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- 1. 1A Quick Safari ThroughQuantum Computation and Algorithms M. Reza Rahimi, Computer Science Department, Sharif University of Technology, Tehran, Iran, August 2005.
- 2. 2Outline Introduction Quantum Physics Quantum Physics Foundations Classical Computation Using Reversible Gates Quantum Gates and Universal Quantum Gates Quantum Complexity Class BQP Case Study: Grovers’ Search Algorithm Conclusion
- 3. 3 Introduction Computation is basically a physical fact. This is the origin of Church-Turing-Markov thesis, which implies that: A Partial function is computable (in any accepted informal sense) if and only if it is computable by some binary Turing machine. In this case, Church-Turing’s thesis is saying that the universe can be simulated by a Turing machine .
- 4. 4 So if we know the rules of the universe, we can make good physical model for computation. One of the first questions that leaded us toward quantum computation was ” What is the minimum energy for computing of a special problem?” In this case we must analysis our program in respect of power consumption. Landauer’s Principle: Erasure of information is necessarily a dissipative process. If the process of erasure is isothermal then the work needed, is at least: W=KTLn2.
- 5. 5 This rule tells us that any physical computation process that erases information, is energy consuming process. ( this fact is derived according to thermodynamics laws.) Charles Bennett found another interesting principle in 1973 that : “Any computation that can be carried out in the reversible process is dissipating no power.” So if we can make reversible gates, we can make computers that dissipate no power. This phenomenon will be very important if we want to make VLSI chip. The generated hit may damage the chip.
- 6. 6 As we know: “the universe is fundamentally quantum mechanics, and the rules of quantum mechanics are reversible in time, what kinds of problems quantum machine can solve for us?” The break of this question was Peter Shor Algorithm about integer factorization in polynomial time. It was not known that integer factorization has a classical polynomial time algorithm or not. The time complexity of Shor algorithm for L- digit number is: O ( L2 Log ( L) Log ( Log ( L))) The best known classical algorithm runs in: 1 2 O (exp(cL Log 3 L)) 3
- 7. 7 So studying quantum computation is useful, For examples :2. In chip design industry,3. In cryptography,4. …… In this talk I focuse on theoretical point of quantum computation. At first general principles of quantum physics and reversible computation is reviewd, then quantum complexity class is defined, and finally I focus on Grover’s search algorithm.
- 8. 8 Quantum Physics Quantum physics phenomena are very odd. Let’s take a look at an example. In figure 1 we have the wall with two slits on it and one electron gun which shoots electrons ( Young’s Experiment). The first experience: Cover one of the slits and compute the number of electrons that collide the wall. Figure 1 shows the results according to our expectation.
- 9. 9 Figure 1: The result of the first experience. The second experience :Now use both of the slits and count the number of electrons that collide the wall. What do you expect?
- 10. 10 Figure 2: The result of the second experience. Very interesting result! it seems that electrons behave like wave. The third experience: Now use one detector in one of the slits and see the movement of electrons. what do you expect to see on the wall?
- 11. 11 The experience shows that in this case we have the result expected in classical physics, that means the similar result drown in figure 1. So it seems that classical physics rules can not describe subatomic phenomena. We need physical framework for subatomic physics.
