An introduction to quantum computing for programmers.

1. Quantum Computing
Programming and Algorithms
Cars.com Tech Tailgate 2016
Stephen Habegger

2. 2
Quantum Computing
•Cryptography
•Authentication
•Data Compression
•Database
•Etc.

3. 3
Quantum Computing
•Mathematics
•Quantum Physics
•Quantum Computer Architecture
•Quantum Algorithms
•Building a Quantum Circuit
• www.research.ibm.com/quantum/
•Quantum Computing Software
•Quantum Computing Hardware
•Questions

4. 4
Mathematics

5. 5
Complex Numbers
● Imaginary
● Complex
● Complex Conjugate
● Modulus

6. 6
Complex Numbers
● Polar Representation
● Euler’s Formula https://en.wikipedia.org/wiki/Complex_number

7. 7
Linear Algebra: Matrix Multiplication

8. 8
Linear Algebra: Tensor Product

9. 9
Quantum Physics

10. 10
Light Polarization
Unpolarized
Light
Vertical
Polarization
Filter
Vertically
Polarized
Light
Horizontal
Polarization
Filter
No
Light

11. 11
Light Polarization
Unpolarized
Light
Vertical
Polarization
Filter
Vertically
Polarized
Light
Horizontal
Polarization
Filter
Horizontally
Polarized
Light
(New Basis)
Diagonal
Polarization
Filter
Diagonally
Polarized
Light
(New Basis)

12. 12
Qubits: The Bloch Sphere

13. 13
Quantum Computer
Architecture

14. 14
State Vectors
The state of a bit/qubit can be represented by a vector.
In the case of a qubit this serves as a complementary
representation to the Bloch sphere, where the
probability of measuring a given state is given by the
modulus of the associated value.

15. 15
State Transitions
•Operations on bits/qubits can be described as
matrices
•Multiplying a matrix which represents an operation
by an input state vector returns the resulting output
state vector

16. 16
Multiple Qubits
Multiple independent qubits (or operations on qubits)
can be represented by the tensor product of the two
state vectors (or operation matrices)

17. 17
Classical Logic Gates
IDENTITY NOT AND
OR XOR

18. 18
Quantum Logic Gates
X (NOT) Z S
T
Y
Hadamard (H) Sqrt NOT

19. 19
Reversible Logic Gates: Toffoli

20. 20
Reversible Logic Gates: Toffoli

21. 21
Quantum AND Gate via Toffoli

22. 22
Quantum Algorithms

23. 23
General Quantum Algorithm
1.Qubits start in some classical state
- e.g. 0000 or 1001
2.System is placed into a superposition of states
3.Act upon qubits using unitary operations
4.Measure some of the qubits (collapsing to a classical
state)

24. 24
Deutsch’s Algorithm
•Determines whether a function is “balanced” or
“constant”
•Operates on functions in the space
•Classical algorithms require 2 calls to the function,
while the quantum algorithm requires 1. This is
achieved by a “change of basis” in the problem
space.

25. 25
Deutsch’s Algorithm
Balanced Function Constant Function
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1

26. 26
Deutsch’s Algorithm
www.research.ibm.com/quantum/
CONSTANT
BALANCED

27. 27
Grover’s Algorithm
Find an element in a set in expected time
1.Start with a selector function
2.Repeat times
a.Perform phase inversion using selector function (selected
element takes opposite phase from other elements)
b.Perform inversion about the mean (amplitude of selected
element is increased)
3.Measure

28. 28
Grover’s Algorithm
Phase Inversion Inversion About Mean

29. 29
Grover’s Algorithm

30. 30
Cryptography
•Public key cryptography (RSA)
• Relies on difficulty of factoring large numbers (with large
prime factors)
• Shor’s algorithm can factor integers in polynomial
expected time
•Quantum private key generation & eavesdropping
• Classical bits randomly generated and transmitted as
qubits in randomly selected bases
• Qubits measured in randomly selected bases
• Subset of qubits transmitted/measured in same bases are
compared; rest are used as private key

31. 31
Quantum Computing
Software

32. 32
Quantum Assembly
•Based on QRAM architecture
• Provably equivalent to Quantum Circuit and Quantum Turing
Machine architectures
•Act upon a register of qubits
•Commands
• INITIALIZE
• SELECT
• APPLY
• CONCAT
• TENSOR
• INVERSE
• MEASURE
•QASM (IBM, MIT, etc.)

33. 33
Higher-Level Quantum Languages
•Types (other than quantum bit sequences)
•Functions (must be reversible)
•QCL
• C-like syntax
• QReg type
• User-defined operators and functions
•Q
•QFC
• Flowchart-type syntax
• Classical control statements

34. 34
Quantum Computing
Hardware

35. 35
Quantum Hardware
•Challenges
• Allow entanglement
• Avoid decoherence
•Ion Trap
• Confine ionized atoms in electromagnetic field
• Excited vs ground state (optical pumping via lasers)
•Optical Quantum Computers
• Light polarization

36. 36
Questions

37. 37
Appendix A: Resources

38. 38
Resources
Yanofsky, Noson S.; Mannucci, Mirco A. (2008-08-11). Quantum Computing for
Computer Scientists. Cambridge University Press - A.
Perry, Riley Tipton (2012-07-11). Quantum Computing from the Ground Up.
Wspc.
IBM Quantum Experience. www.research.ibm.com/quantum/
Microsoft Language-Integrated Quantum Operations: LIQUi|>.
https://www.microsoft.com/en-us/research/project/language-integrated-
quantum-operations-liqui/
Quantum Simulator. https://github.com/shabegger/quantum-emulate

39. 39
Appendix B: Phase Inversion
Calculations

40. 40
Grover Phase Inversion

41. 41
Grover Phase Inversion

42. 42
Grover Phase Inversion

43. 43
Grover Phase Inversion

44. 44
Grover Phase Inversion

45. 45
Grover Phase Inversion

46. 46
Grover Phase Inversion

47. 47
Grover Phase Inversion

48. 48
Grover Phase Inversion