Transcript Turing Machines

Introduction to Quantum Information
and Computation
Anu Venugopalan
Guru Gobind Singh Indraprastha Univeristy
Delhi
_______________________________________________
INTERNATIONAL PROGRAM ON QUANTUM INFORMATION
(IPQI-2010)
Institute of Physics (IOP), Bhubaneswar
January 2010
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Introduction
Quantum
computation
and
quantum
information is the study of the information
processing tasks that can be accomplished
using quantum mechanical systems
Quantum Mechanics
Computer Science
Information Theory
Cryptography
Mathematics
IPQI-2010-Anu Venugopalan
Quantum
information &
computation
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Introduction
Real computers are physical systems
 Stonehenge
 Pebbles and beads
 Tokens, abacus
 Mechanical computers
 Difference Engine I (Charles Babbage)
 Analytical engine
 Electromechanical, electronic (vacuum tubes)
 Electronic (semiconductors)
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Introduction
Real computers are physical systems
Computer technology in the last fifty yearsdramatic miniaturization
Faster and smaller –
- the memory capacity of a chip approximately
doubles every 18 months – clock speeds and
transistor density are rising exponentially...what is
their ultimate fate????
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Moore’s law [www.intel.com]
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Miniaturization and computers
In spite of the dramatic miniaturization in computer
technology in the past five decades, our basic
understanding of how a computer functions – or what
it can do – has not changed. The tiny components
inside all computers today still behave and are
understood according to classical physics.
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Extrapolating Moore’s law
If Moore’s law is extrapolated, by the year 2020
the basic memory component of the chip would be
of the size of an atom – what will be space, time
and energy considerations at these scales (heat
dissipation…)?
At such scales, the laws of quantum physics would
come into play - the laws of quantum physics are
very different from the laws of classical physics everything would change!
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Quantum computation
In anticipation of ultimately hitting atomic scales
in computer technology, the field of quantum
computation was first envisaged
Quantum physics
offers something new and
spectacular.
By exploiting delicate quantum
phenomena that have no classical analogues, it is
possible to do certain computational tasks much
more efficiently than can be done by any classical
computer – even a supercomputer
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Quantum computation
Quantum information and computing offers a new
paradigm and possibilities for computing and will
change the way in which scientists think about
fundamental operations in computing and the
capabilities and ultimate limits to computing
Offers powerful techniques for storage and
manipulation of information
New phenomena - Quantum teleportation
- Quantum cryptography
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Quantum mechanics


At the turn of the last century, several experimental
observations could not be explained by the
established laws of classical physics and called for a
radically different way of thinking --this led to the
development of Quantum Mechanics which is today
regarded as the fundamental theory of Nature
The price to be paid for this powerful tool is that
some of the predictions that Quantum Mechanics
makes are highly counterintuitive and compel us to
reshape our classical (‘common sense’) notions.........
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Introduction to theoretical
computer science
 What are the capabilities of a computer?
 What are the limits of a computer?
 What are the problems that can be solved
efficiently on a computer and what are the ones
that cannot?
 How are these questions related to the actual
physical make up of the computer?
 Does it matter if the computer is made up of
gears and columns, vacuum tubes or integrated
chips?
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Introduction to theoretical
computer science
Alan Turing
(1936)
Alonso Church, Kurt Gödel, Emil Post
- Developed a ‘classical’ mathematical models for
computation which was supposedly ‘free’ of any
assumptions pertaining to the actual physical
mechanism involved in a computer
- on closer examination these models revealed
subtle assumptions that might well break down
when we encounter a new regime of Nature…the
quantum domain.
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Introduction to theoretical
computer science
Why should we spend time investigating classical
computer science (TCS) if we are to study
quantum computation?
• TCS has a vast body of concepts and techniques
that can be applied to and reused in QI and QCmany of the triumphs of QI and QC have come by
combining existing ideas from computer science
with novel ideas from quantum mechanics
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Introduction to theoretical
computer science
One learns to ‘think like a computer scientist’Computer scientists think in a very different style
than does a physicist – anybody wanting a deeper
understanding of QI&QC must learn to think like a
computer scientist (at least some times!)
