General Introduction

1
General Introduction

• Strong Church-Turing thesis states that a
  probabilistic Turing machine (i.e., a classical
  computer that can make fair coin flips) can
  efficiently simulate any realistic model of
  computing
• Therefore if we are interested in which problems
  can be solved efficiently on a realistic model of
  computation, we can restrict attention to a
  probabilistic Turing machine (or an equivalent
  model)

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Physics and Computation

• Information is stored in a physical medium and
  manipulated by physical processes
• Therefore the laws of physics dictate the
  capabilities and limitations of any information
  processor
• The classical laws of physics are (usually) a
  good approximation to the laws of physics

3
Physics and Computation

• The theory of quantum physics is a much better
  approximation to the laws of physics
• The probabilistic Turing machine is implicitly a
  classical device and it is not known in general
  how to use it to simulate efficiently quantum
  mechanical systems Fey82
• A computer designed to exploit the quantum
  features of Nature (a quantum computer) seems to
  violate the Strong Church-Turing thesis

4
Physics and Computation

• Is a quantum computer realistic? Answer seems to
  be YES
• If the quantum computers are a reasonable model
  of computation, and classical devices cannot
  efficiently simulate them, then the strong
  Church-Turing thesis needs to be modified to
  state that a quantum Turing machine can
  efficiently simulate any realistic model of
  computation

5
Quantum Communication and Cryptography

• By exploiting the quantum mechanical behavior of
  the communication medium, we can detect
  eavesdroppers (leading to quantum cryptography)
  and solve distributed computation tasks more
  efficiently.
• Unfortunately, we wont be covering this in
  depth in this course, but we will lay the
  foundation for further reading in quantum
  information theory.

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Probability Amplitude and Measurement
If the photon is measured (with some external
apparatus) when it is in the state then we get
with probability
7
Quantum Operations
The operations are induced by the apparatus
linearly, that is, if and then
8
Quantum Operations
Any linear operation that takes
states satisfying and maps them to
states satisfying must be UNITARY
9
Linear Algebra
corresponds to
corresponds to
corresponds to
10
Linear Algebra
corresponds to
corresponds to
11
Linear Algebra
is unitary if and only if
12
More than one qubit
If we concatenate two qubits
we have a 2-qubit system with 4 basis states
and we can also describe the state as or by
the vector
13
More than one qubit
In general we can have arbitrary
superpositions
where there is no factorization into the tensor
product of two independent qubits. These states
are called entangled.
14
Measuring multi - qubit systems
If we measure both bits of we get with
probability
15
Overview

• Classical Logic Gates
• Reversible Logic
• Quantum Gates
• A taste of quantum algorithms

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Classical Logic Gates

• A gate is a function from m bits to n bits, for
  some fixed numbers m and n

AND
NOT
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Classical Logic Gates

• We glue gates together to make circuits (or
  arrays of gates) which compute Boolean functions

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Classical Logic Gates

• If all physical processes are unitary (and thus
  reversible), a complete description of a physical
  process implementing the AND gate should be
  reversible.
• However the AND gate is not logically reversible.
• Therefore, the (non-reversible) AND gate throws
  away or erases information that would make it
  reversible.

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Classical Logic Gates

• Landauers Principle To erase a single bit of
  information dissipates at least kT log(2) amount
  of energy into the environment
• It was thought that dissipation of energy implied
  fundamental limits on real computation

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Classical Logic Gates

• However Bennett showed that any computation can
  be made reversible and therefore doesnt in
  principle require energy dissipation
• Method Replace each irreversible gate with a
  reversible generalization

21
Irreversible gates from reversible ones

• Note that irreversible gates are really just
  reversible gates where we hardwire some inputs
  and throw away some outputs

22
Some tensor product facts
23
Some tensor product facts
24
Information and Physics

• Information is always stored in a physical medium
  and manipulated by a physical process.
• Any meaningful theory of information processing
  must refer (at least implicitly) to a realistic
  physical theory.

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Quantum Mechanics and Information Processing

• Since physics is quantum mechanical, we need to
  recast the theory of information processing in a
  quantum mechanical framework.

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Implications

• How does this affect computational complexity?

• How does this affect communication complexity?

• How does this affect information security?

• Would you believe a quantum proof?

27
A small computer
(negligible coupling to the environment)
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Information and Physics
(negligible coupling to the environment)
29
Is this realistic?

• We do have a theory of classical linear error
  correction.
• But before we worry about stabilizing this
  system, lets push forward its capabilities.

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Information and Physics
31
A quantum gate
?NOT
?NOT
32
???
What is supposed to mean?
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One thing we know about it
If we measure
we get with probability and
with probability