2008.03.04-gerjuoy.pdf

8/18/2019 2008.03.04-gerjuoy.pdf

http://slidepdf.com/reader/full/20080304-gerjuoypdf 1/21

An Introduction to Quantum Computing

Edward Gerjuoy

University of Pittsburgh

8/18/2019 2008.03.04-gerjuoy.pdf


What is a quantum computer?

“While today’s digital computers processclassical information encoded in bits, a quantumcomputer processes information encoded inquantum bits or qubits.”

-NSF Workshop Report NSF-00-101 (Oct. 28-29, 1999)

What is a bit?

0 = ↑, 1 = ↓

Normalized: |α|² + |β|² = 1

0 1ψ α β = +



What is a bit?

0 = ↑, 1 = ↓




What is a bit?

0 = ↑, 1 = ↓


0 1ψ α β = +



What is a bit?

0 = ↑, 1 = ↓


8/18/2019 2008.03.04-gerjuoy.pdf


Possible Realizations of Qubits

Trapped ions in potential wells

Trapped neutral atoms with electric dipole moments “Photons” interacting with single atom or ion in a

resonant cavity

Photons, either freely moving or bouncing between tworesonant cavities

Quantum dots containing a trapped electron

Quantum dots which can contain an exciton

Superconducting circuit at milli-Kelvin temperature Exotic qubits of various kinds, e.g. employing nonabelian

fractional quantum Hall effect anyons

“A Quantum Information Science and Technology Roadmap Part 1: Quantum Computation,”

U.S. Army Advanced Research and Development Activity (ARDA) Report (April 2, 2004).

8/18/2019 2008.03.04-gerjuoy.pdf


NMR, an aside

Nuclear Magnetic Resonance: Eachmolecule can be a multi-qubit quantum

computer

Unlikely to succeed for non-trivial

calculations

See sources for more information

8/18/2019 2008.03.04-gerjuoy.pdf


What’s the minimum necessary?

Must be able to: Conveniently measure the state of a qubit

Conveniently express the wavefunction in

terms of

Be able to put the qubit in a desired state

0 1ψ α β = +0 1ψ α β = +



terms of




terms of




terms of


8/18/2019 2008.03.04-gerjuoy.pdf


12

9 +

2

F

F

Be QUBITS:

( S electronic ground state)

F = 2, m = -2

F = 1, m = -1

cos( / 2) sin( / 2)

sin( / 2) cos( / 2)

i

i

ie

ie

φ

φ

θ θ

θ θ

−

↓ ≡

↑ ≡

↓ → ↓ + − ↑

↑ → − ↓ + ↑

32

2 P

12

2 P

↑

↓

Laser

two-photon

stimulated-Raman

transitions (vs. μ-waves)

David Wineland, University of Colorado, Boulder

Transitions

8/18/2019 2008.03.04-gerjuoy.pdf


Entanglement

“Entanglement is the feature of the quantumworld that distinguishes it from the classicalworld, saying that for a single pair of systems, adescription of each system’s state is notsufficient to describe the entire state of thesystem…It is also the feature of quantumsystems that makes exponential speedup of

computations possible.”

Nielsen & Chuang, “Quantum Computation and Quantum Information” (Cambridge 2000).

8/18/2019 2008.03.04-gerjuoy.pdf


Entanglement

1 1

00 010 0

10 11

0 0 0 1

0 0 0 0

ij AB A B A B A Bi j

A B A B

c i j c c

c c

ψ = =

= = +

+ +∑∑

2

,

1ij

i j

c =∑

If: , then unentangled. Otherwise, entangled. AB A B

u vψ =

1 1

00 010 0

10 11

0 0 0 1

0 0 0 0

ij AB A B A B A Bi j

A B A B

c i j c c

c c

ψ = =

= = +

+ +∑∑

2

,

1ij

i j

c =∑

If: , then unentangled. Otherwise, entangled. AB A B

u vψ =

8/18/2019 2008.03.04-gerjuoy.pdf


Entanglement: Examples

Unentangled:

