Research notes on universality v2.4 Andrew Childs, Debbie Leung, Laura Mancinska, and Maris Ozols February 17, 2009 Contents 1 Introduction3 1.1 The T gate..............................3 1.2 Definition of universality.......................3 2 T -similarity4 2.1 Basic properties of the T gate....................4 2.2 T -similarity and the pattern.....................5 2.3 Tridiagonal form...........................9 3 Universality conditions 12 3.1 Sufficient conditions for non-universality.............. 12 3.2 The \not so ugly" commutator scheme............... 12 4 Universality for 3 qubits 14 4.1 Universality conditions........................ 14 4.1.1 Central symmetry...................... 14 4.1.2 Block structure........................ 17 4.1.3 Conjugation by U ⊗ U .................... 17 4.2 Invariant subspaces for 3 qubits................... 18 4.2.1 Dimension 1.......................... 19 4.2.2 Dimension 2.......................... 19 4.3 Possible generalizations of T -similarity............... 20 4.4 Unitary transformations that preserve universality........ 20 4.4.1 Centralizer of Sn ....................... 20 4.4.2 Normalizer of Sn ....................... 22 4.4.3 \Monsterizer" of Sn ..................... 22 4.5 Pauli basis............................... 22 4.5.1 From 2 to 3 qubits...................... 22 1 5 Schur basis for Hamiltonians 25 5.1 Decomposition of a 3-qubit Hamiltonian.............. 25 5.2 Schur basis using Pauli matrices.................. 26 5.2.1 Different view of conjugation................ 26 5.2.2 Pauli vector.......................... 26 6 Case studies for 3 qubits 28 6.1 Dimension 15............................. 28 6.1.1 XX; Y Y ............................ 28 6.1.2 XX; Y Z + ZY ........................ 29 6.1.3 XI;YZ + ZY ......................... 29 6.2 Dimension 17............................. 29 6.2.1 XX; Y Z − ZY ........................ 29 6.3 Dimension 28............................. 30 6.3.1 XI;XY ............................ 30 6.4 Dimension 30............................. 31 6.4.1 XI;YZ ............................ 31 6.4.2 XI + IX; Y Z ......................... 31 A Case study (C-NC-U gate) 32 A.1 The Hamiltonian........................... 32 A.2 The big determinant......................... 32 A.3 Universality conditions........................ 33 A.4 Necessity of conditions........................ 34 A.4.1 Zero trace........................... 34 A.4.2 Degenerate eigenvalues.................... 34 A.4.3 Paired eigenvalues...................... 34 B U-similarity 36 C Some simple lemmas 37 D Lie groups and Lie algebras 37 D.1 O(2).................................. 37 D.1.1 Rotation............................ 38 D.1.2 Reflection........................... 38 D.2 O(n).................................. 39 2 1 Introduction 1.1 The T gate Throughout the paper T will denote the gate that swaps two qubits: 01 0 0 01 B0 0 1 0C T := B C : (1) @0 1 0 0A 0 0 0 1 The eigenvalues of T are ±1. The (+1)-eigenspace of T is spanned by1 011 001 001 B0C B1C B0C B C ; B C ; B C : (2) @0A @1A @0A 0 0 1 The (−1)-eigenspace of T is spanned by the singlet state 0 0 1 1 B 1 C jsi := p B C : (3) 2 @−1A 0 Sometimes it will be more convenient to work in a basis where T is diagonal. We will use the following diagonal form: 0−1 0 0 01 B 0 1 0 0C T~ := B C : (4) @ 0 0 1 0A 0 0 0 1 Then the corresponding singlet state is 011 B0C js~i := B C : (5) @0A 0 1.2 Definition of universality Definition. The set generated by U 2 U(4) consists of all finite products of U and TUT . Definition. The set generated by Hamiltonian H consists of all finite products of U(t) := e−iHt and TU(t)T , where any t ≥ 0 can be used for each U(t).2 1Here we could use the Bell basis as well. 2We might sometimes drop the minus sign and define U(t) as eiHt instead. 