1. Quantum Computing MAS 725
Hartmut Klauck NTU

2. Organization Lectures: Mo. 10:00, TR+9 Lecturer: Hartmut Klauck
Office: SPMS-MAS-05-44
Website:

3. Grading Homework (biweekly): 40% Final exam: 60%
Homework must be written individually!
And handed in on time

4. Required Background Linear algebra Some basic probability theory
No background in physics required

5. Textbook
Nielsen/Chuang: Quantum Computation and Quantum Information (Cambridge)

6. Recommended Reading http://homepages.cwi.nl/~rdewolf/qcnotes.pdf

7. Quantum Mechanics
Quantum mechanics is one of the basic theories of physics
Quantum mechanics is concerned with states, and how they evolve/change
Includes many “strange” effects that are different from “classical”, Newtonian mechanics:
Superposition
Entanglement
Such effects usually appear in very small systems

8. Quantum Mechanics and Computing
Moore’s Law: The number of transistors that can be placed on a chip doubles every two years
I.e., the computational power doubles
This trend has been approximately true for more than 50 years
Main way to achieve this is by making smaller transistors!
Even today quantum mechanical effects are important to chip design

9. Quantum Mechanics and Computing
Another problem: heat generation in integrated circuits
This heat is the result of erasing information
Quantum computations are (for the most part) reversible
Reversible computations (ideally) do not generate (much) heat

10. Quantum Mechanics and Computing
Chip designers nowadays mostly “combat” quantum effects
Is it possible to make good use of quantum effects?

11. Quantum Computing First suggested by Feynman and Benioff in the 1980’s
Feynman’s observation:
Simulating quantum systems on classical computers takes exponential time in the ‘size’ of the quantum system
Conclusion: build universal quantum systems
Quantum systems that can simulate all other quantum systems (up to a size)
I.e., quantum computers

12. Quantum Computing
Hence reasons for investigating quantum computing are:
Making good use of quantum effects instead of trying to force microscopic system to adhere to classical physics
There are quantum algorithms and protocols that achieve things that seem to be impossible for classical algorithms/protocols
If the world is quantum mechanical, the ultimate limits of computation are determined by quantum physics

13. Quantum Computing Examples
Some example of tasks that quantum computers can do:
Efficiently factor natural numbers
Break public key cryptosystems like RSA
Search an unordered database in sublinear time
Provide cryptographic protocols that are secure without placing assumptions on the computational power of an eavesdropper

14. Quantum Computing Models
There are several models of quantum computing
E.g. Deutsch (1985) defined Quantum Turing Machines as a universal model of quantum computation
Another (easier to handle) model are quantum circuits
But first we need to understand some basics about quantum mechanics

15. Quantum Mechanics
The double slit experiment for light

16. Quantum Mechanics Perform the “same” experiment with electrons
We observe the same outcome of the experiment
Even when single electrons are emitted
The wave-like behavior is not just statistical

17. Quantum Mechanics
The name “Quantum Mechanics” (coined by Planck) derives from the fact that certain quantities can change only by a discrete amount
E.g. The smallest unit of electromagnetic radiation is a photon (a quantum of light)
It is possible to emit and detect single photons

18. Quantum Mechanics

19. Quantum Mechanics

20. Some History
Development of quantum mechanics: Planck 1900, Schrödinger, Heisenberg, Bohr, Einstein.....
1930’s: von Neumann’s formalism
1935: Einstein, Podolsky, Rosen describe “entanglement” in an attempt to show that quantum mechanics is not a “complete” theory of reality (German “Verschränkung”)
Today quantum mechanics is the best established theory in physics

21. (Quantum) Computer Science
1936: Turing defines a “universal” machine (Church Turing Thesis)
1948: Shannon’s Information Theory
1965: Moore’s Law
1982: Feynman proposes quantum computers (to simulate quantum systems)
1982 Wiesner: first proposal of quantum cryptography published (after more than 10 years)

22. (Quantum) Computer Science
1985: Deutsch finds the first quantum algorithm
1993: quantum teleportation
1994: Shor finds a quantum algorithm for factorization
1996: Grover’s algorithm finds a marked element in a database with n elements in time
Since then the field is steadily growing…

23. Quantum States
Quantum mechanics is an abstract theory of states and transformations on states
Can be derived from certain axioms
Quantum states are vectors in a Hilbert space
Hilbert Space:
A real or complex vector space with an inner product that maps vectors to their length
Must be complete
We will only consider finite dimensional spaces
Usually either Rn or Cn

24. Bits A bit is either the value 0 or the value 1, stored in a register
We will write the state of a bit as |0i, |1i
Examples:
A bit stored in the memory of a computer
The path that a ball took in a giant double slit experiment

25. Bits and Qubits
We identify the states |0i, |1i with the basis vectors of a two dimensional space (say C2): (1,0) and (0,1)
The states of a quantum bit (qubit) are arbitrary unit vectors in C2
Hence all the states of a qubit are: |0i + |1i with ||2 +||2=1

