課程名稱︰量子計算與資訊導論
課程性質︰選修
課程教師︰管希聖
開課學院：理學院
開課系所︰物理所
考試日期（年月日）︰2018/06/07
考試時限（分鐘）：180
試題 :
(註：以'※'表示tensor product)
Problem 1. (20pt)
(a) (7pt)
Prove that non-orthogonal states cannot be reliably distinguished.
(b) (6pt)
Describe briefly the meaning (use) of the Bell inequalities and the basic assumptions to derive the inequality.
(c) (7pt)
Consider the two-qubit states (|01>-|10>)/√2. Show whether this state and the operators Q=Y, R=X, S=(X+Y)/√2, T=(X-Y)/√2 violate the Bell (or more precisely CHSH) inequality, <Q※S>+<R※S>+<R※T>-<Q※T>≦2.
(註：X, Y為Pauli matrices)
Problem 2. (10pt)
Let qubit 1 be the control qubit and qubit 2 be the target qubit of a CNOT gate.
(a) (4pt)
Show that CNOT=(I※H)Cz(I※H), where Cz is a controlled-Z gate.
(b) (6pt)
Prove that CNOT exp(iθZ※Z) CNOT = exp(iθI※Z), where θis a variable (not an operator).
Problem 3. (15pt)
(a) (8pt)
Describe how the quantum teleportation protocol works if the initial entangled state shared by Alice and Bob was(|00>-|11>)/√2.
(b) (4pt)
Draw its corresponding quantum circuit diagram, starting with the above entangled state generated from separable computational basis states, and the Bell's state measurements performed also in terms of the computational basis.
(c) (3pt)
Does the protocol violate the rule saying that information cannot be transmitted faster than light? Give your reason why.
Problem 4. (18pt)
(a) (10pt)
Describe how the BB84 quantum key distribution(QKD) protocol works.
(b) (4pt)
What are key elements or physics laws that make BB84 QKD protocol works?
(c) (4pt)
What is the error rate in the raw key if Eve intercepts every qubit sent by Alice, measures its polarization along X or Z axes and resends it to Bob in a BB84 protocol?
Problem 5. (17pt)
(a) (7pt)
Draw the quantum circuit diagram for the Deutsch's algorithm and describe how it works.
(b) (10pt)
Draw the four-qubit inverse quantum Fourier transform circuit diagram and describe how it works.
Problem 6. (10pt)
Draw the quantum circuit diagram for the phase estimation algorithm, and describe how it works for the case of a phase value that can be expressed exactly in t qubits.
Problem 7. (20pt)
(a) (5pt)
Show the equivalence of factoring and order finding.
(b) (15pt)
Describe how to factorize N=21 using Shor's algorithm in details and draw its corresponding quantum circuit diagram for the case that t=13 qubits for the first register, L=5 for the second register, the accuracy is to n=2L+1 bits, and the randomly chosen number that is coprime to N is x=5.
*Hint: continued fraction expansion:
[a_0,a_1,...a_n]=p_n/q_n
p_0=a_0, q_0=1
P_1=1+a_0*a_1, q_1=a_1,
p_n=a_n*p_{n-1}+p_{n-2}╮
├ for n≧2
q_n=a_n*q_{n-1}+q_{n-2}╯