[試題] 101-2 管希聖 量子計算與量子資訊導論
課程名稱︰量子計算與量子資訊導論
課程性質︰選修
課程教師︰管希聖
開課學院：理學院
開課系所︰物理學研究所、奈米科技學程
考試日期（年月日）︰2013/05/04
考試時限（分鐘）：180 min
試題 :
Midterm Examination
PHYS D3040: Introduction to Quantum Computation and Information
May 4, 2013 Friday
Problem 1.
(a) Describe what the quantum no-cloning theorem is, and give a proof of it.
(b) Describe what the EPR thought experiment(paradox) is.
(c) Describe briefly the meaning(use) of the Bell inequalities and the basic
assumptions to derive the inequality.
(d) Consider the two-qubit state 1/√2 (∣00〉 - ∣11〉). Show that this state
and the operators Q = Y, R = X, S = 1/√2 (X + Y), and T = 1/√2 (X - Y)
violate the Bell inequality (or more precisely, the CHSH) inequality,
〈Q ×S〉+〈R ×S〉+〈R ×T〉-〈Q ×T〉≦2
[NOTE] In this problem, the symbol " ×" indicates the tensor product.
Problem 2.
(a) Let U = exp(i J_x X ×X + i J_y Y ×Y + i J_z Z ×Z) and J_x, J_y, J_z be
some variables (not operators). Show that
(i) X ×X, Y ×Y, and Z ×Z commutes with each others, and
U = exp(i J_x X ×X) exp(i J_y Y ×Y) exp(i J_z Z ×Z);
(ii) (I ×Z) U (I ×Z) = exp(-i J_x X ×X - i J_y Y ×Y +i J_z Z ×Z);
(iii) (H ×H) U (H ×H) = exp(i J_z X ×X + i J_y Y ×Y +i J_x Z ×Z). (note
the places where J_x and J_z appear).
(b) Let qubit 1 be the sontrol qubit and the qubit 2 be the target qubit of a
CNOT gate. Prove the following identities:
(i) CNOT (X ×I) CNOT = X ×X,
(ii) CNOT exp(i θ Z ×Z) CNOT = exp(i θ I ×Z), where θ is a variable
(not an operator).
[NOTE] In this problem, the symbol " ×" indicates the tensor product.
Problem 3.
(a) Describe how the quantum teleportation protocol works if the initial
entanglement state shared by Alice and Bob was 1/√2 (∣01〉 - ∣10〉).
(b) Draw its corresponding quatum circuit diagram, starting with above
entangled state generated from separable computational basis states, and
the Bell's state measurements performed also in terms of the computational
basis.
(c) Does the protocol violate the rule saying that information cannot be
transmitted faster than light? Give your reason why.
Problem 4.
(a) Draw the quantum circuit diagram for the Deutsch' algorithm and describe
how it works.
(b) Draw the four-qubit inverse quantum Fourier transformation circuit diagram
and describe how it works.
Problem 5.
(a) Show the equivalence of factoring and order finding.
(b) Describe how to factor N = 15 using Shor's algorithm in details and draw
its corresponding quantum circuit diagram for tha case t = 11 qubits for
the first register, L = 4 for the second register, the accuracy is to
n = 2L + 1 bits, and the randomly chosen number that is co-prime to N is
x = 8.
[Hint] continuous fraction expansion:
[a_0, a_1, ..., a_n] = p_n / q_n,
p_0 = a_0, p_1 = 1 + a_0 a_1, p_n = a_n p_{n-1} + p_{n-2} for n≧2,
q_0 = 1, q_1 = a_1, q_n = a_n q_{n-1} + q_{n-2} for n≧2.
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