Ph219a/CS219a
Solutions
Nov 8, 2013
Tr
ia
l
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
1
1
1
I
I
I
I
0
i
ud
so that H I and (P) have the bl
Ph219a/CS219a
Solutions of Problem Set 7
March 18, 2009
Tr
ia
l
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
Ph219a/CS219a
Solutions
Nov 8, 2013
Tr
ia
l
Problem 1.1
(a) Since pa = Tr(Ea ) and pa = Tr(Ea ),
N
1
1
|pa pa | = |Tr( ( )Ea )|
2 a=1
2
Writing in its eigen-basis, we have =
i
i |i i|, so that
ud
N
d(p, p)
|Tr( (
=
i |i i|)Ea )|
a=1
i
1
2
N
i| Ea |i ) (Tr
Studying quantum systems using a quantum computer Ph/CS 219, 23 February 2009
Estimating energy eigenvalues and preparing energy eigenstates Ph/CS 219, 2 March 2009 We have argued that a quantum computer can efficiently simulate the time evolution of a qu
de
sk
PD
F
St
ud
io
Tr
ia
l
Ph/CS 219c
2 March 2011
Accessible information
de
F
PD
sk
io
ud
St
l
Tr
ia
de
F
PD
sk
io
ud
St
l
Tr
ia
de
F
PD
sk
io
ud
St
l
Tr
ia
de
F
PD
sk
io
ud
St
l
Tr
ia
de
F
PD
sk
io
ud
St
l
Tr
ia
de
F
PD
sk
io
ud
St
l
Tr
ia
de
F
PD
sk
i
F
PD
sk
de
io
ud
St
Tr
ia
l
F
PD
sk
de
io
ud
St
Tr
ia
l
F
PD
sk
de
io
ud
St
Tr
ia
l
F
PD
sk
de
io
ud
St
Tr
ia
l
F
PD
sk
de
io
ud
St
Tr
ia
l
F
PD
sk
de
io
ud
St
Tr
ia
l
F
PD
sk
de
io
ud
St
Tr
ia
l
F
PD
sk
de
io
ud
St
Tr
ia
l
F
PD
sk
de
io
ud
St
Tr
ia
l
F
P
1
Ph 219c/CS 219c
Exercises
Due: Thursday 22 April 2004
1.1 Good CSS codes
In class we derived the quantum Gilbert-Varshamov bound:
|E (2) | 1 <
22n 1
.
2n+k 2nk
(1)
This is a sucient condition for the existence of a (possibly degenerate)
binary stabilize
Ph219a/CS219a
Solutions of Problem Set 6
March 16, 2014
l
Problem 1
n
n
Xa = (sgn)
u
X
=1
v
Z , u , v cfw_0, 1
=1
io
Then, XA,a XB,a can be written as
n
(XA, XB, )u
n
=1
(1)
(ZA, ZB, )v
ud
XA,a XB,a =
Tr
ia
(a) We can write Xa as a product of X s and Z
Ph219a/CS219a
Solutions
Nov 8, 2013
Tr
ia
l
Problem 2.1
io
(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
Ph219a/CS219a
Solutions 5
Feb 5, 2014
Tr
ia
l
Problem 5.1
io
(a) The derivation of the Gilbert-Varshamov (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
Ph219a/CS219a
Solutions to Hw 4
December 2013
Problem 4.1
sin2 Ar
sin2 r
1
NA
Tr
ia
Prob(y) =
l
(a) Simple trigonometry tells us
1
NA
1
sin r
Further, note that1
sin x
2
x
sin2 x
x2
]
2
4
, x [
, ]
2
2 2
4
, since r [
, ]
2
2 2
St
sin2 (r) (r)2
Thus w
Fault-tolerant quantum gates
Ph/CS 219
2 February 2011
Last time we considered the requirements for fault-tolerant quantum gates that act nontrivially on the codespace of a
quantum error-correcting code. In the special case of a code that corrects t=1 err
Ph/CS 219c, 24 January 2011
Toric code recovery
Last time we discussed the toric code. This is a CSS code, where the qubits are associated with the edges of an
L X L square lattice on a 2D torus (i.e., with periodic boundary conditions). The Z-type genera