Entangled:

10 1 1 0

2

A B A B ABγ ψ ⎡ ⎤= +⎣ ⎦

10 0 2 0 1 2 1 0

5 AB A B A B A Bδ ψ ⎡ ⎤= + −⎣ ⎦

Unentangled:

Entangled:

Unentangled:

Entangled:

10 1 1 0

2

A B A B ABγ ψ ⎡ ⎤= +⎣ ⎦

10 0 2 0 1 2 1 0

5 AB A B A B A Bδ ψ ⎡ ⎤= + −⎣ ⎦

0 1 ; AB A Bα ψ =

1 2 1 30 1 0 1

3 2 43 A A B B AB

i i β ψ ⎡ ⎤ ⎡ ⎤

= + −⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

0 1 ; AB A Bα ψ =

1 2 1 30 1 0 1

3 2 43 A A B B AB

i i β ψ ⎡ ⎤ ⎡ ⎤

= + −⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

0 1 ; AB A Bα ψ =

1 2 1 30 1 0 1

3 2 43 A A B B AB

i i β ψ ⎡ ⎤ ⎡ ⎤

= + −⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

8/18/2019 2008.03.04-gerjuoy.pdf



12 0 0 5 3 1 1 5 0 1 2 15 1 0

12t

A B A B A B A B

ψ ⎡ ⎤= − − + +

⎣ ⎦1

2 0 0 5 3 1 1 5 0 1 2 15 1 012 A B A B A B A Bμ ψ ⎡ ⎤= − + +⎣ ⎦

00 01

10 11

0?c c

c c=

... ..., unentangled. AB A B C u v wψ =... ..., unentangled. AB A B C u v wψ =

?? ??

Does

If

?? ??

Does

If

8/18/2019 2008.03.04-gerjuoy.pdf



Unentangled:

Entangled:

No-cloning theorem

10 1 1 0

2

A B A B ABγ ψ ⎡ ⎤= +⎣ ⎦1

0 1 1 0

2

A B A B ABγ ψ ⎡ ⎤= +⎣ ⎦

1 2 1 30 1 0 1 0 1

3 2 43 AB A B A A B B AB

i iα β ψ ψ ⎡ ⎤ ⎡ ⎤

= = + −⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

Wootters and Zurek, Nature 299, 802 (1982).

1 2 1 30 1 0 1 0 1

3 2 43 AB A B A A B B AB

i iα β ψ ψ ⎡ ⎤ ⎡ ⎤

= = + −⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

Unentangled:

Entangled:

No-cloning theorem

10 1 1 0

2

A B A B ABγ ψ ⎡ ⎤= +⎣ ⎦

Unentangled:

Entangled:

No-cloning theorem

8/18/2019 2008.03.04-gerjuoy.pdf


What is a quantum computer?

8/18/2019 2008.03.04-gerjuoy.pdf


Permitted Quantum Computer Operations

In either case, whether U1 or U2, U is unitary,

Let ( 0); ( )i f

i H t tψ ψ ψ ψ ψ ψ = ≡ = ≡

1 , is time-independentiHt

f i ie U H ψ ψ ψ

−

= ≡

20

( )

time-ordered ,

time dependent

t

i i

i H t dt U

f e

H

ψ ψ

ψ −

≡∫=

1t t UU U U = =

8/18/2019 2008.03.04-gerjuoy.pdf


Definition of Quantum Computation

Any particular quantum computation involves finding

a sequence of unitary operations that will transform

some chosen initial computer wave function into a

final wave function wherein there is a non-negligible

probability of measuring those combinations of qubitstates that can be interpreted as yielding the outcome

of the computation.

