QUANTUM COMPUTING Carlile Lavor clavor@ime.unicamp.br UNICAMP, - PDF document

QUANTUM COMPUTING Carlile Lavor clavor@ime.unicamp.br UNICAMP, Brazil

1 The postulates of quantum mechanics � Postulate 1: there is a complex vector space with inner product associated to any closed physical system, where a state of this system is described by a unit vector. � System: Quantum Bit (qubit) � Vector Space: C 2

" # 1 � An orthonormal basis for C 2 can be given by 0 " # 0 and , which will be represented by the Dirac 1 notation: " # 1 j 0 i = ; 0 " # 0 j 1 i = : 1 � A general state j i of a qubit can be given by j i = � j 0 i + � j 1 i ; where j � j 2 + j � j 2 = 1 ( �; � 2 C ). � The basis fj 0 i ; j 1 ig is called the computational basis and the vector j i is called a superposition of the states j 0 i and j 1 i , with amplitudes � and � .

� Postulate 2: the evolution of a closed quantum system is described by a linear operator which pre- serves the inner product (unitary operator ) . That is, j 2 i = U j 1 i ; where j 1 i is the state of the system at time t 1 , j 2 i is the state at time t 2 , and U is a unitary operator. � There is a unitary operator which transforms j 0 i in j 1 i and vice versa. It is denoted by X and its matrix representation, in the computational basis, is given by " # 0 1 X = : 1 0 � Another example is the operator Z : " # 1 0 Z = : 0 � 1

� It is easy to see that X j 0 i = j 1 i ; Z j 0 i = j 0 i ; and, for j i = � j 0 i + � j 1 i , = � j 0 i + � j 1 i ; X j i Z j i = � j 0 i � � j 1 i : � However, note that for the Hadamard operator, given by " # 1 1 1 H = ; 1 � 1 2 1 = 2 we obtain 1 H j 0 i = 2 1 = 2 ( j 0 i + j 1 i ) :

� The dual of j ' i 2 C 2 , denoted by h ' j , is de�ned by h ' j = j ' i y : � Given j ' i ; j i 2 C 2 , the inner product h ' j i and the outer product j ' ih j are de�ned, respectively, by j ' i y j i ; h ' j i = j ' ij i y : j ' ih j = � Example: h 0 j 1 i = 0 and " # 0 1 j 0 ih 1 j = : 0 0

� Postulate 3: a measurement of a quantum sys- tem is described by a hermitian operator M ( M y = M ), where the possible outcomes of the measure- ment correspond to the eigenvalues � i of M . � Upon measuring the state j i , the probability of getting result � i is given by p � i = h j ( j i ih i j ) j i ; where fj i ig is an orthonormal basis of eigenvec- tors associated to f � i g . � Given that outcome � i occured, the state of the system immediately after the measurement is j � i i = ( j i ih i j ) j i : p 1 = 2 � i

� Example: consider the hermitian operator Z , " # 1 0 Z = ; 0 � 1 which can be written as Z = j 0 ih 0 j � j 1 ih 1 j : � Suppose that the state being measured is j i = � j 0 i + � j 1 i : Then, j � j 2 ; p 1 = � j 1 i = j � jj 0 i ; and j � j 2 ; p � 1 = � j � 1 i = j � jj 1 i :

� Postulate 4 : the joint state of a system with com- ponents j 1 i ; j 2 i ; :::; j n i is the tensor product j 1 i � j 2 i � ::: � j n i . � For A 2 C m � n and B 2 C p � q , we de�ne the tensor product A � B by: 2 3 A 11 B A 12 B � � � A 1 n B 6 7 A 21 B A 22 B � � � A 2 n B 6 7 A � B = 5 : 6 7 . . . ... . . . 4 . . . A m 1 B A m 2 B � � � A mn B � Example: 2 3 0 " # " # 6 7 1 0 1 6 7 j 0 i � j 1 i = � = 6 7 0 1 4 0 5 0 and 2 3 0 " # " # 6 7 0 1 0 6 7 j 1 i � j 0 i = � = 5 : 6 7 1 0 4 1 0

2 Grover�s algorithm � Problem: given an unstructured list with N ele- ment, �nd a speci�c one. � Suppose that the list is f 0 ; 1 ; :::; N � 1 g , where N = 2 n , and that the function that recognizes the searched element i 0 is given by f : f 0 ; 1 ; :::; N � 1 g ! f 0 ; 1 g ; where ( 1 ; if i = i 0 f ( i ) = : 0 ; if i 6 = i 0

3 The �rst Grover�s operator � For each element of the list f 0 ; 1 ; :::; N � 1 g , we associate the state j i i n of n qubits. � We search for an operator U f which transforms j i i n into j f ( i ) i 1 : � Since U f must be unitary, consider U f j i i n j 0 i 1 � ! j i i n j f ( i ) i 1 :

� Then, ( j i ij 1 i ; if i = i 0 U f ( j i ij 0 i ) = : j i ij 0 i ; if i 6 = i 0 � If the second register is j 1 i , we de�ne ( j i ij 0 i ; se i = i 0 U f ( j i ij 1 i ) = : j i ij 1 i ; se i 6 = i 0 � In a more compact form, we have U f ( j i ij j i ) = j i ij j � f ( i ) i ; where � is the sum modulo 2 (note that U f 2 C 2 n +1 � 2 n +1 ).

4 Superposition of the elements of f 0 ; 1 ; :::; N � 1 g � The �rst and second registers are initialized on the states j 0 i n and j 1 i 1 , respectively. � If we apply the operator H on each qubit of these registers, we obtain that 2 n � 1 X 1 j i = ( H j 0 i ) � n = j i i 2 n= 2 i =0 and 1 j�i = H j 1 i = 2 1 = 2 ( j 0 i � j 1 i ) : � Now, applying the operator U f on j ij�i , we get 0 1 N � 1 X 1 ( � 1) f ( i ) j i i A j�i : @ U f ( j ij�i ) = N 1 = 2 i =0

5 The second Grover�s operator � The next step should be to increase the amplitude of the searched element, which can be obtained using another unitary operator de�ned by 2 j ih j � I: � Applying this operator on the state N � 1 X 1 ( � 1) f ( i ) j i i N 1 = 2 i =0 and measuring the �rst register, the probability of getting the searched element is � � 2 � 3 N � 4 � � : � N 3 = 2 � The composition of the operators U f and 2 j ih j� I is called Grover�s operator G , that is, G = ((2 j ih j � I ) � I ) U f :

6 Complexity of Grover�s algorithm � It can be proved that the resulting action of the operator G k ( k 2 N ) rotates j i towards j i 0 i by k� rad, in the subspace spanned by j i and j i 0 i , where � is the angle between j i and G j i . � It can also be proved that the number of times k that the operator G must be applied so that the angle between j i 0 i and G k j i becomes zero is � 1 � � � N � 2 �� � 1 k = arccos arccos ; N N which implies that N 1 = 2 = � k 4 ) k = O ( N 1 = 2 ) : lim N !1