- very useful for studying QI and QC
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Introduction to theoretical
computer science
Key concept of computer science – Algorithm
An algorithm is a precise recipe for performing some
task – e.g. adding two numbers
The fundamental model for algorithms is
the Turing Machine
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The Turing Machine
The most influential computer model due to Alan
Turing (1936) – captures in a mathematical
definition, what we mean when we use the intuitive
concept of an algorithm
It is said to have been Turing’s response to David
Hilbert’s challenge (‘Entscheidungsproblem’) and is
also regarded as a computational analog of Gödel’s
Incompleteness Theorem in Logic.
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The Turing Machine
The proof process
If one were to look over a mathematician's
shoulders during a proof derivation, what would
one see in his/her notes?
Turing abstracted the process appearing in these
notes into four principle ingredients
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The Turing Machine
1.
2.
3.
4.
A set of transformational rules
A method for recording each step in the
proof
A method to go back and forth
A mechanism for deciding which rule to apply
at a given moment
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The Turing Machine
- the four steps listed are simplified so that a
machine could be made to implement them –
- translating these steps in terms of symbols (0,1)
on a one-dimensional tape with the read/write
concept
- In this way Turing translated the mechanistic
analogues of the human thought process into a
mathematical form
The deterministic Turing machine
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The Turing Machine
Infinite tape
1
1
1
0
1
0
b
b
--------
Read/write head
Internal states/program
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The deterministic Turing Machine
- the four main elements of the DTM are
1.
Finite State Control
2.
Tape
3.
A read/write tape head
4.
Program
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DTM- Finite State Control (FSC)
- Finite State Control
The FSC for a TM can be visualized as a stripped
down microprocessor which coordinates the
other operations of the machines
A finite set of m internal states: q1 , q2, ……….. qm
qs : starting state
qh: halting state
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The Tape
-
the tape is a one dimensional strip which
stretches off to infinity in one direction – the
tape squares are labelled and each contains one
symbol drawn from some alphabet
e.g., 0,1 and b (blank)
marks the left hand edge of the tape
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The read/write head
-
the read/write tape head identifies a
single square on the DTM tape as the
square that is being currently accessed
by the machine
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Successor Program
[adapted from www1.cs.columbi.edu/~zeph/3261/L14/L14.ppt]
Sample Rules:
If read 1, write 0, go right, repeat
If read 0, write 1, HALT!
If read •, write 1, HALT!
Let’s see how they are carried out on a
piece of paper that contains the reverse
binary representation of 47:
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Successor Program
[adapted from www1.cs.columbi.edu/~zeph/3261/L14/L14.ppt]
Program
qs , , q1, ,1
 q1 ,0, qh ,1,0
q1 ,1, q1 ,0,1
 q1 , b, qh ,1,0
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Successor Program
[adapted from
www1.cs.columbi.edu/~zeph/3261/L14/L14.ppt]
1
1
1
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1
0
1
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Successor Program
[adapted from
www1.cs.columbi.edu/~zeph/3261/L14/L14.ppt]
0
1
1
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1
0
1
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Successor Program
[adapted from
www1.cs.columbi.edu/~zeph/3261/L14/L14.ppt]
0
0
1
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1
0
1
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Successor Program
[adapted from
www1.cs.columbi.edu/~zeph/3261/L14/L14.ppt]
0
0
0
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1
0
1
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Successor Program
[adapted from
www1.cs.columbi.edu/~zeph/3261/L14/L14.ppt]
0
0
0
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0
0
1
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Successor Program
[adapted from
www1.cs.columbi.edu/~zeph/3261/L14/L14.ppt]
0
0
0
0
1
1
So the successor’s output on 111101 was 000011
which is the reverse binary representation of
48.
