Ph219a/CS219a
Solutions to Hw 7 May 5, 2009
Problem 1
(a) We can recall the action of the Hadamard (H = H-1 ) and the phase gates (P) on the Pauli operators from an earlier Hw problem (HW4, Problem 3a), where we had shown HXH-1 PXP
-1
= Z , HYH-1 = -Y , H
Ph219a/CS219a
Solutions to Hw 8 May 5, 2009
Problem 1
(a) We can write Xa as a product of X 's and Z 's:
n n
Xa = (sgn)
=1
u X
=1
Z , u , v cfw_0, 1
v
(1)
Then, XA,a XB,a can be written as
n
(XA, XB, )u
n
(ZA, ZB, )v (2)
XA,a XB,a =
=1
=1
Theref
Ph219a/CS219a
Solutions to Hw 9 June 5, 2009
Problem 1
(a) Clearly for x = 1, lnx = x - 1 = 0. Since the function f (x) = lnx is strictly concave, lnx x - 1 for x = 1, if x - 1 is a tangent at x = 1. It is easy to see that this is indeed the 1 case, since