3 Claim. For each t < 0 there exists t0 > 0, such that U(t) = U(t0). Therefore we can drop the restriction t ≥ 0 in the previous definition. Lemma 1. The set generated by Hamiltonian H is a group. Proof. We can generated the identity, since U(0) = I. We can also get the inverse of any element, since U(t)−1 = U(−t). Lemma 2. The group generated by U is closed under conjugation by T . Proof. Let U0 = U and U1 = TUT . If we can generate a matrix V , then Qn n Qn Qn V = i=1 Uxi for some x 2 f0; 1g . Then TVT = i=1 TUxi T = i=1 Ux¯i corresponds tox ¯ { the bitwise negation of x. Hence we are interested only in the subgroups of U(4) that are closed under conjugation by T . Note that the Lie algebra of such subgroup is also closed under conjugation by T . Definition. We say that U 2 U(4) is universal, if it generates a set that is dense in U(4). In other words, U and TUT can be used to approximate any unitary V 2 U(4) to any desired accuracy. The universality of Hamiltonian H, is defined similarly. 2 T -similarity 2.1 Basic properties of the T gate Let H be a Hamiltonian. Lemma 3. The following statements are equivalent: (1) H commutes with T , (2) there is an orthonormal basis, such that both H and T are diagonal, (3) jsi is an eigenvector of H, (4) H is of the form 0 1 x1 x7 + ix8 x7 + ix8 x5 + ix6 Bx7 − ix8 x2 x4 x9 + ix10C B C (6) @x7 − ix8 x4 x2 x9 + ix10A x5 − ix6 x9 − ix10 x9 − ix10 x3 for some x1; : : : ; x10 2 R. Lemma 4. The following statements are equivalent: (1) U and T have a common non-trivial invariant subspace, 4 (2) U and T have a common eigenvector, (3) U has an eigenvector orthogonal to jsi. Lemma 5. E ⊆ Cn is an invariant subspace of M 2 Cn×n if and only if n E = span fjv1i ;:::; jvkig, where jv1i ;:::; jvki 2 C are eigenvectors of M and k = dim E. 2.2 T -similarity and the pattern Observe that all unitary matrices that commute with T form a group. Let us denote this group by C: C := fU 2 U(4)jUT = TUg : (7) Definition. Matrices A and B are called T -similar, if there is P 2 C, such that B = P AP y. We will also say that two sets of matrices are T -similar if one set can be obtained from other by conjugating all elements with some fixed P 2 C. Observe that T -similarity is an equivalence relation and it partitions the set U(4) into equivalence classes. The notion of T -similarity is central to our problem, since T -similar matrices generate the same subgroup, up to a change of basis. Claim. If U and V are T -similar, then they generate T -similar subgroups. Definition. Assume M 2 U(4) has non-degenerate eigenvalues λi with corre- 2 sponding eigenvectors j ii. We call si = jhsj iij the overlap of j ii with jsi. We define the pattern of M to be λ λ λ λ 1 2 3 4 : (8) s1 s2 s3 s4 We will use patterns only of the matrices that do not have degenerate eigen- values. In such case the pattern is well defined (up to permutations of columns). Note that s1 + s2 + s3 + s4 = 1. Theorem 1. Assume U and V does not have degenerate eigenvalues. Then U and V are T -similar if and only if they have the same patterns. [old version] (T1) U and V have the same eigenvalues, (T2) the eigenvectors of U and V corresponding to the same eigenvalue have equal absolute values of the inner product with jsi. Proof. It is clear that conditions (T1) and (T2) are necessary. We will show that they are also sufficient. Assume (T1) and (T2) hold. There exists P 2 U(4) such that PUP y = V , since U and V have the same eigenvalues. Let ei'i be the eigenvalues of U and V 5 and juii and jvii be the corresponding eigenvectors. Let ri := jhsjuiij = jhsjviij. We can express the singlet state jsi in the eigenbasis of U: X iαi jsi = rie juii : (9) i Since P juii is an eigenvector of V , we have jhsj P juiij = ri and iβi hsj P juii = rie (10) for some phase βi. Define 0 X −i(αi+βi) P := P e juii huij : (11) i We claim that (a) P 0UP 0y = V , (b) jhsj P 0 jsij = 1. (a) We have: 0 1 ! 0 0y X −i(αi+βi) X i'j P UP = P e juii huij @ e juji hujjA (12) i j ! X i(αk+βk) y e juki hukj P (13) k 0 1 X B −i(αi+βi)+i'j +i(αk+βk) C y =P @ e juii huijuji hujjuki hukjA P i;j;k | {z } δi;j;k (14) ! X i'i y y =P e juii huij P = PUP = V: (15) i (b) We have: 0 X iαi 0 hsj P jsi = e ri hsj P juii (16) i X iαi X −i(αj +βj ) = e ri hsj P e juji hujjuii (17) i j X iαi −i(αi+βi) X −iβi = e ri hsj P e juii = e ri hsj P juii : (18) i i By applying (10) we get 0 X 2 hsj P jsi = ri = 1: (19) i 6 Part (a) tells us that U and V are similar via P 0. From (b) it follows that jsi is an eigenvector of P 0. Thus according to Lemma3, P 0 commutes with T . Hence, U and V are T -similar. Theorem 2.