CSI 789 002 Quantum Computation

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Quantum Computation
Dr. Richard B. Gomez rgomez_at_gmu.edu

Introduction to Quantum Computing
Lecture 4
George Mason University School of Computational
Sciences
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Last Lecture Topics

• Quantum Logic Gates
• Quantum Dots
• Quantum Error Correction

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Review Quantum Gates

• There are two types of quantum logic gates

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OUTLINE

• Quantum Cryptography
• Quantum Algorithms

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Quantum Cryptography
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Secure Cipher

• A secure cryptosystem can be produced from a
  random key which is as long as a message.
• Process The cipher text is the XOR of the
  random bits with the plain text bits

Plaintext 0 1 1 0 1 0 1 0 1 1 0 1 0 0 0
KEY 1 1 0 0 1 1 1 0 0 1 0 0 1 0 1
Ciphertext 1 0 1 0 0 1 0 0 1 0 0 1 1 0 1
KEY 1 1 0 0 1 1 1 0 0 1 0 0 1 0 1
Plaintext 0 1 1 0 1 0 1 0 1 1 0 1 0 0 0
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Key Distribution

• The problem with a stream cipher of this form is
  the distribution of a key
• If the key is short so everyone can easily
  remember it, then it is also easy to break
• If the key is long, so it has to sent between the
  users, it could be intercepted and compromised

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Quantum Key Distribution

• Quantum effects can be used to distribute a long
  key with assurances that the key has not be
  intercepted
• To understand this process
• Consider another form of a qubit
• The standard key distribution format
• Quantum effects

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Photons

• Another form of a qubit could be a photon
• A photon is an electromagnetic wave (we know it
  as light)
• A photon consists of an oscillating electric
  field and an oscillating magnetic field which lie
  in perpendicular planes

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Polarization

• One property of a photon that is of interest is
  its polarization
• Polarization is a relative term that describes
  the plane of the electrical field
• If the photon is traveling along the x-axis then
  the electric field can be in any plane

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Photon Qubits

• Define a reference line
• A photon which is polarized at either 0 or 45
  degrees of the line is a 0
• A photon which is polarized at either 90 or 135
  degrees of the line is a 1

0 and 90 degrees are called rectilinear
polarization 45 and 135 degrees are called
diagonal polarization
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Crypto Definitions

• It is a standard in cryptography to define the
  sender, receiver, and interceptor as
• Alice is the one who sends the ciphertext
• Bob is the one who receives the ciphertext
• Eve is the (evil) one who tries to steal the
  plaintext or key

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Quantum Key Distribution I

• Alice and Bob want to exchange a key on a quantum
  channel and ciphertext on a normal channel

If photons are exchanged, then the quantum
channel could be a fiber optics link
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Set Up

• Alice selects a random sequence of bits
• Out of this sequence, Alice and Bob will
  ultimately construct a common key
• Example

1 1 1 1 1 0 0 1 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0
0 1 0 1 1
Alice must encode these into polarized photons
and send them to Bob along the quantum channel
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Encoding Process

• Alice chooses to encode each bit in either the
  rectilinear polarization () form or the diagonal
  polarization (x) form
• Summary of polarization forms

1 x / - 0 x \
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Alices Choice

• Say Alice sends the random bits with the
  following choice of polarization

1 1 1 1 1 0 0 1 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0
0 1 0 1 1
x x x x x x x x x x x x x
x x x x
\ - \ \ \ / / - / \ \ / / - / /
/ \ \ \
Alice sends this sequence of polarized photons to
Bob
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Bobs Task

• Bob must measure the direction of the
  polarization of Alices photons to reconstruct
  the set of bits
• However, Bob does not know when Alice used
  rectilinear or diagonal polarization so he has to
  guess
• If his guess is correct then he will recover the
  correct bit
• If his guess is wrong, he has only a 50-50 chance
  of recovering the correct bit

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Decoding

• Bob receives Alices polarized photons

\ - \ \ \ / / - / \ \ / / - / /
/ \ \ \


x x x x x x x x x x x x x
x
0
1
1 1 1 0 0 0 1 1 0 0 0 1 0 0 0 0 0 1 1 1 1 0 1 1 1
0 1 0
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Eavesdropping Test

• Now Alice tells Bob the polarizer orientation for
  a subset of the bits and Bob tells Alice the
  orientations he used on that same subset
• For those cases where Alice and Bob agreed, Alice
  tells him what bit values he should have received

x x x x x x
x x
x x x x

1 0 0
0
1 0 0
0
They agree so there was no eavesdropping
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Common Key

• Since the channel is secure, Alice sends Bob the
  polarization orientation for another subset of
  the bits
• Bob compares the actual polarization with his
  guess
• He only uses the bits for which the two
  polarizations match (and so does Alice)

x x x x x x x x x
x x x
x x x x x x x x
x
1 0 0 0
1
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The Effect of Eve

• If Eve intercepts and measures the photons, she
  has to send her measured values on to Bob

0 1 1 1 1 0 0 1 0 0 0 0 1 0 0 0 0 0 1 1 1 0 1 1 0
0 1 0 1 0
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Bobs Test

• Bob, unaware of Eves presence, decodes the
  photons

0 1 1 1 1 1 0 1 1 1 0 0 1 0 0 0 0 0 1 1 1 1 1 0 0
0 1 0 1 0
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Quantum Algorithms
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Quantum Algorithms

• There have been a number of applications of
  quantum computing that have been developed over
  the last few years
• High speed parallel search
• Factoring
• Key Exchange
• Logic Gate Structures
• . . .

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Factoring

• Given two large prime numbers p and q it is easy
  to calculate their product
• p 15485863 and q 15485867 thenp x q
  239813014798221
• On the other hand, given a large number n it is
  very difficult to find two integers p and q such
  that n p x q

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Number Theory Trick

• Given an integer n to factor, create a function
• fn(a) xa mod n
• x is a random integer such that gcd(x,n) 1
• It turns out that fn(a) is periodic
• For successive inputs a 0, 1, 2, . . . The
  function values fn(0), fn(1), . . . will repeat
  (different x values will produce different
  patterns)
• For a given x, the period of the pattern is r

There is a very good chance that the gcd(n,xr/2
1) is a factor of n
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Example

• Find a factor of 15

Select x 8 then f15(a) 8a mod 15
82 1 63
Find the gcd(63, 15)
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3 is a factor of 15
This does not always work.
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Quantum Approach

• Goal Find the period of fn(a)
• PROCESS construct a single quantum register
  then partition it into two parts
• R1 and R2
• Store a superposition of all values of a in R1
• Evaluate fn(a) and place the result in R2

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Effect

• Now R2 is a superposition of all possible
  function values (it only took 1 evaluation)
• Measure R2 this causes it to collapse to a
  single value, say k
• This means that for some a, xa mod n k
• Because R1 and R2 are entangled, R1 now contains
  a superposition of only those values of a such
  that xa mod n k

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Final Effort

• Perform a Fourier Transform on R1 to find the
  period r
• Calculate the gcd to find a possible factor