Introduction to Quantum Computing

1
Introduction toQuantum Computing
Guest lecture at Universita della Calabria,
Cosenza ,ITALY and Politechnika Gdanska,
Gdansk, POLAND

• Prof. Dr. Janusz Kowalik
• Seattle, Washington, USA
• October, 2003

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QUANTUM COMPUTATION

• This new discipline of informatics combines
• information sciences
• and quantum mechanics.

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Part 1 Fundamentals

• Quantum Computing (QC) definition
• Quantum mechanics (QM) accomplishments
• Mathematical background
• Part 2
• Applications
• 1 Factoring problem
• ( cryptography )
• 2 Inverse search
• ( searching unstructured databases )

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Definition

• Quantum Computing is the study of information
  processing tasks that can be accomplished
  using principles of quantum mechanics.
• Comment.
• Every computing device is based on
• the principles of some field of physics.

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Quantum Mechanics

• The universe is quantum mechanical
• QM is an extremely precise tool for predicting
  behavior of the physical world
• This has been verified beyond any reasonable
  doubt.
• Thanks to QM we understand the structure of
  atoms, nuclear fusion, superconductors, the
  structure of DNA etc
• But QM is often counterintuitive
• QM can be used as a basis for communication
  ,computation and cryptography.

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Computational Complexity

• A fundamental question in informatics
• How much information resources are needed to
  perform a specific information processing task?
• Resource metrics algorithmic steps,
• memory locations ,amount
• Of communication.
• The required number of algorithmic steps,
• memory access steps and communication determines
  the total computation time required for solution.

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Example Factoring Problem

• How many steps are required to find the prime
  factors p and q of a 300-digit number n?
• A) The best classical algorithm known would take
  about
• Assuming a TeraHertz computer this is equivalent
  to 150,000 years of computing time.
• B) A quantum computing algorithm running on a
  quantum computer executing at the same Hertz
  speed would take
• Less than a second!
• 
  TeraHertz
• .
• Current processors are about 1000 times slower.

steps
steps
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Why such a difference?

• Both computers have equal Hertz speed!
• The difference is algorithmic!
• Quantum computers execute highly parallel
  algorithms
• The quantum computing algorithm for factoring
  composite numbers has a much lower computational
  complexity ( measured by the number of steps ).
• It requires fewer parallel steps.
• This parallelism is not based on multiple CPUs
  but on the principle of superposition

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Questions??????

• Can we use conventional computers to execute
  better quantum algorithm?
• In other words, can we simulate quantum computer
  using conventional computers?
• The answer is NO ( for any large problems).
• The main reason is a huge amount of required
  memory technologically impossible now and in the
  foreseeable future.
• The key advantages of quantum computing are
  faster algorithms, ( massively parallel
  computation and the ability to handle huge
  amounts of data )

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Conventional computing BITS
1
0
OR
All information is encoded using two binary
states BITS
There is no intermediate state combining 0 and 1

Boolean logic gates manipulate bits Logic gates
NOT,AND,NAND,OR,XOR,NOR
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QUBITS

• The quantum version of a bit

The information in a qubit must be extracted by a
measurement. The result is an ordinary bit ,0 or
1.
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Qubit

• The classic computing is based on processing bits
  0 and 1
• by logic gates.
• Similarly quantum computer is built from circuits
  containing wires and quantum gates.
• Dirac notation for two opposite quantum states

Where a and b are complex numbers that satisfy
the equation
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Qubit

• Qubit is a linear combination of two basic states
• If it is measured it collapses to one of them
  with the probability
• or sum of which is 1.
• Qubit exists in a two-dimensional complex space
  called the Hilbert space.

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Matrix representation

• Two basic states can be represented by linearly
  independent vectors

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Example

• Note that the sum of squares of the amplitudes (
  numbers that multiply basis vectors ) is 1 .
  In this example the amplitudes are real numbers.

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Processing a qubitQuantum Single Qubit Not-Gate

• It changes
• to

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Quantum Not Gate
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General Quantum Gate
Gate
Where
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General Quantum Gates

• Both, the input qubit to the gate and the output
  qubit from the gate are unit vectors.

