QUANTUM COMPUTING - EXAMPLE 32.5 : In quantum teleportation, let (C0 + D1) (00 + 11) = C000 + C011 + D100 + D111. Ba = 00 + 11, Bc = 10 + 01, Be = 00 - 11, Bm = 10 - 01. (a) Find the values of 00, 01, 10 and 11 in term of Ba, Bc, Be and Bm. (b) Prove by calculation that (C0 + D1) (00 + 11) = 0.5 [ Ba (C0 + D1) + Bc (C1 + D0) + Be (C0 - D1) + Bm (-C1 + D0) ].
QUANTUM COMPUTING - ANSWER 32.5 : (a) Ba + Be = (00 + 11) + (00 - 11) = 2 (00), 00 = (Ba + Be) / 2. Bc - Bm = (10 + 01) - (10 - 01) = 2 (01), 01 = (Bc - Bm) / 2. Bc + Bm = (10 + 01) + (10 - 01) = 2 (10), 10 = (Bc + Bm) / 2. Ba - Be = (00 + 11) - (00 - 11) = 2 (11), 11 = (Ba - Be) / 2. (b) (C0 + D1) (00 + 11) = C000 + C011 + D100 + D111 = (00) C0 + (01) C1 + (10) D0 + (11) D1 = (1/2) (Ba + Be) C0 + (1/2) (Bc - Bm) C1 + (1/2) (Bc + Bm) D0 + (1/2) (Ba - Be) D1 = (1/2) [ (Ba + Be) C0 + (Bc - Bm) C1 + (Bc + Bm) D0 + (Ba - Be) D1 ] = 0.5 ( Ba C0 + Be C0 + Bc C1 - Bm C1 + Bc D0 + Bm D0 + Ba D1 - Be D1) = 0.5 [ Ba C0 + Ba D1 + Bc C1 + Bc D0 + Be C0 - Be D1 + Bm (-C1) + Bm D0 ] = 0.5 [ Ba (C0 + D1) + Bc (C1 + D0) + Be (C0 - D1) + Bm (-C1 + D0) ] (Proven). The answer is given by Kang Chuen Tat; PO Box 6263, Dandenong, Victoria VIC 3175, Australia; SMS +61405421706; email@example.com; http://kangchuentat.wordpress.com.
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QUANTUM COMPUTING - EXAMPLE 32.3 : A system of linear congruences consists of 3 equations : X ≡ 1 (mod 3), X ≡ 3 (mod 5), X ≡ 4 (mod 6). X has positive values. (a) List the values of these equations from 1 to 35. Then find the minimum value of X. (b)(i) Find the least common multiple (LCM) of b = 3, 5 and 6 where X ≡ a (mod b). (ii) If b - a has the same value of all equations above, then X + (b - a) is divisible by LCM. Find the value of minimum value of X via LCM division.
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