Unformatted text preview: ﬁrst calculate the L(Hnof the Hadamard transform
in the form of a it helps transform Uf 2 effect +1 , H) (Fig. III.23).
on a state xi. By checking the cases x = 0 and x = 1 separately we see that for a single
p
P
qubit H xi = z ( 1)xz z i/ 2. Thus
¶3. We are told only that fP either constant or balanced, which means
is
x1 z1 +·· +xn zn
(
z1 , . , zn i
that it is 0non , half xn i = z1 ,...,zand1)1 on the other. .half. .Our task is to
p
(1.49)
H x1 . . . , its domain n
n
2
determine into which class a given f falls. This can be summarized more succinctly in the very useful equation
P
¶4. Classical: Consider ﬁrst the classical·zsituation. We can try di↵erent
( 1)x z i
n
,
(1.50)
H xi = z p
input bit strings x.
2n We might (if we’re lucky) discover after the second query of f that it where x · z constant.
is not is the bitwise inner product of x and z , modulo 2. Using this equation
and (1.48) we can now evaluate  3 i,
But we might require as many as 2n /2+1 queries to answer the question.
X ) function (x) z i 0i
So we’re facing O(2n 1 X ( 1)x·z+fevaluations.1i
p
 3i =
.
(1.51)
2n
2
z
x ¶5. Initial state:...
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 Fall '13
 BruceMacLennan
 Quantum computer, Qubit, Theoretical Computer Science, Quantum information science

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