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Last revised 4/5/06
LECTURE NOTES ON QUANTUM COMPUTATION
Cornell University, Physics 481681, CS 483; Spring, 2006
c
±
2006, N. David Mermin
II. Quantum Computation: General features and some simple examples
A. The general computational process
We would like a suitably programmed quantum computer to act on a number
x
to
produce another number
f
(
x
) for some speciFed function
f
.
Appropriately interpreted,
with an accuracy that increases with increasing
k
, we can treat all such numbers as non
negative integers less than 2
k
. Each integer is represented in the quantum computer by
the corresponding computationalbasis state of
k
Qbits.
If we specify
x
as an
n
bit integer and
f
(
x
) as an
m
bit integer, then we shall need at
least
n
+
m
Qbits: a set of
n
Qbits, called the
input register
, to represent
x
, and another set
of
m
Qbits, called the
output register
, to represent
f
(
x
). Qbits being a scarce commodity,
you might wonder why we need separate registers for input and output. One important
reason is that if
f
(
x
) assigns the same value to di±erent values of
x
, as many interesting
functions do, then the computation cannot be inverted if its only e±ect is to transform
the contents of a single register from
x
to
f
(
x
). Having separate registers for input and
output is standard practice in the classical theory of reversible computation.
Since (as
remarked in Chapter 1 and expanded on in Section C of this chapter) quantum computers
must operate reversibly (except for measurement gates) to perform their magic, quantum
computers are generally designed to operate with both input and output registers.
We
shall Fnd that this dualregister procedure can be exploited by a quantum computer in
some strikingly nonclassical ways.
The computational process will generally require many Qbits besides the
n
+
m
in
the input and output registers, but we shall ignore these additional Qbits for the moment,
viewing a computation of
f
as doing nothing more than applying a unitary transformation,
U
f
to the
n
+
m
Qbits of the input and output registers. We shall return to the fundamental
question of why the additional Qbits can be ignored in Section C, only noting for now that
it will be the reversibility of the computation that makes it possible for them to be ignored.
To deFne unitary transformations
U
it is enough to deFne their action on any basis,
since any other state

Ψ
i
must be a linear superposition of basis states, and therefore,
since unitary transformations are linear, their action on

Ψ
i
is entirely determined by their
action on the basis.
In most cases of interest one takes the basis in which
U
is deFned
to be the computational (classical) basis.
In many cases (such as the deFnition below
of
U
f
) the action of
U
on each computational basis state is simply to produce another
computational basis state. Since
U
must have an inverse, it therefore acts on the entire
1
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View Full Document computational basis as a permutation, just as a classical transformation does. Since linear
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This note was uploaded on 02/01/2008 for the course CS 483 taught by Professor Ginsparg during the Spring '08 term at Cornell University (Engineering School).
 Spring '08
 Ginsparg

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