University of Queensland Quantum Information and Computation Summer School

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University of Queensland Quantum Information
and Computation Summer School
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Overview of the school
Monday Basic background on quantum mechanics,
computer science, and information theory.
Tuesday, Wednesday, Thursday (morning), Friday
(morning) Pedagogical lectures on quantum
information and computation.
Monday, Tuesday, Wednesday Afternoon tutorial
sessions.
Worked problems
Q A
The name of your tutor, and a map showing your
tutorial room is in your summer school materials.
Wednesday evening Public lecture by Bob Clark
(UNSW).
Thursday, Friday Afternoon research seminars.
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Required Background
What you dont need quantum mechanics, computer
science, or information theory.
!
Dont be concerned if you dont know all this
stuff!
What you do need elementary linear algebra,
mathematical maturity of a third or fourth year
undergrad.
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Philosophy and goals of the school
Pedagogical lectures given by a small number of
lecturers.
Focus of the pedagogical lectures is on theory.
Worldwide, theory is ahead of experiment.
In Australia, experiment is ahead.
Too many experimental proposals to do justice.
Focus in pedagogical lectures is general
principles (Andrew White).
Australias experimental effort discussed in the
research seminars.
Broad aim is to bring you up to the cutting edge
of research in some parts of the field, and to
make it easier to get to the edge in other parts
of the field.
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Miscellany
A hardcopy of the viewgraphs will be handed out
each morning.
Many viewgraphs will be available
at http//www.physics.uq.edu.au/qicss
Lab tours Tuesday White Wednesday
Rubinsztein-Dunlop/Heckenberg Thursday Meredith
Computers available in computer room, level 1 of
Physics Annexe (building 6) between 1200pm and
230pm. Username qct1, qct2, qct3,
qct4 Password changEmE
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Questions?
Tutors Michael Bremner Paul Cochrane Chris
Dawson Jennifer Dodd Alexei Gilchrist Sarah
Morrison Duncan Mortimer Tobias Osborne Rodney
Polkinghorne Damian Pope
Organizing committee Michael Bremner Jennifer
Dodd Anna Rogers
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Quantum Mechanics IBasic Principles
Michael A. Nielsen University of Queensland
I aint no physicist but I know what matters -
Popeye the Sailor
Goal of this and the next lecture to introduce
all the basic elements of quantum mechanics,
using examples drawn from quantum information
science.
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What is quantum mechanics?
It is a framework for the development of physical
theories.
It is not a complete physical theory in its own
right.
Applications software
Operating system
Specific rules
Quantum electrodynamics (QED)
Quantum mechanics
QM consists of four mathematical postulates which
lay the ground rules for our description of the
world.
Newtonian gravitation
Newtons laws of motion
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How successful is quantum mechanics?
It is unbelievably successful.
Not just for the small stuff!
QM crucial to explain why stars shine, how the
Universe formed, and the stability of matter.
No deviations from quantum mechanics are known
Most physicists believe that any theory of
everything will be a quantum mechanical theory
A conceptual issue, the so-called measurement
problem, remains to be clarified.
Attempts to describe gravitation in the
framework of quantum mechanics have (so far)
failed.
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The structure of quantum mechanics
linear algebra
Dirac notation
4 postulates of quantum mechanics
1. How to describe quantum states of a closed
system.
state vectors and state space
2. How to describe quantum dynamics.
unitary evolution
3. How to describe measurements of a quantum
system.
projective measurements
4. How to describe quantum state of a composite
system.
tensor products
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Example qubits (two-level quantum systems)
photons electron spin nuclear spin etcetera
Normalization
All we do is draw little arrows on a piece of
paper - that's all. - Richard Feynman
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Postulate 1 Rough Form
Associated to any quantum system is a complex
vector space known as state space.
The state of a closed quantum system is a unit
vector in state space.
Example well work mainly with qubits, which
have state space C2.
Quantum mechanics does not prescribe the state
spaces of specific systems, such as electrons.
Thats the job of a physical theory like quantum
electrodynamics.
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A few conventions
This is the ket notation.
nearly
v
We always assume that our physical systems have
finite-dimensional state spaces.
Qudit
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Dynamics quantum logic gates
Quantum not gate
Input qubit
Output qubit
Matrix representation
General dynamics of a closed quantum
system (including logic gates) can be represented
as a unitary matrix.
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Unitary matrices
Hermitian conjugation taking the adjoint
A is said to be unitary if
We usually write unitary matrices as U.
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Nomenclature tips
matrix (linear) operator (linear)
transformation (linear) map quantum gate
(modulo unitarity)
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Postulate 2
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Why unitaries?
Unitary maps are the only linear maps that
preserve normalization.
Exercise prove that unitary evolution preserves
normalization.
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Pauli gates
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Exercise prove that XYiZ
Exercise prove that X2Y2Z2I
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Measuring a qubit a rough and ready prescription
Measuring in the computational basis
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Measuring a qubit
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More general measurements
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Qubit example
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Inner products and duals
Young man, in mathematics you dont
understand things, you just get used to them. -
John von Neumann
Example
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Duals as row vectors
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Postulate 3 rough form
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The measurement problem
Quantum system
Measuring apparatus
Rest of the Universe
?
Postulate 3
Postulates 1 and 2
Research problem solve the measurement problem.
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Irrelevance of global phase
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Revised postulate 1
Associated to any quantum system is a complex
inner product space known as state space. The
state of a closed quantum system is a unit
vector in state space.
Note These inner product spaces are often
called Hilbert spaces.
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Multiple-qubit systems
Measurement in the computational basis
General state of n qubits
Hilbert space is a big place - Carlton Caves
Perhaps we need a mathematical theory of
quantum automata. the quantum state space has
far greater capacity than the classical one
in the quantum case we get the exponential growth
the quantum behavior of the system might be
much more complex than its classical simulation.
Yu Manin (1980)
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Postulate 4
The state space of a composite physical system is
the tensor product of the state spaces of the
component systems.
Example
Properties
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Some conventions implicit in Postulate 4
Alice
Bob
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Examples
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Quantum entanglement
Alice
Bob
Schroedinger (1935) I would not
call entanglement one but rather the
characteristic trait of quantum mechanics, the
one that enforces its entire departure from
classical lines of thought.
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Summary
Postulate 1 A closed quantum system is described
by a unit vector in a complex inner product
space known as state space.
Postulate 2 The evolution of a closed quantum
system is described by a unitary transformation.
Postulate 4 The state space of a composite
physical system is the tensor product of the
state spaces of the component systems.