1. Quantum Algorithms Oracles
Artur Ekert

2. Query Scenario f BLACK BOX, ORACLE
An ORACLE is very precious, you are charged
some fixed amount of money each time you use it
Typical scenario:
Given an ORACLE that computes f your goal is to determine some properties of f making as few queries to the ORACLE as possible, i.e. you want to minimize your expenditure. f
You are not allowed to look inside the ORACLE, but you can embed it into any Boolean network composed of any logic gates of your choice, we assume you are not charged for extra logic gates

3. Asymptotic notation for comparisons

4. Three Query Scenarios Deutsch’s problem (1985) Grover’s search (1996)
We analyse three scenarios in which we gain if we
use quantum rather then classical oracles.
Deutsch’s problem (1985)
quantum oracles outperform classical oracles
Grover’s search (1996)
quadratic separation
Simon’s problem (1994)
exponential separation

5. ? Deutsch’s Problem f Given is f constant or balanced
David Deutsch
four possible oracles f
CONSTANT
BALANCED

6. Deutsch’s Problem f f H H H f Classical 2 queries + 1 auxiliary
operation f f
Quantum
CONSTANT H H
1 query +
3 auxiliary
operations
BALANCED H f

7. Deutsch’s Problem revisited
CLASSICAL
COMPLEXITY:
INPUT:
queries
PROMISE:
either constant or balanced
OUTPUT:
determine whether constant or balanced H H
00000
CONSTANT H H H H
any other output
BALANCED H H H H f

8. Deutsch’s Problem revisitedf

9. Fair comparison? classical deterministic:
classical probabilistic with error prob. :
quantum : 1
FAIR COMPARISON
Query in k places, if the queries had at least one 0 and one 1 then the function
is balanced, otherwise assume it is constant. Probability that it is balanced when
declared constant is

10. Bernstein-Vazirani Problem
INPUT:
PROMISE:
is of the form
OUTPUT:
binary string H H H H H H H H H H f

11. Bernstein-Vazirani ProblemH H f

12. Search Problem binary string INPUT: PROMISE: OUTPUT: Classical
Complexity:
PROMISE:
OUTPUT:
binary string
Searching large and unsorted database containing 2n items
Example of a sorted database:
a phone book if you are given a name and looking for a telephone number
n lookups suffice
Example of an unsorted database:
a phone book if you are given a number and looking for a name
you need to check 2n items before you succeed with probability P=1
you need to check 2n-1 items before you succeed with probability P=0.5

13. Grover’s algorithm binary string H H H H H H H H H H H H H H H H H H H
INPUT:
Quantum
Complexity:
PROMISE:
OUTPUT:
binary string
ITERATION 1
ITERATION 2 … H H H H H H H H H H H H H H H H H H H H f f0 f f0

14. Grover’s algorithm H H H H H H H H f f0 ITERATION
When we add a C-gate, which can stand for the interaction with an external system (environment, rest of the universe, measuring device), the interference is no longer perfect...
(a|0>+b|1>)m0 -> a|0> m0 +b|1>m1 , m1<->U(m0)
< m0 |m1><->0 (perfect measurement)

15. Grover’s algorithm H H H H H H H H f f0 ITERATION
When we add a C-gate, which can stand for the interaction with an external system (environment, rest of the universe, measuring device), the interference is no longer perfect...
(a|0>+b|1>)m0 -> a|0> m0 +b|1>m1 , m1<->U(m0)
< m0 |m1><->0 (perfect measurement)

16. Grover’s algorithm H H H H H H H H f f0 reflection about hyperplane
orthogonal to
reflection about
hyperplane
orthogonal to H H H H f f0
When we add a C-gate, which can stand for the interaction with an external system (environment, rest of the universe, measuring device), the interference is no longer perfect...
(a|0>+b|1>)m0 -> a|0> m0 +b|1>m1 , m1<->U(m0)
< m0 |m1><->0 (perfect measurement)

17. Grover’s algorithm H H H H H H H H reflection about reflection about
hyperplane
orthogonal to
reflection about
hyperplane
orthogonal to H H H H H H

18. Grover’s algorithm two reflections about the planes at angle
rotate the vector by

19. Grover’s algorithm H H H H H H H H

20. Grover’s algorithm H H H H H H H H H H H H H H H H H H H H f f0 f f0
ITERATION 1
ITERATION 2 … H H H H H H H H H H H H H H H H H H H H f f0 f f0

21. Grover’s algorithm After r iteration the state is rotated by
from the hyperplane
for large n
We iterate until

22. Query complexity classical probabilistic: quantum :
Quadratic speedup compared to classical search algorithms
Cryptanalysis: Attack on classical cryptographic schemes such as DES
(the Data Encryption Standard) essentially requires a search among
256=7 £ 1016 possible keys. If these can be checked at a rate of, say,
one million keys per second, a classical computer would need over a
thousand years to discover the correct key while a quantum computer
using Grover's algorithm would do it in less than four minutes.

23. Simon’s Problem period INPUT: PROMISE: OUTPUT: Example:
Classical
Complexity:
PROMISE:
OUTPUT:
period
Example:
s=110 is the period (in the group)
000
001
010
011
100
101
110
111
111
010
100
110

24. Simon’s algorithm H H
n qubits
n qubits

25. Simon’s algorithm H H n qubits n qubits
Solve the system of linear equations
Probability of failure of generating
linearly independent vectors y is less
than 0.75
Needs roughly n queries. Quantum complexity

26. Classical Complexity Analysis
Classical approach:
Randomly choose:
Evaluate:
Search for collisions:
Average number of
collisions:
Probability of at least
one collision:
Number of queries in a classical probabilistic approach :
CLASSICAL

27. Quantum Complexity Analysis 1

28. Quantum Complexity Analysis 2

29. Summary H f H H H H f0 f0 f f H H f
Deutsch (1985), Deutsch and Jozsa (92):
The first indication that quantum computers can
perform better H f
Grover: Polynomial separation H H H H f0 f0 f f
classical
quantum
Simon: Exponential separation H H f
classical
quantum