- 12. 12 Quantum Physics Foundations States: A state is a complete description of a physical system. In quantum mechanics a state is a ray in Hilbert Space . We use the following Dirac Ket Notation. ϕ = ∑αi i , αi ∈C i Observables: The observable is a property of a physical system that in principle can be measured. In quantum mechanics an observable is a self- adjoint operator: A = At
- 13. 13 Measurement: In quantum mechanics the numerical outcome of a measurement of observable A is an eigenvalue of A, and the state of it is eigenstate. Briefly we have: 2 2 ϕ =α 0 + β 1 α + β =1 Which α2 and β are the probability of system 2 to be in state 0 or 1 after measurement. Dynamics: Time evolution of the system is unitary we have Schrödinger equation. d ϕ(t ) = −iH ϕ(t ) → ϕ(t ) = U (t ) ϕ(0) dt U (t ) t U (t ) = I
- 14. 14 Examples: State of the n-qubit quantum register is: ∑α ∑α 2 ϕ = s S s =1 n s∈ 0 ,1} n { s∈ 0 ,1} { Suppose that we observe one qubit of quantum register and see it is 0. what is the state of the register after observation? 1 ∑α s 0 ⊗ S + βs 1 ⊗ S 0observed in qreg[1]→ is ∑α s 0 ⊗S ∑α 2s∈{0 ,1}n −1 s∈{0 ,1}n −1 s s∈{0 ,1} n −1
- 15. 15 For 1- qubit system we have: ϕ =α 0 +β 1 α + β =1 2 2 The above expression means that:Pr[ After measurement the 1-qubit is in state 0]= α 2Pr[ After measurement the 1-qubit is in state 1]= β 2 For 2-qubit system we have: 2 2 2 2 ϕ = α 00 00 + α 01 01 + α 10 10 + α 11 11 , α 00 + α 01 + α 10 + α 11 = 1 2Pr[ After measurement the 2-qubit is in state 00]= α 00 2Pr[ After measurement the 2-qubit is in state 01]= α01 2Pr[ After measurement the 2-qubit is in state 10]= α10 2Pr[ After measurement the 2-qubit is in state 11]= α11
- 16. 16Classical Computation Using Reversible Gates As stated before if we want to achieve minimum energy, we must use reversible gates. f : {0,1}n −1→ 0,1}n { 1 For example classical AND and OR gates are not reversible. One of the most popular reversible gates is Fredkin gate. The definition of this gate is as follow:
- 17. 17 f(a, b, c) = (a, if(a) then b else c, if(a) then c else b). It is easy to check that F(F(a,b,c))=(a,b,c). AND, OR, and NOT gates can be easily made up of Fredkin gate as follow: a a a^b b Fredkin Gate 0 ¬a^b Figure 3: AND gate Implementation. a a 0 ¬a Fredkin Gate 1 a Figure 4: NOT gate Implementation.
- 18. 18 So we can implement any logical circuit with Universal Fredkin gate. (in linear size with some control input bits). Input Bits Output Bits Fredkin Circuit Control Input Bits Some Junk output
- 19. 19Quantum Gates and Universal Quantum Gates As it was said quantum gates are unitary matrices. For example: u00 u01 1-input quantum gate is: U = u , UU t = I 10 u11 α0 0 + β0 1 u u01 α 0 + β 1 U = 00 u 10 u11 U (α0 0 + β0 1 ) = (α 0 + β 1 ).
- 20. 20 For 2-input quantum gate we have: u00 u01 u02 u03 α 00 00 + α 01 01 + α10 10 + α11 11 u u11 u12 u13 β 00 00 + β 01 01 + β10 10 + β11 11 U = 10 , UU = I t u u21 u22 u23 20 u u31 u32 u33 30 Generally for n-input quantum gate the matrix size is: 2 × 2 n n
- 21. 21 There are some examples of famous quantum gates: 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 1 F = 0 0 1 0 0 0 0 H= 1 − 1 0 0 0 0 1 0 0 2 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 8×8 Fredkin Quantum Gate Hadamard Quantum Gate Note that the Fredkin gate is permutation matrix and Hadamard gate has this property that if input is in state 0 or 1 the output state will be symmetric. (Both are Unitary Matrix).
- 22. 22 Suppose we have n-qbit register and on the first qbit U operates. what is the new state of the system? U n-qbit n-qbit (U ⊗ I )( 0 ⊗ ϕ 0 + 1 ⊗ ϕ 1 ) = U 0 ⊗ I ϕ 0 + U 1 ⊗ I ϕ 1 a00 B a01B ... A ⊗ B = a10 B ... ... ϕ ⊗ φ = ∑αiβ j i ⊗ j i, j ... ... ...