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Successor Program
[adapted from
www1.cs.columbi.edu/~zeph/3261/L14/L14.ppt]
Similarly, the successor of 127 should be 128, as one
can see in the following:
1
1
1
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1
1
1
1
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Successor Program
[adapted from
www1.cs.columbi.edu/~zeph/3261/L14/L14.ppt]
0
1
1
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1
1
1
1
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Successor Program
[adapted from
www1.cs.columbi.edu/~zeph/3261/L14/L14.ppt]
0
0
1
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1
1
1
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Successor Program
[adapted from
www1.cs.columbi.edu/~zeph/3261/L14/L14.ppt]
0
0
0
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1
1
1
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Successor Program
[adapted from
www1.cs.columbi.edu/~zeph/3261/L14/L14.ppt]
0
0
0
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0
1
1
1
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Successor Program
[adapted from
www1.cs.columbi.edu/~zeph/3261/L14/L14.ppt]
0
0
0
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0
0
1
1
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Successor Program
[adapted from
www1.cs.columbi.edu/~zeph/3261/L14/L14.ppt]
0
0
0
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0
0
0
1
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Successor Program
[adapted from
www1.cs.columbi.edu/~zeph/3261/L14/L14.ppt]
0
0
0
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0
0
0
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Successor Program
[adapted from
www1.cs.columbi.edu/~zeph/3261/L14/L14.ppt]
0
0
0
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0
0
0
0
1
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The Church-Turing Thesis
Church-Turing thesis:
.
Any algorithmic process can be simulated on a Turing
machine
A Turing Machine – an idealized and rigorously defined
mathematical model of a computing device.
Many different models of computation are equivalent
to the Turing machine (TM).
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The Church-Turing Thesis
The class of functions computable by a Turing Machine
.
corresponds exactly to the class of functions which we
would naturally regard as being computable by an
algorithm
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The Church-Turing Thesis
Despite its lack of adornments, the TM model has
proved to be remarkably durable in all the 70 years of
its existence. Though computer technology has
.
advanced dramatically, our qualitative understanding of
the computation process remains the same
- in the strict theoretical sense, all computers are the
same.
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Computation
Real computers are finite devices, not
infinite, like the Turing machine – they can be
understood by a circuit model of computation
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Circuits
x
y
1x y
NAND
x
x
x
x
x
0
NOT
0
AND
NAND
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x 1
NOT
x
1
y
x
y x
controlled-not gate (CNOT)
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Universality
The NAND gate can be used to simulate the AND,
XOR and NOT gates, provided wires, fanout, and
ancilla are available.
wire – “memory”
Fanout – e.g
x
x
x
x
Ancilla- bits in pre-prepared
states.
The NAND gate, wires, fanout and ancilla form a
universal set of operations for computation.
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Irreversibility
0
1
NAND
1
1
0
NAND
1
From the output of the NAND gate it is impossible to
determine if the input was (0,1), (1,0), or (0,0)
The NAND gate is irreversible
- there is no logic gate capable of inverting the NAND.
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Irreversibility
Computing machines inevitably involve devices which
perform logical functions that do not have a single-valued
inverse.
This logical irreversibility is associated with physical
irreversibility and requires a minimal heat generation, per
machine cycle ~ order of kT for each irreversible function.
Landauer’s principle: Any irreversible operation in a
circuit is necessarily accompanied by the dissipation
of heat.
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Irreversibility
•As the densities and switching speeds of our
computational
devices
continue
to
increase
exponentially, the amount of energy dissipated by
these devices must remain at a certain level, otherwise
economically impractical cooling apparatus is required.
•Conventional computers perform thermodynamically
irreversible logic operations.
•Information, in the form of bits, is erased.
•This bit erasure represents entropy, which is
correlated to heat dissipation.
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Irreversibility
How can we compute without dissipating heat?
1. Lower the temperature of our computers

2. Develop thermodynamically reversible computers which do not
generate entropy and therefore do not dissipate nearly as much
heat as conventional, irreversible computers.
Use only reversible circuit elements
quantum gates are the most natural candidates for
reversible gates.
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Some reversible circuit elements
x
x
y x
y
x
x
y
x
y
y x
The Toffoli gate (or controlled-controlled-not).
x
x
x
y
y
y
z
z x y
z
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x
y
z x y
x
y
z
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Computing using reversible circuit elements
Example: The reversible NAND gate.
