯ i 11i¯ i-1¯ i¯ i 1ii1-i¯ i1 1-ii¯ i¯ i10¯ i0-10i0 1¯ ii-¯ i11¯ i 10i0010¯ i 1-00-100 1-00¯ ii00 00001-00 1¯ i00-i00 1¯ i00¯ i-00 00001¯ i00 01000-00 01000¯ i00 00000100 0-001000 1i00¯ i-00 10000¯ i00 1-00¯ i¯ i00 1¯ i00-¯ i00 1100-100 11-1---1 11¯ ii--¯ ii 111-¯ i¯ ii¯ i 00i¯ i1100 11i¯ i¯ i¯ i-1 110000-1 1i1¯ i-¯ i1¯ i 1i¯ i--¯ i¯ i1i-i¯ i1¯ i1i0000¯ i1ii1¯ i1-i 00-i1i00 100¯ i-00¯ i 1001-001 100i¯ i00000-1000 100-¯ i00¯ i 1000000¯ i 1--11--1 1-¯ ii1-¯ ii 001-0011¯ i-i1¯ i-i 1¯ i¯ i-1¯ i¯ i001¯ i001¯ i 010-010010¯ i010¯ i 00010001 0-100-10 1i¯ i-1i¯ i100¯ i100¯ i 1-¯ i¯ i1-¯ i¯ i 1¯ i-¯ i1¯ i-¯ i 11-111-1 10-00101¯ i-i¯ i1i1¯ ii-¯ i1-i 01-010010¯ i0010¯ i 1¯ i¯ i-¯ i1-¯ i 1i1¯ ii1¯ i1 100¯ i01¯ i0 00100001 001¯ i00¯ i1 1-11-111 1-¯ i¯ i-1¯ i¯ i 0i0¯ i10-0 11--¯ iii¯ i 1-¯ i¯ iii-1 0i-0100¯ i 0i0110¯ i0 11¯ i¯ i¯ ii-1 11-1¯ ii¯ i¯ i 10010i¯ i0 000i0010 001100¯ ii 1¯ i-¯ i-i-¯ i 1¯ i¯ i1-i¯ i1 1i-¯ i1¯ i-i 1--111-1---11-1 1¯ i-¯ i1i-i 1i¯ i11¯ i¯ i1-¯ ii11¯ i¯ i 11¯ ii1-¯ i¯ i 1i¯ i-1¯ i¯ i1 001i001¯ i 001-0011 0i100¯ i10 0-100110 1-000011-¯ ii¯ ii11¯ i00001¯ i 1¯ i¯ i-¯ i-1¯ i 01000001 010¯ i0¯ i01 1-1111-1 1¯ ii11i¯ i1 1-¯ i¯ i¯ i¯ i-1 10¯ i00¯ i01 1¯ i1ii1¯ i1 1¯ i0000¯ i1 0¯ i1001¯ i0 00100100 1¯ i-¯ i-¯ i-i 11¯ i¯ i-1¯ ii 1¯ i¯ i1¯ i1-i 010i10¯ i0 1---i¯ iii 00ii1-00 01100i¯ i0 0i000010 0i10100¯ i 1-i¯ iii11 0-101001 1i¯ i1¯ i-1¯ i 1-iii¯ i11 1-000011 1¯ ii-i11i 001i1¯ i00 00i¯ i1-00 1-1-¯ iii¯ i 00i11¯ i00 1¯ i1¯ i¯ i-i1 000i0100 01010¯ i0i 11i¯ i-1ii 1¯ i1¯ i-¯ i1i 11-1¯ iiii

