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1 The state 2 ( 00 + 11 ) Is entangled, to prove it we assume the contrary 1 ( 00 + 11 ) = (a0 0 + a1 1 )(b0 0 + b1 1 ) = 2 ! = a0b0 00 + a0b1 01 + a1b0 10 + a1b1 11 " a0b0 = 1 2 a0b1 = 0 a1b0 = 0 1 a1b1 = 2 contradiction ! Toffoli gate T : F23 " F23 ,T(x1, x 2 , x 3 ) = (x1, x 2 , x1 # x 2 $ x 3 ) This gate is called Toffoli gate Toffoli gate does not change bits x1 and x2 It computes the not-operation on x3 only if x1=1 and x2=1 ! Symbol for Toffoli gate 1 All Boolean circuits can be simulated by using only reversible gates Not gates are reversible And gate are simulated by Toffoli gate with x3=0 T(x1, x 2 , x 3 ) = (x1, x 2 , x1 " x 2 # x 3 ) T(x1, x 2 ,0) = (x1, x 2 , x1 " x 2 ) x1∨x2=¬(¬x1∧¬x2) Fanout (multiple wires leaving a gate) is simulated by the controlled not-gate with x2=0 ! C(x1, x 2 ) = (x1, x1 " x 2 ) C(x1,0) = (x1, x1 ) ! Spectral Representation of Unitary Operators Functions eiT defined on a self-adjoint operator defined by spectral representation T = "1 x1 x1 + "2 x 2 x 2 + L + "n x n x n e iT = e i"1 x1 x1 + e i"2 x 2 x 2 + L + e i"n x n x n iT * ! ! ! (e ) = e"i#1 x1 x1 + e"i#2 x 2 x 2 + L + e"i#n x n x n = (e iT ) eiT is unitary "1 Each unitary mapping can be represented as eiT, where T is a self-adjoint operator 2 Shor’s Algorithm Shor’s quantum algorithm for factoring relies upon a result from number theory Relates the period of a particular periodic function to the factor of an integer Given an integer n (number to be factored) construct a function fn(a)=xa mod n where x is an integer chosen at random that is a coprime to n Coprime, means that the greatest common divisor of x and n is 1, gcd(x,n)=1 3 Why is this function interesting with respect to the problem of factoring n It turns out that fn(a) is periodic For a=0,1,2,3,.. the values of the function fn(0),fn(1),fn(2),fn(3),.. fall into repeating pattern eventually Different values of x give rise to different patterns The number of values in between the repeating pattern, for a particular value x is called period of x modulo n indicated by r xr=1 mod n Shor’s Algorithm for factoring n 1) Pick a number q (with small prime factors) such that 2n 2 " q " 3n 2 ! 2) Pick a random integer x that is coprime to n 3) Repeat steps labeled (a) through (g) order log(q) times, using the same random number x each time 4 (a) Create a quantum memory register and partition the qubits into two sets, called Register1 and Register2 If the qubits in Register1 are in the state reg1 and those in Register2 are in the state reg2, we represent the joint state of both registers as (decimally) reg1,reg2 ! (c) Apply exploiting quantum parallelism the transformation xa mod n to each number in Register1 and Place the results in Register2 q#1 1 " = a, x a mod n $ q a= 0 ! 5 (d) Measure the state of Register2 obtaining some result k This has the effect of projecting out the state of Register1 to be a superposition of just those values of a such that xa mod n=k " = 1 A $ a',k a' #A Where A={a’: xa mod n=k} and ||A|| is the number of elements in this set ! How to find the period r of xa mod n=k? We will compute the Fourier transform of |a’> Fourier transform can be represented by an unitary operator (Quantum Fourier transform) • Unitary because of the Parseval’s identity States corresponding to integer multiplies of the inverse period, and these close to them have a higher value (greater amplitude) 6 (e) We compute the discrete Fourier transform of the projected state inRegister1 |a’> is mapped into q%1 1 a' " e 2 #ia$ c / q c & q c= 0 ' = 1 A q%1 & a' (A 1 e 2 #ia$ c / q c,k & q c= 0 ! (f) Measure the state of Register1 • The discrete Fourier transform is sampled This returns some number c’ c' # " q r ! 