C191
Problem Set 2
Out: January 24, 2012
1. Consider the state = 1/ 2 0 + ei / 2 1 . To estimate the phase , we measure in the
sign basis. Analyze the probability that the outcome of the measurement is + .
2. Prove that the Bell state is rotationally inva
1
The secular approximation
1.1
Rotating frame
Dene
h
| (t) = eiH1 t/ |(t)
We will seek an equation governing time evolution for | (t) . First, lets write
the ordinary Schrodinger equation in terms of | (t) :
i
h
d
|(t) =(H0 + H1 )|(t)
dt
d iH1 t/
h
h
(e
1
Unitarity of a Fourier Transform
The Fourier transform mod N is the N N matrix given by
1
1
1
1
FTN = .
N .
.
1
2
1
2
4
.
.
.
.
.
.
1
N 1
2(N 1)
.
.
.
.
.
,
(1)
2
1 N 1 2(N 1) (N 1)
where = e2i/N is a primitive N th root of unity. That is, the i, jt
C191 - Lecture 2 - Quantum states and observables
I.
ENTANGLED STATES
We saw last time that quantum mechanics allows for systems to be in superpositions of basis states. Many of these
superpositions possess a uniquely quantum feature known as entanglement
Spin
1
Introduction
For the past few weeks you have been learning about the mathematical and algorithmic background of
quantum computation. Over the course of the next couple of lectures, well discuss the physics of making
measurements and performing qubi
Spin Resonance (ESR, NMR, SR, etc.)
1
Electron Spin Resonance
Nearly every important concept in quantum computing can be illustrated with nuclear magnetic resonance
(NMR). The rst quantum factoring algorithm with implemented with NMR quantum computing, an
CS191 Fall 2014
Homework 7 solutions
1. Error correction for a mixture of errors. Suppose | = |000 + |111 is a general single qubit state
encoded in the bit ip code. Then, due to errors it is mapped to the following mixed state:
p
= (1 p)0 + (X1 0 X1 + X
CS191 Fall 2014
Homework 6 solutions
1. Gaussian integral. Show that:
1
2 2
e
(0 )2
2 2
it
d = e
(t)2
2
i0 t
(1)
We begin by completing the square in the exponent
( 0 )2
it
2 2
1
2
2 2 20 + i2t 2 + 0
2
1
2
2 2 2(0 it 2 ) + 0 + (t2 4 i2t0 2 ) (t2 4 i2t0
1
The Cherno bound
1.1
Dene the random variable Xi to represent the i-th coin toss. Let Xi = 1 when
the coin is heads, and Xi = 0 when the coin is tails. Then for n coin ips, the
sum X =
Xi = 2n/3.
If less than half of the ips come out heads, then X < n/2
CS191 Fall 2014
Homework 4 solutions
1. Fidelity calculation.
I
= p + (1 p)| |
d
(a) F (, | |) =
| =
p
d
p
1 (p d )
+1p=
(b) For xed p this delity decreases as d increases. Intuitively this is because as the dimension of the state
space increases, there
CS191 Fall 2014
Homework 5 solutions
1. The six state protocol. Let the tuple (a, b, e) represent the basis that Alice, Bob and Eve measure/prepare in.
After sifting the only events that are kept (out of the 27 possible events) have one of the following c
Chapter 1
Qubits and Quantum
Measurement
1.1
The Double Slit Experiment
A great deal of insight into the quantum theory can be gleaned by addressing
the question, is light transmitted by particles or waves? Until quite recently,
the evidence strongly favo
Chapter 3
Observables
3.1
Observables
An observable is an operator that corresponds to a physical quantity, such as
energy, spin, or position, that can be measured; think of a measuring device
with a pointer from which you can read o a real number which i
C191 - Lecture 1
I.
