Quantum Circuits

Question 1

Quantum Logic

Answer 1

A or B /= B or A
(1)Measure in the vertical direction if +1 stop the proposition is true
(2)else measure in the horizontal direction if +1 it is true
else it is false
Consider the electron goes upward then:
(1) is true
but if we measure in the horizontal direction (2) is true with 1/2 probability if false we measure (1) the spin has been destroyed by the measurement and there is 1/2 to be upward or downward. Overall 0.25 to be false !
=> A or B /= B or A

A and B uncertain principle

Question 2

Operations on entangled states

Answer 2

although entangled space are not separable, separable operations can be applied to them

Question 3

Quantum circuits

Answer 3

Represent the evolution of the quantum system
State of system described by tensor product of the components. The operators can be applied on each state by tensor of identity matrices and unitary matrices: (X (x) I ) |phi>
initial state
Wires are the lines and it indicate information flow of qbits: left to right
Boxes are the unitary operators
then measurement gate: on 1 or all qbits

Question 4

CNOT operator circuit

Answer 4

Applied on 2 qbits, 1 is useless
Control bit is black dot
Target bit is (+)
The control bit is unchanged
The target bit is XOR of x, y : x(+)y. If control = 0, unchanged, if control = 1, target switched
NOT(x,y)->(x,x(+)y)
3 CNOT makes a swap operator

Question 5

CNOT operator circuit non adjacent bits

Answer 5

SWAP apply CNOT and SWAP again
if target is on 1st qbit and target on 3rd qbit, swap 2nd and 3rd:
(I2(x)SWAP)(CNOT(x)I2)(I2(x)SWAP)
1st unchanged, 3rd unchanged, 1st unchanged
SWAP and CNOT are a unitary transformation on 2 bits

Question 6

Quantum gates

Answer 6

can do everything with just H and CNOT (+ T to change base)

Question 7

Gottesman Kill Theorem

Answer 7

X,Y,Z,H,CNOT quantum circuits can be efficiently simulated by a classical machine