7 (g) To determine the period r we need to estimate λ Accomplished by computing the convergent of the fraction expansion of c’/q and retaining the closest such fraction of λ/r Grover’s Amplification Operators which we will use: We need a query operator which calls for value fy uses n qubits for the source register and one target bit y " F2n V f x = ("1) f (x ) x ! ! # 1,if x = y f y (x) = $ %0, otherwise We need a quantum operator Rn defined on n qubits and operating as Rn 0 = " 0 ! and Rn x = x ,x # 0 8 Amplitude Amplification Finding y by the quantum operator • Gn=-HnRnHnVf • Working on n qubits representing elements x • HnRnHn can be written as a 2nx2n matrix # 2 %1" 2 n % 2 %" n % 2 H n Rn H n = % 2 " % 2n % M %% " 2 $ 2n 2 2n 2 1" n 2 2 " n 2 M 2 " n 2 " 2 2n 2 " n 2 2 1" n 2 M 2 " n 2 " L L L O L 2 & 2n ( 2 ( " n ( 2 ( 2 " n ( 2 ( M ( 2 1" n (( 2 ' " ! HnRnHn can be also expressed as HnRnHn=I-2P Where I is a 2nx2n identity matrix and P is a 2nx2n projection matrix whose every entry is 1/2n 9 In this example we consider function y " F2n # 1,if x = y f 5 (x) = $ %0, otherwise • The search begins with superposition ! 1 2n #x x "F2n c 0 = c1 = c 2 = Lc 2 n $1 = 1 2n ! Vf5 is applied o change the sign of x=y Those amplitudes that are coefficients of a vector |x> satisfying f5(x)=1 become negative, c5 becomes negative 10 The average of the amplitude is now A= ! 1# n 1 1 & 1 # 2& 2 "1) n " n ( = 1" % n %( n( 2 $ 2 2 ' 2n $ 2 ' Inversion about the average-operator -HnRnHn will perform a transformation 1 2 " n a 2A " 1 2 n a 2A + 1 2 1 # n 1 2 2n # 3$ n 1 2n ! 1 2 " n a 2A " 1 2 n a 2A + 1 2 n # 1 2 n 1 2n # 3$ 1 2n The probability to find the answer is 9/2n by a single query, 4.5 times better than a classical randomized search can do ! 11 Iterative use of the mapping Gn=-HnRnHnVf Instead of a blackbox function that assumes only one solution, we will study a general function f having k solutions By using a quantum circuit, any problem in NP can be solved with a nonvanishing correctness probability in time ( O 2 n p(n) ! ) Where p is polynomial depending on the particular problem 12 No teleportation theorem A classical information channel can not transmit quantum information Remember: no cloning theorem? Quantum states can not be copied! A quantum state cannot be determined via a single measurement Once converted to classical information, quantum information cannot be recovered The entangled bits or qubits of a state are called an ebit An ebit is a shared resource An ebit is allways disrtributed between two particles (qubits) 1 An ebit provides a channel 11 ) ( 00for+ communication 2 Once either particle comprising the ebit is measured, the states of both particles become definite ! 13 Bell basis describes four orthogonal states 1 + ( 00 2 1 "# = ( 00 2 1 $+ = ( 01 2 1 $# = ( 01 2 " = + 11 ) # 11 ) + 10 ) # 10 ) ! The three particle state shown above thus becomes the following four-term superposition in the new basis: %# ( %$ ( %+$ ( 0 1 - + %# ( + + + " + " + , + , / 2 ' * ' * ' * ' * part We2 have& $done a change of basis on Alice's of the )B &+$ ) B &# ) B & # )B 1 . system into the orthogonal basis (Bell state) • No operation has been performed and the three particles are still in the same state ! 14 Given the density matrix p, von Neumann defined the entropy as S( p) = "Tr( pln p) ! It is a proper extension of the Gibbs entropy (and the Shannon entropy) to the quantum case We note that the entropy S(p) times the Boltzmann constant equals the thermodynamical or physical entropy It can be shown that if two observables are measured simultaneously, the uncertainty in their joint values must always obey the inequality (Heisenberg Uncertainty) ˆ ˆ 1 ˆ ˆ "A"B # [ ] A, B 2 1 "Aˆ "Bˆ # x Aˆ , Bˆ x 2 1 Var$ (A)Var$ (B) # x Aˆ , Bˆ x 4 [ ] [ ] ! 2 15 16