π/2.59 π/2.59 ⊥ π/2.59 π/2.59 ⊥ π/2.59 π/2.59 π/2.59 ⊥ π/2.59 π/3 π/3 π/3 ⊥ π/3 π/3 ⊥ ⊥ ⊥ ⊥ ⊥ π/3 π/4 π/3 π/3 π/3 π/2.59 π/2.59 π/2.59 ⊥ π/2.59 π/3 π/2.59 π/2.59 π/2.59 π/3 π/2.59 ⊥ π/3 π/3 π/3 ⊥ π/3 π/4 π/2.59 π/2.59 ⊥ π/2.59 π/2.59 ⊥ ⊥ ⊥ ⊥ ⊥ π/2.59 π/3 π/2.59 π/2.59 π/2.59 π/3 π/2.59 π/2.59 ⊥ π/3 π/2.59 π/2.59 π/3 ⊥ ⊥ π/2.59 π/2.59 ⊥ π/2.59 π/2.59 ⊥ ⊥ π/2.59 π/2.59 π/3 ⊥ ⊥ π/2.59 π/2.59 π/2.59 π/2.59 π/2.59 π/2.59 π/2.59 π/2.59 π/2.59 π/2.59 ⊥ ⊥ ⊥ ⊥ π/3 π/2.59 π/3 π/2.59 ⊥ ⊥ π/2.59 π/2.59 π/2.59 π/3 π/2.59 π/3 ⊥ ⊥ π/2.59 π/2.59 π/2.59 ⊥ π/2.59 ⊥ ⊥ ⊥ ⊥ π/2.59 ⊥ π/2.59 π/2.59 π/3 π/2.59 ⊥ ⊥ π/2.59 ⊥ π/2.59 ⊥ ⊥ π/2.59 π/2.59 π/2.59

10-00i0i 1i¯ i-1¯ ii11--1-11i-i¯ i-i0-0-10-0 100-0¯ ii0 1-1-¯ i¯ iii 100¯ i01i0 1i-¯ ii1i1 1-i¯ i1-¯ ii 1--11-11¯ ii11¯ i¯ i1¯ i-i1¯ i1¯ i 010i010¯ i 010-0101 1--1i¯ i¯ ii 1-¯ iii¯ i1001-00i¯ i 1¯ i-ii1¯ i1¯ i¯ i-i11¯ i 001¯ i00i1 010-0i0¯ i 010¯ i0i01 0001000i 0-100¯ ii0 1i¯ i-i-1¯ i 100¯ ii001 1-¯ i¯ ii¯ i11 1¯ i-¯ ii1¯ i1 11-1ii¯ ii 0¯ i0i10-0 1i-¯ i-i1¯ i 1i¯ i--i¯ i1 100-0i¯ i0 0¯ i0-10¯ i0 1i¯ i1-ii1 1¯ i-¯ i1i1¯ i 0¯ i-0100¯ i 000¯ i0010 001i00-i 1---¯ ii¯ i¯ i 1-ii¯ ii-10-00-01 1--1--11 11i¯ i1-¯ i¯ i 100¯ i0-¯ i0 10¯ i00-0i 1-¯ ii--ii 1-11--101-01001 0010000001-00-1¯ i1i¯ i-i1¯ ii-¯ i--¯ i 1¯ i-ii-¯ i1 11--i¯ i¯ ii 11-1i¯ i¯ i¯ i 1i-ii1¯ i1 1¯ i¯ i-i-1i 11¯ i¯ ii¯ i11-¯ i¯ iii11¯ i¯ i1i-1¯ i 001¯ i00i001100i¯ i 0¯ i100-i0 01100¯ ii0 00¯ ii1-00 1-i¯ i-1i¯ i 00¯ i-1¯ i00 1¯ ii1-ii1 000¯ i0100 010i0-0i 11-1i¯ iii 1i¯ i1i1-i 11¯ ii-1ii 0-0¯ i10¯ i0 1¯ i-¯ i1¯ i1i 00-¯ i1¯ i00 0i100i-0 0¯ i000010 1i1¯ i¯ i-i1-i¯ i¯ i¯ i-1i¯ i--ii10¯ i00i01-111--1-0000-0-100--0 00100-00 100¯ i0-i0 11¯ i¯ i1-i¯ i 10010¯ ii0 1¯ ii1-¯ ii1-¯ i¯ i1-ii 00¯ i¯ i1-00 1¯ i¯ i11¯ ii1¯ i0000i1-0000-1 1--1-1-1 1¯ i0000-i 1¯ i-i-i-i 0100000010-0-01-¯ i¯ i¯ i¯ i11i-¯ i¯ i-¯ i1-----1010110-0 1¯ ii-i-1¯ i 1-1-ii¯ i¯ i 1¯ i-¯ i-¯ i1¯ i 10-00¯ i0¯ i 0-1010011---1-1 0¯ i10100¯ i 1¯ i1¯ i1i-¯ i 1-¯ iii¯ i-1 1-1-i¯ i¯ ii 1¯ i¯ i-i1-i 1¯ i1¯ ii1¯ i010¯ i0i001010i0¯ i 1--10000 1-¯ ii0000 001-0000 1¯ i-i0000 1¯ i¯ i-0000 001¯ i0000 010-0000 010¯ i0000 00010000 0-100000 1i¯ i-0000 100¯ i0000 1-¯ i¯ i0000 1¯ i-¯ i0000 11-10000

π/3 π/2.59 π/2.59 π/2.59 ⊥ π/3 π/2.59 π/3 π/2.59 π/2.59 π/2.59 π/2.59 π/2.59 ⊥ ⊥ π/2.59 π/2.59 ⊥ π/2.59 π/2.59 ⊥ ⊥ ⊥ ⊥ ⊥ π/2.59 π/3 π/2.59 π/2.59 π/2.59 ⊥ π/2.59 π/2.59 π/3 ⊥ π/2.59 π/2.59 ⊥ ⊥ ⊥ π/2.59 π/2.59 π/3 π/2.59 π/2.59 π/3 π/3 π/2.59 π/2.59 ⊥ ⊥ ⊥ π/2.59 π/2.59 π/2.59 π/2.59 π/2.59 π/2.59 π/2.59 π/2.59 π/2.59 π/2.59 ⊥ ⊥ ⊥ ⊥ ⊥ π/2.59 ⊥ π/2.59 ⊥ ⊥ π/2.59 π/2.59 π/2.59 ⊥ π/2.59 ⊥ ⊥ ⊥ π/2.59 π/2.59 π/2.59 π/3 π/2.59 π/3 ⊥ ⊥ π/3 π/2.59 π/3 π/2.59 π/2.59 ⊥ π/2.59 π/3 π/3 π/2.59 π/3 π/2.59 ⊥ ⊥ π/2.59 π/2.59 π/2.59 ⊥ π/2.59 π/2.59 π/2.59 π/3 ⊥ π/2.59 ⊥ π/2.59 π/2.59 π/2.59 π/2.59 π/2.59 ⊥ ⊥ π/3 π/3 ⊥ π/3 π/3 ⊥ ⊥ ⊥ ⊥ ⊥ π/3 π/4 π/3 π/3 π/3

1--1-111--1¯ iii¯ i 00001--1 1-¯ ii-1i¯ i 1-¯ ii¯ ii-1 00001-¯ ii 001-00-1 001-00¯ ii 000000100-11-00 1-i¯ i¯ ii-1 1-0000¯ ii 1--1¯ ii¯ ii 1-¯ ii-1¯ ii 1-1--111¯ i-i-i1¯ i 1¯ i-i¯ i-i1 00001¯ i-i 1¯ i¯ i--ii1 1¯ i¯ i-¯ i--i 00001¯ i¯ i001¯ i00-i 001¯ i00¯ i0000001¯ i 00-i1¯ i00 1¯ ii1¯ i--i 1¯ i0000¯ i1¯ i-i¯ i-¯ i1¯ i¯ i--i¯ i1¯ i1¯ i-i1¯ i 010-0-01 010-0¯ i0i 0000010010¯ i0-0i 010¯ i0¯ i00000010¯ i 00010000001000¯ i 00000001 000-0100 010i0¯ i00100000¯ i 010-0¯ i0¯ i 010¯ i0-0¯ i 01010-01 0-0110-0 1i-¯ i¯ i-i1 10-00¯ i0i 1--1¯ i¯ iii 1¯ i-i-¯ i1i 11---110-1001-0 0-100i¯ i0 00000-10 1i¯ i--¯ ii1 1i¯ i-¯ i1-i 