課程名稱︰量子演算法
課程性質︰選修
課程教師︰呂學一
開課學院：電機資訊
開課系所︰資訊工程
考試日期（年月日）︰2016/4/11
考試時限（分鐘）：120
試題 :
ps. 以下用(╳)表tensor product,用_表下標
共六題每題廿分，可按任何順序答題。每題難度不同，請審慎判斷恰當的解題順序。
第一題
Let (U,〈東,西〉),(V,〈南,北〉)∈H_2
Suppose that the composed system U (╳) V is in the state
2
iπ iπ ie
e e ie 1
── 東(╳)南 ＋ ── 東(╳)北 ＋ ─── 西(╳)南 ＋ ── 西(╳)北
2 √3 √6 2
What are the probabilities of the following events?
1. Getting 東(╳)北 from observing U(╳)V.
2. First getting 北 from observing V and then getting 東 from
observing U.
3. Under the condition of getting 西 from observing U, getting 南 from
observing V.
4. With probability 0.5 to observe U (respectively, V) first and then
observing V (respectively, U), getting 西 from observing U or
getting 南 from observing V.
第二題
Describe and explain the mechanism of teleporting the state of a
quantum bit via an existing EPR-pair by sending two classical bits.
第三題
Let (U,α) ∈ H_m, and (V,β) ∈ H_n. Prove that if T_U is an operation
of U and T_V is an operation of V, then T_U (╳) T_V is an operation of
U (╳) V.
第四題
Recall that the order of integer a in Z_n is the smallest positive
r
integer r such that a ≡ 1 (mod n). Run Peter Shor's order-finding
algorithm on F_m (╳) F_n for n = 15, a = 4, and m = 16 (although
2 2
n ≦ m ＜ 2n does not hold) and show the probability of outputing
the correct answer r = 2 for each possible observered state ∣p ﹥in
the first quantum system F_m.
第五題
Prove that β is an orthonormal basis of F_n, where for each
j = 1,...,n, β_j is the mapping from Z_n to C defined by
2πijk
───
def 1 n
β_j(k) ＝ ── e , for each k ∈ Z_n
√n
第六題
Let A and B be two complex square matrices. Prove or disprove that
if A (╳) B ＝ B (╳) A , then at least one of A and B is an all-zero
matrix.