Quantum Computation and Algorithms
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Quantum Computation and Algorithms

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Presented as a short talk in IPM (Institute for Research in Fundamental Sciences), Tehran.

Presented as a short talk in IPM (Institute for Research in Fundamental Sciences), Tehran.

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Quantum Computation and Algorithms Quantum Computation and Algorithms Presentation Transcript

  • 1A Quick Safari ThroughQuantum Computation and Algorithms M. Reza Rahimi, Computer Science Department, Sharif University of Technology, Tehran, Iran, August 2005.
  • 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 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 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 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 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 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 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 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 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 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 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 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 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 0observed 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 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
  • 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 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 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
  • 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 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 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 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 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.
  • 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 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.
  • 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.
  • 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 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 ... 0D = 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 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  α 1   β1       α2   β2   .  =  .  → β = − 2 α +α = −2µ + α . N D     i ∑ j i N j =1 i  .   .  α   β   N  Nµ µ αi αj αi αj µ βi βj
  • 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 ).
  • 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 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