8/18/2019 2008.03.04-gerjuoy.pdf


Circuit Model of Quantum Computation

[ ]0 0

1 1

2 2

3 3 2

: 0 0 0 0 0 00000

1: 0 1 0 1 0 1 0 0

2 2000 001 010 0111

0 0100 101 110 1112 2

0000 0010 0101 01111: 0

1000 1010 1101 11112 2

:

a b c d e abcde

a a b b c c d e

d e

abc

eabcd

abc

t

t

t

t U

ψ

ψ

ψ

ψ ψ

= ≡

⎡ ⎤ ⎡ ⎤ ⎡ ⎤= + + +⎣ ⎦ ⎣ ⎦ ⎣ ⎦

+ + +⎡ ⎤= ⎢ ⎥+ + + +⎣ ⎦

+ + +⎡ ⎤=

⎢ ⎥+ + + +⎣ ⎦=

0a

b

c

d

e

0

0

0

0

Measure

Hadamard

CNOT

t0 t1 t2 t3

0a

b

c

d

e

0

0

0

0

H

H

HH

H

H

1H 0 0 1

2

CNOT 00 00CNOT 01 01

CNOT 10 10

CNOT 11 11

bd bd

bd bd

bd bd

bd bd

= ⎡ + ⎤⎣ ⎦

=

=

=

=


CNOT 10 10

CNOT 1

1H 0 0 1

2

1 11

bd bd

bd bd

bd bd

bd bd

= ⎡ + ⎤⎣

=

⎦

=

=

=

[ ]

1 1

2 2

3 3 2

0 0

1: 0 1 0 1 0 1 0 0

2 2000 001 010 0111

0 010

: 0 0 0 0 0 00000

0 101 110 1112 2

0000 0010 0101 01111: 0

1000 1010 1101 11112 2

:

a a b b c c d e

d e

abc

eabcd

a b c d e abcd

a

e

bc

t

t

t U

t ψ

ψ

ψ

ψ ψ

⎡ ⎤ ⎡ ⎤ ⎡ ⎤= + + +⎣ ⎦ ⎣ ⎦ ⎣ ⎦

+ + +⎡ ⎤= ⎢ ⎥+ + + +⎣ ⎦

+ + +⎡ ⎤=

⎢ ⎥+ + + +⎣ ⎦=

= ≡ [ ]

2 2

0 0

1 1

3 3 2

000 001 010 0111 0 0

100 101 110 1112 2

0000 0010 0101 0

: 0 0 0 0 0 00000

1: 0 1 0 1 0 1

1111: 0

1000 1010 1101 11112

0 0

2

2

:

2

a b c d e abcde

a a b

d e

abc

eabcd

b c

abc

c d e

t

t

t

t

U

ψ

ψ

ψ

ψ

ψ

+ + +⎡ ⎤= ⎢ ⎥+ + + +⎣ ⎦

= ≡

⎡ ⎤ ⎡ ⎤ ⎡ ⎤= + + +⎣ ⎦ ⎣ ⎦ ⎣ ⎦

+ + +⎡ ⎤=

⎢ ⎥+ + + +⎣ ⎦=

[ ]

2 2

3 3

0

1 1

2

0: 0 0 0 0 0 00000

1: 0 1 0 1 0 1 0 0

2 2000 001 010 0111

0 0100 101

0000 0010 0101 01111: 0

1000 1010 1101 1111

11

2 2

:

0 1112 2

a b c d e

eabcd

a

ab

bc

cd e

a a b b c c d e

d e

abc

t

t

t

t U

ψ

ψ

ψ

ψ

ψ

= ≡

⎡ ⎤ ⎡ ⎤ ⎡ ⎤= +

+ + +⎡ ⎤=

⎢ ⎥

+ +⎣ ⎦ ⎣ ⎦ ⎣ ⎦

+ + +⎡ ⎤= ⎢ ⎥

+ + + +⎣ ⎦=

+ + + +⎣ ⎦

[ ]0 0

1 1

2 2

3 3 2

: 0 0 0 0 0 00000

1: 0 1 0 1 0 1 0 0

2 2000 001 010 0111

0 0100 101 110 1112 2

0000 0010 0101 01111: 0

1000 1010 1101 11112 2

:

