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 U113. A particular algorithm for a quantum computer is comprised of 150 single qubit gates and 37 two-qubit gates. None of the gates evolve qubits simultaneously in time, that is, each gate evolution occurs in series in a different period of time. For evolution times, 100 single qubits gates require Ins, 50 require 1.5ns, and all two-qubit gates require 2.6 ns. Experimental results indicate that 97.5% of the time this program executes without a decoherence event occurring. Therefore, you can assume that the probability of decoherence for a particular quantum computer executing this program is 0.975. You should assume that the Redfield model applies for decoherence properties. a) (5 points) What is the decoherence time constant, 72, for this particular quantum program on this particular quantum computer?b) (5 points) It is further determined that the time constant for a dephasing event, To is one-fifth that of time constant, 71, for a depolarization event. Find the value of /1 for this particular quantum computer executing this particular program.c) (5 points) What is the probability that a decoherence event happens at time t = 100ns? d) (5 points) What is the probability that a depolarization event happens at time t = 100ns? e) (5 points) What is the probability that a dephasing event happens at time t = 100ns