Ph219a/CS219a
Solutions
Nov 8, 2013
Problem 2.1
(a) Consider how the protocol works for the special case n = 1, ie. when Alice encrypts a single
qubit state . The eect of Alices encryption protocol can then be viewed as a quantum channel
which applies one
1
Ph 219a/CS 219a
Exercises
Due: Wednesday 23 October 2013
1.1 How far apart are two quantum states?
Consider two quantum states described by density operators and
in an N dimensional Hilbert space, and consider the complete or
thogonal measurement cfw_
1
Ph 219b/CS 219b
Exercises
Due: Wednesday 4 December 2013
4.1 The peak in the Fourier transform
In the period nding algorithm we prepared the periodic state
1
A
A1
j=0
x0 + jr ,
(1)
where A is the least integer greater than N/r; then we performed the
qu
Ph219a/CS219a
Solutions to Hw 4
December 2013
Problem 4.1
(a) Simple trigonometry tells us
Prob(y) =
sin2 Ar
sin2 r
1
NA
1
NA
1
sin r
(1)
2
Further, note that1
sin x
2
x
sin2 x
x2
]
2
4
, x [
, ]
2
2 2
1 , x [0,
Therefore,
sin2 (r) (r)2
(2)
4
, since
Ph219a/CS219a
Solutions
Nov 8, 2013
Problem 1.1
(a) Since pa = Tr(Ea ) and pa = Tr(Ea ),
N
1
1
pa pa  = Tr( ( )Ea )
2 a=1
2
d(p, p) =
Writing in its eigenbasis, we have =
i
(1)
i i i, so that
N
d(p, p)
Tr( (
=
i i i)Ea )
a=1
i
N
=
=
1

2 a=1
1
Ph219a/CS219a
Solutions of Problem Set 7
March 18, 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,
Ph219a/CS219a
Solutions of Problem Set 6
March 16, 2014
Problem 1
(a) We can write Xa as a product of X s and Z s:
n
n
Xa = (sgn)
u
X
=1
v
Z , u , v cfw_0, 1
(1)
=1
Then, XA,a XB,a can be written as
n
XA,a XB,a =
(XA, XB, )u
=1
n
(ZA, ZB, )v
(2)
=1
Ther
Ph219a/CS219a
Solutions
Nov 8, 2013
Problem 3.1
(a) Constructing the unitary transformation U1 as given in the circuit, we have U1 = (H
I)(P)(H I). In the standard basis, H and P are given by
1
H=
2
1
1
1
1
P =
1
1
0
i
(1)
so that H I and (P) have the b
1
Ph 219b/CS 219b
Exercises
Due: Wednesday 20 November 2013
3.1 Universal quantum gates I
In this exercise and the two that follow, we will establish that several
simple sets of gates are universal for quantum computation.
The Hadamard transformation H is
1
Ph 219a/CS 219a
Exercises
Due: Wednesday 6 November 2013
2.1 The price of quantum state encryption
Alice and Bob are working on a top secret project. I cant tell you
exactly what the project is, but I will reveal that Alice and Bob are
connected by a pe
Ph219a/CS219a
Solutions 5
Feb 5, 2014
Problem 5.1
(a) The derivation of the GilbertVarshamov (GV) bound for CSS codes follows closely the
argument discussed in class for the general GV bound, except that we want to include the
specic property that CSS co