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### Homework2

Course: ECE 227, Fall 2009
School: Duke
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Word Count: 650

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227/PHY272 ECE Quantum Information Science Homework #2 Due in Class 2/11/2009 1. Bell States are maximally entangled states of two particles. Using two basis states 0 and 1 , one of the Bell states reads = 0 1 0 2 + 1 1 1 (a) Express the Bell state using the states 1, 2 ( 2 ) 2 ( 00 + 11 ) 1, 2 2. =0 ( 1 1, 2 ) 1, 2 2 as the bases states. (b) Express the Bell State using the states 1, 2 = cos 0...

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227/PHY272 ECE Quantum Information Science Homework #2 Due in Class 2/11/2009 1. Bell States are maximally entangled states of two particles. Using two basis states 0 and 1 , one of the Bell states reads = 0 1 0 2 + 1 1 1 (a) Express the Bell state using the states 1, 2 ( 2 ) 2 ( 00 + 11 ) 1, 2 2. =0 ( 1 1, 2 ) 1, 2 2 as the bases states. (b) Express the Bell State using the states 1, 2 = cos 0 + sin 1 1, 2 and 1, 2 = sin 0 1, 2 + cos 1 1, 2 as the bases states. (c) Express the Bell State L 1, 2 =0 ( 1, 2 i 1 1, 2 ) using the states R 1, 2 =0 ( 1, 2 + i 1 1, 2 ) 2 and 2 as the basis states. 2. Asymptotic Notations are useful in quantifying the computational resources needed to run an algorithm. Some of the important properties of functions are for you to verify in this exercise. Show that (a) log n is O(nk) for any k > 0. (Textbook Exercise 3.11) (b) nk is O(nlog n) for any k, but that nlog n is lever O(nk). (Textbook Exercise 3.12) (c) cn is (nlog n) for any c > 1, but that nlog n is lever (cn). (Textbook Exercise 3.13) 3. Fredkin and Toffoli gates are elementary reversible logic gates that are universal: i.e. one of these gates could be used to construct any reversible circuit. Fredkin gate is sometimes called controlled-SWAP, and Toffolli controlled-controlled-NOT. (a) Construct a reversible circuit which, when two bits x and y are input, outputs ( x, y, c, x y ) , where c is the carry bit when x and y are added. (Textbook Exercise 3.31) (b) Construct a circuit simulating a Fredkin gate, using smallest number of Toffoli gates. (Textbook Exercise 3.32) (c) Prove that there are reversible Boolean functions which cannot be computed using only one and two bit reversible gates and ancilla bits. (Textbook Problem 3.5) 4. Single qubit gates are 2x2 unitary operators. (a) Show that an arbitrary 2x2 unitary matrix U can be written in the form i ( 2 2 ) cos ei ( 2+ 2 ) sin e 2 2 . (Textbook Exercise 4.9) U = i ( + 2 2 ) i ( + 2+ 2 ) sin cos e e 2 2 (b) From definition of rotation matrices and matrix verify multiplication, that this expression is equivalent to Z-Y decomposition (Eq. 4.11 in textbook). 5. Rotation Operators are exponentiated Pauli operators. Consider a qubit state = cos 2 0 + i sin 2 1 on the y-z plane of a Bloch sphere, and apply the rotational operator R X ( ) = exp[iX / 2] . Show that this operator corresponds to rotating the qubit state around the x-axis by an angle . 6. Quantum Fredkin and Toffoli Gates are quantum mechanical analogs of classical Fredkin and Toffoli gates. Unlike the classical case, we can construct a quantum Fredkin gate using only two-qubit gates and single qubit gates (qubit rotations). We will explicitly construct one here. (a) Give a quantum circuit which uses three Toffoli gates to construct a Fredkin gate. (Textbook Exercise 4.25) (b) Replace the Toffoli gates with CNOT and two-qubit controlled gate C(V) to obtain a Fredkin gate using only six two-qubit gates. (c) Can you come up with a much simpler construction? What is the minimum number of two-qubit gates with which you can construct a Fredkin gate (this is an open competition!!) (d) Verify the operation of the circuit shown in Fig. 4.9 of the textbook. 7. Matrix notations of two-qubit gates take time to get used to. In this exerci...

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