Quantum GATE
Input qubit
Output qubit
Quantum gates rotate qubit vectors but preserve
their lengths. Quantum qubits are normalized to
1.
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Hadamard Gate
Input qubit
Changed to
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Alternatively we get
the output qubit
It is easy to check that this is a unit vector
since
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Properties of Quantum gate matrices

• Observe that applying the Hadamard gate twice we
  change nothing since

Where I is the unit matrix
Observe also that H is symmetric and its inverse
is itself
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More general properties of the quantum gate
matrices

• Each quantum gate operation matrix is UNITARY

Where the superscript means transposition and
conjugation. For real matrices we have only
transposition,
Quantum computation can be reversed Conventional
computation is NOT reversible Unitary matrices
are invertible
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Unitary matrices

• If U is a unitary matrix its inverse is the
  transpose and conjugated U.
• The fundamental property of a Unitary matrix is

Unitary matrix preserves the length of the
vector x.
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Quantum gate

• Each quantum gate can be represented by a unitary
  matrix
• Quantum computation is a series of quantum gates
  acting upon a qubit or multiple qubits.
• An algorithm can be described by specifying a
  sequence of quantum gates.

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An Example of a real Unitary matrix
The transpose is
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Real Unitary Matrix

• Multiplying this matrix by its transpose gives

If
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Can classical computer simulate quantum
computation?

• Yes , for small problems.
• No , for larger problems.
• For example, solving a problem with k qubits we
  would need to store matrices of size

For k 100 the amount of storage required is
well beyond current and even future classical
computers.
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Part 2 Applications

• Computational Complexity
• The factoring problem
• Unstructured search
• Quantum parallelism
• Conclusions

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Quantum computing algorithms

• Several recently discovered quantum computing
  algorithms have superior computational complexity
  compared to the computational complexity of the
  best classical computing algorithms for the same
  problems..
• Examples include
• Factoring a composite number,
• Unstructured Search

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1. The Factoring Problem

• Peter Shor (ATT) in 1994
• Proposed a QC algorithm that can factor
  efficiently large composite n
  p x qwhere p and q are large primes
• The best known conventional algorithm for
  factoring requires time
• For p q 65 digits longT(n) is approximately
  one month on a cluster of hundreds of
  workstations

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Factoring Problem

• For p q 200 digits long
• T(n) 1010 years
• the time from the big bang
• This makes the public key cryptosystem RSA safe
• Shors QC algorithm
• for n 400 digits long ,T(n) lt 3 years assuming
  the same Hertz computing speed

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Unstructured Search

• Given a search space of size N and no prior
  knowledge of the information in it, find an
  element which satisfies some known property.
• Classic computation requires O(N) operations,
• Quantum Computing can do it using
• steps

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Examples of Unstructured Search

• 1. Inverse telephone search
• Have the number want to find the owner
• 2. Searching for keys of a Crypto
• System.
• Assume that keys are 56 bit long binary numbers.
• The number of keys to be checked
• is

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Crypto keys search

• Assume your computer checks 100 Million keys per
  second
• The classical search would take 23 years of
  computation
• An equally fast !! Quantum computer would need
  only 4 years.
• Why? Because quantum computer needs fewer steps (
  Its time complexity function is proportional to
• compared with the linear time complexity
  function for conventional machine).

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How Doesa QC Algorithm Look ?

• Create initial qubit vectors
• Apply a sequence of gates
• Measure all the resulting qubits after projecting
  them on the 0gt , 1gt basis

• All QC algorithms are probabilistic !
• Applying an algorithm twice to the same input
  will not necessarily produce the same result
• All results are only highly probable

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Parallelism

• Classical parallelism can be accomplished by
  multiple processors
• Quantum parallelism is accomplished by a single
  gate
• This is possible because quantum gates can act
  upon superposition of states

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Quantum Computers

• D. Gottesman from Microsoft
• I.LChuang from IBM
• A general purpose quantum computer can be built
  out of three components
• 1. entangled particles,
• 2. teleporters,
• 3.quantum gates that operate on a single
• qubit at a time

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Conclusions

• (1) QC is capable of massive parallelism
• (2) QC cannot be simulated on classical computers
  for large number of qubits
• (3) The source of computational power of QC is in
  its ability to handle superposition states
• (4) QC algorithms are probabilistic
• (5) Several recently discovered QC algorithms
  reduce computational complexity from exponential
  to polynomial

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Literature

• (1) Michael A. Nielson and Isaac L.Chuang
• Quantum Computation and quantum
• Information,
• Cambridge University Press,2000
• (2)Arthur O. Pittenger
• An introduction to Quantum Computing
• Algorithms
• Birkhauser, 1999