AN INCOMPLETE LIST OF THE AXIOMS OF QUANTUM MECHANICS
1. Every physical system is associated with a Hilbert space. A Hilbert space is a vector space (collection of objects
that can be added together and multiplied by scalars) together
C191
Problem Set 5
Out: February 14, 2012
1
1. The Pauli spin matrices represent the spin observable for a spin- 2 particle along the x, y and z axes of
physical 3-space:
Find the eigenvectors, eigenvalues, and diagonal representations of the Pauli matri
CS 191 Homework Set - Two Level Systems and The NMR Quantum Computer
Here are a few things you might need for this assignment. The Pauli matrices are
0 1
1 0
x X =
,
y Y =
0 i
i 0
,
z Z =
1 0
0 1
.
The Bloch sphere representation is dened in terms of the
C191
Problem Set 7
Out: February 28, 2012
1. Consider a system of 3 qubits. If we apply a CNOT gate to the rst two qubits and a phase
ip Z to the last qubit, write the unitary transformation that is applied to the composite
system.
2. Suppose you have an
C191
Problem Set 4
Out: February 6, 2012
You should think of this problem set as review problems for Midterm I. It is due Friday February 10 at 5
pm.
1. Suppose you have an unlimited supply of qubits in the state = 0 + 1 and qubits in the
state 0 . Give q
C191
Problem Set 6
Out: February 21, 2012
1. Evaluate the commutators [A, B] for the following pairs of operators A and B:
d2
,B=x
dx2
d
d
= dx x, B = dx
i) A =
ii) A
+x
2. Consider operators acting on a set of functions fn (x) that are dened on the in
C191
Problem Set 1
Out: January 17, 2012
1
1. Write the state 2 0 + 1+2 2i 1 in standard vector notation. What is the result of measuring this state
in the standard ( 0 , 1 ) basis? Repeat for a measurement in the sign ( + , ) basis.
2. The goal of this q
Fourier Sampling
Consider a quantum circuit acting on n qubits, which applies a Hadamard gate to each qubit. i.e. the circuit
applies the unitary transformation H n , or H tensored with itself n times.
Another way to dene this unitary transformation H2n i
Simons Algorithm
Suppose we are given function 2 1 f : cfw_0, 1n cfw_0, 1n , specied by a black box, with the promise that
there is an a cfw_0, 1n with a = 0n such that
For all x f (x + a) = f (x).
If f (x) = f (y) then either x = y or y = x + a.
The ch
Reversible Computation
A quantum circuit acting on n qubits is described by an 2n 2n unitary operator U . Since U is unitary,
UU = U U = I. This implies that each quantum circuit has an inverse circuit which is the mirror image of
the original circuit and
Building an NMR Quantum Computer
Spin, the Stern-Gerlach Experiment, and the Bloch Sphere
Kevin Young
Berkeley Center for Quantum Information and Computation,
University of California, Berkeley, CA 94720
Scalable and Secure Systems Research and Developmen
Building an NMR Quantum Computer
Pauli Matrices and Spin Precession
Kevin Young
Berkeley Center for Quantum Information and Computation,
University of California, Berkeley, CA 94720
Scalable and Secure Systems Research and Development,
Sandia National Lab
C191
Problem Set 9
Out: March 13, 2012
1. In this question we will work through an example of QFTM for M = 6.
What is ?
What is QFT6 of
What is QFT6 of
What is QFT6 of
What is QFT6 of
2. Let =
1
M
1
(
2
1
(
2
1
(
3
1
(
3
0 + 3 )?
1 + 4 )?
0 + 2 +
C191
Problem Set 1
Out: January 17, 2012
1. Write the state 1 |0 + 1+2 2i |1 in standard vector notation. What is the result of measuring this state
2
in the standard (|0, |1) basis? Repeat for a measurement in the sign (|+, |) basis.
Solution:
Let | = 1
C191
1.
Problem Set 3
Out: January 31, 2012
(a) Consider a CNOT gate whose control qubit (rst input) is 1/ 2 0 + 1/ 2 1 and target qubit
(second input) is 1/ 2 0 1/ 2 1 . What are the states of the two output qubits? Repeat
when the control qubit is = 1/
C191
Problem Set 8
Out: February 28, 2012
1. Let f : cfw_0, 1n cfw_0, 1n be a bijection on the n-bit strings. We showed in class that if there is an
efcient classical circuit, say of size m (where size is the total number of gates in the circuit), for
com