EE 599 - Quantum Information and Quantum Computation Topics for Individual Projects Communication Protocols Quantum channel capacity. This is several problems: how much classical information can be sent using quantum channels? How much if we also all
UNIVERSITY OF SOUTHERN CALIFORNIA DEPARTMENT OF ELECTRICAL ENGINEERING EE 520 Introduction to Quantum Information Processing Phone: (213) 740-3503 Email: tbrun@usc.edu
Instructor: Prof. Todd A. Brun Ofce: EEB 502 Ofce hours: Mon 2-4, Tue 10-11:30 Te
Measurement and Interference
Let us recall what we learned about projective
measurements. Any ideal physical measurement on a
system with a D-dimensional Hilbert space
corresponds to an observable.
=O
.
An observable is a D D Hermitian operator O
Such op
LC Circuits
A type of circuit that is well-known from classical circuit
theory is the LC circuit, in which an inductor and a
capacitor cause oscillations in the flux of a circuit loop:
The energy function for this circuit can be written
Q2 2
1
H=
.
+
, =
Ensembles and incomplete information
So far in this course, we have described quantum
systems by states that are normalized vectors in a
complex Hilbert space. This works so long as (a) the
system is not entangled with anything else, and (b) we
have compl
Phase estimation
Last time we saw how the quantum Fourier transform
made it possible to find the period of a function by
repeated measurements and the greatest common
divisor (GCD) algorithm. We will now look at this same
problem again, but using the QFT
Historical Overview of QM
Quantum Mechanics is now over 100 years old, and is
one of the most successful scientific theories ever
created. We believe it to be the underpinning of all
physical laws. But at ordinary human scales, its effects
are almost tota
Requirements for scaleable QIP
These requirements were presented in a very influential
paper by David Divincenzo, and are widely used to
determine if a particular physical system could potentially
be used to build a scaleable quantum computer.
1. Existenc
The Stern-Gerlach experiment and spin
Experiments in the early 1920s discovered a new aspect of
nature, and at the same time found the simplest quantum
system in existence. In the Stern-Gerlach experiment, a
beam of hot atoms is passed through a nonunifor
Generating Arbitrary Unitaries
In the quantum circuit model, all computations are done
by applying a succession of quantum gates. These are
unitary transformations acting on one, two or three
q-bits at a time. By applying them successively, one
builds up
The Postulates of Quantum Mechanics
We have reviewed the mathematics (complex linear
algebra) necessary to understand quantum mechanics.
We will now see how the physics of quantum mechanics
fits into this mathematical framework.
We are really defining the
Complex Linear Algebra
The basic mathematical objects in quantum mechanics
are state vectors and linear operators (matrices). Because
the theory is fundamentally linear, and the probability
amplitudes are complex numbers, the mathematics
underlying quantu
Classical error correction
The most general classical single-bit error is the bit-flip:
0 1. We will assume a simple error model (the binary
symmetric channel) in which bit flips errors occur on each
bit independently with probablility p per unit time. We
QIP with light
One of the most fruitful avenues for quantum
information so far has been the use of photons to carry
quantum information. This draws upon the field of
quantum optics, where some of the most important
experiments to date have been performed.
Other Topics in Quantum Information
In a course like this there is only a limited time, and only a
limited number of topics can be covered. Some additional
topics will be covered in the class projects. In this lecture, I
will go (briefly) through several
Topics for EE 520 Midterm Exam Spin-1/2 particles and q-bits. Dirac notation. Bases. Inner and outer products. Operators: unitary, Hermitian, projectors, positive. Tensor products of states and operators. Postulates of quantum mechanics. Quantum regi
EE 520: Quantum Information Processing Solution to Homework # 1
Exercise 2.2
In terms of the basis {|0 , |1 } A= 01 10 = |1 0| + |0 1|.
We can transform by choosing a dierent basis. For instance, in the basis A | = (|0 |1 )/ 2 A= 10 0 1 = |+ +|
Modular arithmetic
Much of modern number theory, and many practical
problems (including problems in cryptography and
computer science), are concerned with modular arithmetic.
While this is probably familiar to most people taking this
course, I will review
More advanced codes
The Shor code was the first general-purpose quantum
error-correcting code, but since then many others have
been discovered. An important example, discovered
independently of the Shor code, is the seven-bit Steane code:
1
|0i |0000000i
Classical Information and Bits
The simplest variable that can carry information is the
bit, which can take only two values, 0 or 1. Any
ensemble, description, or set of discrete values can be
quantified by the number of bits needed to express it
(as can c
Quantum trajectories
A quantum system undergoing decoherence can be
described as evolving by a sequence of completely
positive maps:
X
X
Ak Ak
Akn Ak1 Ak1 Akn ,
k
k1 ,.,kn
where the set of operators cfw_Ak satisfy
X
Ak Ak = I.
k
This condition is just
Quantum Fourier Transform
This lecture will concentrate almost entirely upon a
single unitary transformation: the quantum Fourier
transform. This is a discrete Fourier transform, not upon
the data stored in the system state, but upon the state
itself.
Let
Searching an unordered database
In searching for a needle in a haystack, one might hope
that it pays to be systematic. Unfortunately, it does not.
Let f (x) be a function whose argument is an integer
0 x N 1, and which returns 1 for exactly one value
x1 ;
Indirect measurement
Because in practice it is often difficult to directly measure a
quantum system without destroying it, most actual
measurement are done indirectly.
1. Prepare an extra system (an ancilla) in a known initial state.
2. Have the system an
Unitary time evolution
Time evolution of quantum systems is always given by
Unitary Transformations. If the state of a quantum system is
|i, then at a later time
|i U |i.
Exactly what this operator U is will depend on the
particular system and the interac
Decoherence
The Schrdinger equation describes the evolution of
quantum systems in isolation. These closed systems have
a well-defined Hamiltonian, which gives complete
information about how these systems evolve. The
resulting evolution, as we have seen, i
Quantum Circuits
We have already seen the notation for quantum circuits,
quite analogous to that for classical Boolean circuits.
Like classical reversible circuits, the number of quantum
bits in must equal the number out. (The exception is if
we allow des