Appendix A: Quantum Gate Names, Symbols and Transfer Matrices Gate Name Gate Symbol Transfer Matrix Pauli-X X or NOT x = Q al Pauli-Y Y or Y Y = [ ] Pauli-Z or 7. Phase S or S s = [ 9 7/8 or 22.5" T OF T T= | Hadamard or H Square-root- of-NOT V V =: 1 1+i 1-1 or V Square-root- of-NOT transpose i 1+i or V vt = zhi+i 0 0 0 Controlled-U Cu = 10 1 0 0 or L 0 101 0 U10 U115. The following circuit represents a Bell state generator. Unfortunately: the Bell state generator is constructed with a Hadaniard gate, Had\"! that cannot be ideally realized The transfer matrix for Harm\" is: H _ l 1+ED 1+Em] \"mmVIE 1+5\": 1+e11 A quantlun engineer wishes to model the Bell state generator with the following circuit where the E gate is a fault model that accounts for the non-ideal Hadarnard gate. a) (5 points) Find the transfer matrix E for the fault model gate. For full credit, show your derivation and give the explicit form of E. b) (5 points) Given that: -0.03 0.1 Eadd = 0.1 0.03 Find the probability that [ P(t3)) = 10) given the following circuit. - - - - - 14(to)) = 10) H. I'(t3)) Z - - - - 10(to)) = 10)c) (5 points) All actual gates with a systematic error. such as Humm. must obey the following identity in order to satisfy the principles of quantum mechanics. As an example of one such principle, all actual gates must be unitary. Hrrctuul : HE : Ell + Exx + 51!? + 62: Given that E _ U.03 0.1 \"5\" _ 0.1 003' Find the values of {ED em 6\" 6:}. For full credit. nd a solution where as many of the El: values are zero (0) as possible