a b c d e abcde

a a b b c c d e

d e

abc

eabcd

abc

t

t

t

t U

ψ

ψ

ψ

ψ ψ

= ≡

⎡ ⎤ ⎡ ⎤ ⎡ ⎤= + + +⎣ ⎦ ⎣ ⎦ ⎣ ⎦

+ + +⎡ ⎤= ⎢ ⎥+ + + +⎣ ⎦

+ + +⎡ ⎤=

⎢ ⎥+ + + +⎣ ⎦=

1H 0 0 1

2


CNOT 10 10

CNOT 11 11

bd bd

bd bd

bd bd

bd bd

= ⎡ + ⎤⎣ ⎦

=

=

=

=

Uabc

Uabc

8/18/2019 2008.03.04-gerjuoy.pdf


Any unitary operator which is greater than4x4 can be reduced to a succession of 2-qubit and 1-qubit operations.

Not all 4x4 matrices are needed. There areso called universal gates. CNOT is one ofthem.

Any greater than 2x2 unitary operator can bereduced to the product of one qubit gates andCNOT gates.

Any unitary operator which is greater than4x4 can be reduced to a succession of 2-qubit and 1-qubit operations.

Not all 4x4 matrices are needed. There areso called universal gates. CNOT is one ofthem.

Any greater than 2x2 unitary operator can bereduced to the product of one qubit gates andCNOT gates.

1-Qubit and 2-Qubit Gates

8/18/2019 2008.03.04-gerjuoy.pdf


What is Shor’s Algorithm?

Let N=pq, where p and q denote different odd primes.

Then Shor’s algorithm enables a quantum computer tofind the prime factors p and q of N.

Moreover, for very large N, the quantum computer using

Shor’s algorithm will factor N with less computationaleffort than a classical computer.

However, the meaning of the phrase, “less computationaleffort” must be carefully defined.

P.W. Shor, SIAM J. Comp. 26, 1484 (1997).

“Progress in Quantum Algorithms” (Sept. 14, 2005). http://www-math.mit.edu/~shor/

E. Gerjuoy, “Shor’s factoring algorithm…,” Am. J. Phys. 73 (2005), p. 521.

http://www-math.mit.edu/~shor/

http://www-math.mit.edu/~shor/

8/18/2019 2008.03.04-gerjuoy.pdf


Shor’s Algorithm

In this simulation, the first register is composed of 8 qubits, which can

represent 256 different binary numbers S, with each S corresponding to adifferent state of the first register.

The period r = 26. The probability P(S) is plotted for S = 0 to about 60.

The probabilities for larger S manifest similar peaks. Each peak occurs

at the values of S which most nearly satisfies S/256 = d/r, where d is aninteger < r.

Probability the first register will be in the state, S, after the Quantum

Fourier Transform

8/18/2019 2008.03.04-gerjuoy.pdf


Factoring N=pq with a classical computer

Crudest Attempt: Divide N by the sequence of

integers 2, 3, 4,…, up to √ N . This must find one

of the prime factors of N .

Suppose N is a number of order 10100, and thatthe average time to perform on of these divisions

is 10-12 sec. Then the total time to find a factor

of N will be ~ 10-12 × 1050 = 1038 sec.

The age of the universe is only 12 billion years =

3.8 × 1017 sec.

8/18/2019 2008.03.04-gerjuoy.pdf


Modern Developments

Other Models Measurement model, cluster model, graph

state

Adiabatic quantum computing

Topological computer


state




state




state



8/18/2019 2008.03.04-gerjuoy.pdf


Closing

Counter-factual quantum computation

“The logic underlying the coherent nature of quantuminformation processing often deviates from intuitivereasoning, leading to surprising effects. Counterfactualcomputation constitutes a striking example: the potentialoutcome of a quantum computation can be inferred, evenif the computer is not run…Here we demonstratecounterfactual computation, implementing Grover’s

search algorithm with an all-optical approach.”

O. Hosten et al., Nature, 439, 949 (2006). On counterfactual computation. See also J. Dowling,

Nature, 439, 919 (2006) and L. Vaidman, Phys. Rev. Lett. 98, 160403 (4/25/07).

Documents

2008.03.04-gerjuoy.pdf