1
Ph 219b/CS 219b
Exercises Due: Friday 3 February 2006 5.1 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 the
1
Ph 219a/CS 219a
Exercises Due: Wednesday 26 October 2005 1.1 Alice does Bob a favor Alice, in Anaheim, and Bob, in Boston, share a bipartite pure state  , which can be expressed in the Schmidt form  =
i
pi i i ,
(1)
where cfw_i is an orthonormal b
Errata list for Quantum Computation and
Quantum Information
Michael A. Nielsen and Isaac L. Chuang
October 7, 2004
Thanks to all the people whove contributed to this errata list: Scott
Aaronson, Marcus Curty Alonso, Andris Ambainis, Dave Bacon, Mahesh
Ban
1
Ph 219a/CS 219a
Exercises Due: Wednesday 12 November 2008 2.1 Which state did Alice make? Consider a game in which Alice prepares one of two possible states: either 1 with a priori probability p1 , or 2 with a priori probability p2 = 1  p1 . Bob is to
1
Ph 219c/CS 219c
Exercises Due: Monday 1 June 2009 9.1 Positivity of quantum relative entropy b) The (classical) relative entropy of a probability distribution cfw_p(x) relative to cfw_q(x) is defined as H(p Show that H(p q) 0 , (2) with equality iff the
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Ph/CS 219, 4 March 2009 "Quantum MerlinArthur and the local Hamiltonian problem"
As we have discussed, we expect that quantum computers can compute the ground state energy of local Hamiltonians in cases where the problem is hard classically; this may be
Nonabelian Hidden Subgroup Problem Ph/CS 219, 9 February 2009
Last time we discussed the HSP for finitely generated abelian groups. For a black box function f: G > X that is constant and distinct on the cosets of H < G, classically it takes
Now a natural
Ph/CS 219c
2 March 2011
Accessible information
Ph/CS 219
2 March 2011
Jointly typical sequences
Here is a little more detail about the proof of the Noisy Channel Coding Theorem, specifically the proof
that the mutual information is an achievable rate. (Ba
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 Ztype genera
Ph/CS 219
2 March 2011
Jointly typical sequences
Here is a little more detail about the proof of the Noisy Channel Coding Theorem, specifically the proof
that the mutual information is an achievable rate. (Based on Chapter 8 of Cover and Thomas.)
A noisy
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
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
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 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
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
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
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 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 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 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 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
1
Ph 219c/CS 219c
Exercises
Due: Wednesday 12 March 2014
7.1 Positivity of quantum relative entropy
a) Show that ln x x1 for all positive real x, with equality i x = 1.
b) The (classical) relative entropy of a probability distribution cfw_p(x)
relative to
1
Ph 219b/CS 219b
Exercises
Due: Wednesday 5 February 2014
5.1 Good CSS codes
In class we derived the quantum GilbertVarshamov bound:
E (2) 1 <
22n 1
.
2n+k 2nk
(1)
This is a sucient condition for the existence of a (possibly degenerate)
binary stabili