1. Richard Cleve DC 3524 cleve@cs.uwaterloo.ca
Introduction to Quantum Information Processing CS 467 / CS 667 Phys 667 / Phys 767 C&O 481 / C&O 681
Lecture 1 (2005)
Richard Cleve
DC 3524

2. Overview

3. Moore’s Law Following trend … atomic scale in 15-20 years
1975
1980
1985
1990
1995
2000
2005
104
105
106
107
108
109
number of
transistors
year
Following trend … atomic scale in years
Quantum mechanical effects occur at this scale:
Measuring a state (e.g. position) disturbs it
Quantum systems sometimes seem to behave as if they are in several states at once
Different evolutions can interfere with each other

4. Quantum mechanical effects
Additional nuisances to overcome? or
New types of behavior to make use of?
[Shor, 1994]: polynomial-time algorithm for
factoring integers on a quantum computer
This could be used to break most of the existing public-key cryptosystems on the internet, such as RSA

5. Quantum algorithms Classical deterministic: Classical probabilistic:
START
FINISH
Classical probabilistic:
FINISH p1 p3 p2
START
Quantum:
FINISH α1 α3 α2
START
POSITIVE
NEGATIVE

6. Also with quantum information:
Faster algorithms for combinatorial search [Grover ’96]
Unbreakable codes with short keys [Bennett, Brassard ’84]
Communication savings in distributed systems [C, Buhrman ’97]
More efficient “proof systems” [Watrous ’99]
classical
information
theory
quantum information
… and an extensive quantum information theory arises, which generalizes classical information theory
For example: a theory of quantum error-correcting codes

7. This course covers the basics of quantum information processing
Topics include:
Quantum algorithms and complexity theory
Quantum information theory
Quantum error-correcting codes*
Physical implementations*
Quantum cryptography
Quantum nonlocality and communication complexity
* Jonathan Baugh

8. General course information
Background:
classical algorithms and complexity
linear algebra
probability theory
Evaluation:
3 assignments (15% each)
midterm exam (20%)
written project (35%)
Recommended text:
“Quantum Computation and Quantum Information”
by Nielsen and Chuang (available at the UW Bookstore)

9. Basic framework of quantum information

10. Types of information is quantum information digital or analog?
probabilistic
digital: 1
analog: 1
r  [0,1] r p q
Probabilities p, q  0, p + q = 1
Cannot explicitly extract p and q
(only statistical inference)
In any concrete setting, explicit
state is 0 or 1
Issue of precision (imperfect ok)
Can explicitly extract r
Issue of precision for
setting & reading state
Precision need not be
perfect to be useful

11. Quantum (digital) information1  
Amplitudes ,   C, ||2 + ||2 = 1
Explicit state is
Cannot explicitly extract  and 
(only statistical inference)
Issue of precision (imperfect ok)

12. Dirac bra/ket notation
Ket: ψ always denotes a column vector, e.g.
Convention:
Bra: ψ always denotes a row vector that is the conjugate transpose of ψ, e.g. [ *1 *2  *d ]
Bracket: φψ denotes φψ, the inner product of φ and ψ

13. Basic operations on qubits (I)
(0) Initialize qubit to |0 or to |1
(1) Apply a unitary operation U (U†U = I )
Examples:
Rotation:
Hadamard:
Phase flip:
NOT (bit flip):

14. Basic operations on qubits (II)
1
ψ′
(3) Apply a “standard” measurement:
0 + 1
ψ
||2
0
||2
… and the quantum state collapses
() There exist other quantum operations, but they can all be “simulated” by the aforementioned types
Example: measurement with respect to a different orthonormal basis {ψ, ψ′}

15. Distinguishing between two states
Let be in state or
Question 1: can we distinguish between the two cases?
Distinguishing procedure:
apply H
measure
This works because H + = 0 and H − = 1
Question 2: can we distinguish between 0 and +?
Since they’re not orthogonal, they cannot be perfectly distinguished …

16. n-qubit systems Probabilistic states: Quantum states:
Dirac notation: |000, |001, |010, …, |111 are basis vectors, so

17. Operations on n-qubit states
Unitary operations:
(U†U = I ) ?
Measurements:   
… and the quantum state collapses

18. Entanglement Product state (tensor/Kronecker product):
Example of an entangled state:
… can exhibit interesting “nonlocal” correlations:

19. Structure among subsystems
qubits: #2 #1 #4 #3
time V U W
unitary operations
measurements

20. Quantum computations Quantum circuits: 0 1 1
“Feasible” if circuit-size scales polynomially

21. Example of a one-qubit gate applied to a two-qubit system
(do nothing)
The resulting 4x4 matrix is
00  0U0 01  0U1 10  1U0 11  1U1
Maps basis states as:

22. Controlled-U gates U Resulting 4x4 matrix is controlled-U =
Maps basis states as:
00  00 01  01 10  1U0 11  1U1

23. Controlled-NOT (CNOT)
a
b
ab X ≡
Note: “control” qubit may change on some input states
0 + 1
0 − 1

24. THE END