26. Qubits |0i, |1i are two basis vectors in C2
Qubits have states:  |0i+ |1i with ||2 +||2=1
, are called amplitudes
Qubit states are unit vectors under the euclidean norm

27. Comparison to Probability Theory
Suppose we have a random bit (say a coin flip)
Then we need to specify the probability of 1 and 0 (coin may not be fair)
For example 0 has probability p, 1 has probability 1-p ) probability distributions on bits are unit vectors under 1-Norm
Qubits: , are complex numbers, possibly negative!
The squares of the absolute values of the amplitudes form a probability distribution

28. The Quantum Formalism Quantum states are vectors in a Hilbert space
The Hilbert space corresponds to a register that can hold a quantum state
Hilbert space here: Ck with the inner product h (vi) | (wi) i = i=1…k vi* wi x*: complex conjugate

29. Dirac Notation h  | “BRA” row vector |  i “KET” column vector
h  |  i inner product (product of a row and a column vector)
|ÁihÃ| outer (matrix valued) product

30. Many Qubits To hold k qubits we need a Hilbert space of dimension 2k
I.e. 2k basis vectors (corresponding to the 2k values of k bits)
First notation: |ii, i=1,...,2k. Unit vectors are of the form i i |ii; i=1....2k with i |i |2 = 1
Better notation: identify i=1...2k with x2{0,1}k
Basis states are |xi, x2 {0,1}k
Basis states correspond to classical values a register can hold
General quantum states are linear combinations of the 2k classical (basis) states
Also called “superpositions”

31. Tensor Product
For Hilbert spaces H, K, with dimensions dH and dK their tensor product H­K is a Hilbert space of dimension dH¢dK
Tensor product of vectors: (a1,..., al) ­ (b1,...,br)= (a1b1,a1b2,...,a1br,a2 b1,......,albr)
Example: |0i = (10)T; |1i= (01)T and |01i= |0i­|1i = (0100)T
A basis of H­K: all |xi­|yi =|xyi where |xi,|yi are basis vectors of H,K
Not all vectors in H­K are tensor products of vectors in H and K

32. Example Basis of C4: |00i, |01i, |10i, |11i Another basis:
(|00i + |11i) (|00i - |11i) (|01i + |10i) (|01i - |10i) (scaled by square root of 2)
None of these are tensor products of vectors in C2

33. What can we do with one or more qubits?
Quantum systems evolve according to the Schrödinger equation
The result can be described as the application of a unitary transformation to the quantum state
Additionally quantum states can be measured
This leads to observable output
Need some background from linear algebra…

34. Linear Algebra
Linear transformations: A(x+y)=Ax + Ay
x,y: vectors in Ck, A: k£k matrix (complex entries)
Over the reals a linear transformation O is orthogonal, if OOT=I
Over the complex numbers a matrix U is unitary, if UUy =I U*: take the complex conjugate of all entries Uy = (U*)T
Unitary transformation preserve the euclidean length of vectors
Transformations in QM: unitary (i.e., reversible and length preserving)

35. Examples On one qubit: classical transformations: identity, negation
Hadamard Transformation:

36. Applying Hadamard

37. Applying Hadamard

38. Applying Hadamard

39. Unitary Transformations
Define U |xi for all x2 {0,1}k ) U is defined. The U|xi need to be unit vectors and U|xi? U|yi for all xy
Tensor product for matrices:
A ­ B=

40. Unitary Transformations

41. Hadamard Transformation
x,z2{0,1}n and x¢z= xizi
For any x we have H­n |xi= 1/2n/2 (|0i +(-1)x(1) |1i)­­(|0i +(-1)x(n) |1i)

42. Applying many unitary transformations
Later we will construct unitary transformations as the product of many “simple” unitary transformation
First applying a unitary U, and then a unitary V is the same as applying the product VU.
Note that the product of two unitary matrices is unitary
Careful: matrix multiplication is not commutative!
The exact sequence of multiplications matters

43. Measurements
Quantum states (unit vectors in Ck) can be changes by applying a unitary transformation
Computations on quantum states consist of unitary transformations and measurements
Measurements allow us to access the result of a computation
What happens if we measure i i |ii ?
The result will be i with probability |i|2
i |i2|=1 is very helpful now!
After measuring the value i the state “collapses” to |ii

44. Example Measuring the state
Will result in the outcome 0 or 1, each with probability ½
If we measured 1, the resulting state after the measurement will be

45. Overview
Quantum states: unit vectors in a Hilbert space, the log of the dimension corresponds to the number of qubits
States in a Hilbert space of dimension 2k correspond to superpositions of strings of length k and the space is a register of k qubits
Evolution: by applying unity transformations
Measurement:  i |ii results in output i with probability |i|2, the state collapses to |ii if i is the measurement result