- 23. 23 As it is clear the set of all quantum gates are uncountable, So one may ask are there any small sets of universal quantum gates? The answer to this question is Yes. Researchers have shown that there are some universal quantum gates that we can make every quantum circuit with good approximation. (for example Tofolli and Hadamard Gates is universal set) But for now we only use Hadamard and Fredkin gates.
- 24. 24Quantum Complexity Class BQP Definition: A language L ⊂ {0,1}∗ is in BQP iff there is a set of quantum circuit {Cn }of size n k that: 2 x ∈ L ⇒ Pr{C ( x)1 = 1} ≥ 3 1 x ∉ L ⇒ Pr{C ( x)1 = 1} ≤ 3 Also the circuit must be uniform which means that a n deterministic polynomial time Turing machine with input 1 writes the description of the circuit {Cn } . Note! the Turing machine writes the approximation of the circuit because each gate can have complex numbers and for complex numbers we need generally infinite precession.
- 25. 25 Theorems: 1. P ⊆ BQP 2. BPP ⊆ BQP 3. BQP ⊆ PSPACE For proving the first one we know that every language in P has Polynomial size circuit, we can easily replace it with Fredkin gate. For the second one we know that BPP has polynomial size circuit with random control bits.
- 26. 26x1 |x1> Out Outx2 |x2>x3 Transformation to Quantum circuit |x3>x4 Classical Circuit |x4> Fredkin Circuitxn |xn>rand1 |0> Hrand2 |0> Hrandn |0> H For the proof of the last one you can see references.
- 27. 27Case Study: Grovers Search Algorithm Problem Statement: There is quantum space of size N we want the target state a . This Problem is usually called Quantum Database Search. For solving this problem we use Grover Search Algorithm. Before presentation of algorithm lets define some basic unitary operators.
- 28. 28 1 0 ... 0 0 1 ... 0 . . a = −1 0 ii 0 ... 0 1 N ×N Phase shift operator which changes the sign of the i th element . It is Obvious that this operator is unitary. −1 0 ... 0 0 1 ... 0D = HN H N , H N = H 2× 2 ⊗ H 2× 2 ⊗ ... ⊗ H 2× 2 ... ... ... ... Log 2 N Times 0 0 0 1 D is called diffusion operator .
- 29. 29 Lemma: Diffusion operator has two properties: It is unitary and can be efficiently realized. It can be interpreted as “inversion about the mean”. Proof: − 1 0 ... 0 0 1 ... 0 D =H N ... HN ... ... ... 0 0 ... 1 − 2 0 ... 0 0 0 ... 0 =H N ... +I N H ... ... ... 0 0 0 0 − 2 N − N 2 ... − N 2 − 2 N − N 2 ... − N 2 = +I ... ... ... ... 2 N − − N 2 − N 2 − N 2 2 N + − 1 − N 2 ... − N 2 − N 2 − N + 2 1 ... − N 2 = ... ... ... ... − N 2 − N 2 ... − N + 2 1
- 30. 30 α 1 β1 α2 β2 . = . → β = − 2 α +α = −2µ + α . N D i ∑ j i N j =1 i . . α β N Nµ µ αi αj αi αj µ βi βj
- 31. 31 11. Start state is ϕ =∑ x . x N2. Invert the phase of a using phase shift operator.3. Then invert about the mean using D.4. Repeat step 2 and 3 N times.5. Measure. According to the last relation it is obvious that after N we can measure a with probability at least 0.5. Running Time of The Grover Search Algorithm = O ( N ).
- 32. 32Conclusion In this talk we took a glance at quantum computation. It is clear that quantum computing can solve some problems that are hard for classical computers. Some people may ask “what is the philosophical source of the power for quantum machines?” Really the sources of the power of quantum machines are quantum superposition and quantum entanglement.
- 33. 33 Talking about these properties is a little long and deep so for more information you can see books in quantum mechanics. Nowadays researchers spend a lot of time working on theoretical and practical aspects of quantum machines. The END

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