00001i¯ i1i¯ i-¯ i-1i 100-0-10 1¯ ii--¯ ii1 1-1---11 0i¯ i010011--¯ ii¯ ii 100¯ i-00i 100¯ i¯ i000000100¯ i 1-¯ i¯ i-1ii 1-¯ i¯ i¯ ii-00001-¯ i¯ i 11i¯ ii¯ i11 100¯ i0¯ i10 1-¯ i¯ i11i¯ i 1i-¯ i1¯ i1¯ i 0-¯ i0100¯ i 1¯ i1¯ ii-¯ i1 1¯ i-¯ i-i1i 1¯ i-¯ i¯ i-i00001¯ i-¯ i 11-1--111-1¯ i¯ ii¯ i 000011-1 11-1-111 1i-i-i1i 1---¯ i¯ ii¯ i 0¯ i0¯ i10-0 1¯ i-¯ i¯ i1i1 10-00101 0-0i10¯ i0 1i¯ i1¯ i--i 10¯ i00¯ i01-¯ ii¯ i¯ i-1¯ i¯ i--¯ ii11¯ i¯ i-1i¯ i 1¯ i-¯ ii-i1 0-i0100¯ i 1i1¯ i1¯ i-¯ i 1-i¯ i11¯ i¯ i 100¯ i0¯ i-0 11¯ i¯ ii¯ i-1 1-11¯ i¯ ii¯ i 0¯ ii01001 11-1-1-1i¯ i1-i¯ i100101-0 1¯ ii1¯ i1-¯ i 1-¯ i¯ i--i¯ i 1¯ i¯ i1-¯ ii1 11¯ ii¯ ii-0i0-10¯ i0 1i¯ i-¯ i--¯ i 10¯ i00-0¯ i 000-0010 001i00¯ i0010000¯ i 001-00¯ i¯ i 001¯ i00-¯ i 001100-1 1---1-11 1-ii1-¯ i¯ i 1-11i¯ i¯ i¯ i 1-0000¯ i¯ i 1-¯ i¯ ii¯ i-00--1-00 1¯ i1i1¯ i-¯ i 1¯ ii-1¯ i¯ i1 1¯ i-¯ ii1i00i-1¯ i00 1¯ i¯ i1i1-¯ i 1¯ i0000-¯ i 0¯ i100¯ i-0 011001-0 0i100-¯ i0 0-000010 0-100¯ i¯ i0 00100¯ i00

π/2.59 π/2.59 ⊥ π/2.59 π/2.59 ⊥ ⊥ ⊥ ⊥ ⊥ π/2.59 π/3 π/2.59 π/2.59 π/2.59 π/2.59 π/2.59 ⊥ π/2.59 π/2.59 ⊥ ⊥ ⊥ ⊥ ⊥ π/2.59 π/3 π/2.59 π/2.59 π/2.59 ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ π/2.59 π/3 π/2.59 π/2.59 π/2.59 ⊥ ⊥ ⊥ π/2.59 π/2.59 ⊥ π/2.59 π/3 π/2.59 π/2.59 ⊥ π/2.59 π/3 π/3 ⊥ π/2.59 π/2.59 ⊥ π/2.59 π/3 π/2.59 π/2.59 ⊥ π/2.59 π/2.59 π/2.59 ⊥ π/2.59 π/2.59 ⊥ π/2.59 π/2.59 π/2.59 ⊥ π/2.59 π/3 ⊥ π/2.59 π/3 π/2.59 π/2.59 π/2.59 π/2.59 ⊥ π/2.59 π/2.59 π/3 π/2.59 π/2.59 ⊥ π/2.59 π/2.59 π/3 π/2.59 π/2.59 π/2.59 π/2.59 ⊥ π/2.59 π/3 ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ π/2.59 π/2.59 π/2.59 π/3 π/2.59 ⊥ π/2.59 π/2.59 π/2.59 ⊥ π/2.59 π/3 ⊥ ⊥ ⊥ ⊥ ⊥ ⊥

14