x
x
y
y
1
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1 x  y
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Computing using reversible circuit elements
x
x
0
x
Fanout
quantum gates are natural candidates for reversible
gates as any isolated quantum system has a dynamical
evolution which is reversible………
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The Quantum Turing Machine
• 1973: Charles Bennett – suggested the concept of a
reversible Turing Machine
• !980: Paul Benioff - any isolated quantum system had a
dynamical evolution which was reversible in the exact sense –
and could mimic a reversible TM
• 1982: Richard Feynman - no classical TM could simulate
certain quantum phenomena without incurring an exponential
slowdown
• 1985: David Deutsch: Described the first Quantum Turing
Machine
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The Quantum Turing Machine
In a QTM, all operations –
e.g. read, write. shift etc. are
accomplished by quantum
mechanical interactions
• The tape could exist in
states that were highly
‘nonclassical’
• A QTM would encode data
not just in bits of 0 and I but
in superposition states –
qubits
• Quantum parallelism
•
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1
0
Qubit
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Complexity
• The fact that a computer can solve a particular
kind of problem in principle does not guarantee
that it can solve it in practice
• If the running time is too long and the memory
requirements are too large, an apparently feasible
computation can still lay beyond the reach of any
practicable computer
• ‘efficiency’
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complexity classes
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Complexity classes
•
How does the computational cost incurred in
solving a problem scale up as the size of the
problem increases?
• ‘Measure’ of efficiency – rate of growth of
time/memory requirements to solve a problem at
the size increases
• ‘Size’: number of bits, L, needed to state the
problem to the computer
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Complexity classes
• ‘Size’: number of bits, L, needed to state the
problem to the computer. E.g. If N is a large
integer, then the size of the integer in binary
representation is L=Log2N
• Is the ‘cost’ polynomial in L or exponential in L?
• An exponential growth exceeds polynomial growth
regardless of the order of the polynomial
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Complexity classes
eL
Example
The factoring problem
L2
10433 x 16453=-------(polynomial, L2)
L
---- x ---- =200949083
(exponential)
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Complexity classes
If you had a very fast (classical) computer which
could do,
1010 divisions per second, then to
factorize:
L=20 --- one second
L=34 ---- one year
L=60 ----- 1017 seconds > age of the Universe!
Multiplying two L=60 numbers would take only a
few seconds.
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Complexity classes
•
Multiplication is ‘easy’ as costs grow only
polynomially in size of the problem
• Factoring is ‘hard’ as costs grow exponentially in
size of the problem
• The difficulty of factoring large numbers – lies at
the heart of public key cryptosystems
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RSA-129
•
In 1994 Rivest, Shamir and Adleman , using a
network of more than 1600 computers solved the
most famous cryptography challenge in existence,
a challenge that was thought to be unbreakable:
finding the prime factors of a 129 digit number
• Number field sieve (Lenstra 1990)
Cost grows as exp[L2 log(L)2/3]
Sub exponential, but super polynomial
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Complexity classes
• Problems that can be solved in polynomial time are
termed tractable and belong to
Complexity class P
e.g. Multiplication
• Problems that cannot be solved in polynomial time
are termed intractable and may belong to one of
several classes, e.g.
Complexity classes NP, ZPP, BPP
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Complexity classes
Complexity class NP
Nondeterministic polynomial time
- problems where the computational costs incurred is
exponential in L but once a candidate solution is
found, its correctness can be tested ‘efficiently’,
i.e., in polynomial time – this means there is an
efficient nondeterministic algorithm for solving
the problem
e.g. the factoring problem
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Complexity classes
Complexity class P
e.g. Multiplication
Complexity class NP e.g. the factoring problem
Complexity class NP complete
e.g. the traveling salesman problem
A subset of problems
in NP that can be
mapped into one
another in polynomial
time
Is P=NP?
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The power of quantum computing
One of the biggest motivations for studying
Quantum Computation is the ability (as shown by
Peter Shor in 1994) of a potential quantum
computer to perform a computational task like
factoring a large number far more efficiently than
any conventional computer
– an ability to break 'unbreakable codes' like the
most secure public key cryptosystem in the world
today- RSAIPQI-2010-Anu Venugopalan
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The power of quantum computing
o
o
In public key cryptosystems like the RSA system, the
public key is broadcast/published – it contains a
number, which if factored would reveal the
private/secret key
Factoring is hard – it is exponentially difficult for
classical (conventional) computers....
But a Quantum algorithm implemented on a quantum
computer could factor a large number very fast (Peter
Shor
1994)-A
Computation
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Killer
Application
for
Quantum
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The power of quantum computing
Today, it seems technologically infeasible to build a
working quantum computer, but its risky to think
that this will be the position forever
At the pace of technological progress that we see, a
quantum computer might become a reality sooner
than later....
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