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Dirac Notation and Basic Linear Algebra for Quantum Computing
Dave Bacon
“Mathematicians tend to despise Dirac notation, because it can prevent them from making important distinctions,
but physicists love it, because they are always forgetting such distinctions exist and the notation liberates them from
having to remember.”
 David Mermin
In quantum theory, the basic mathematical structure we deal with is a complex Hilbert space.
A complex
Hilbert space is a complex vector space with an inner product and which is also complete with respect to the norm
deﬁned by the inner product (complete here means that every Cauchy sequence of vectors converges to a vector
where convergence is measured by the norm.) In quantum computing we will be dealing almost exclusively with
the case where this Hilbert space is the vector space of complex
N
dimensional vectors,
C
N
and the inner product
between the vectors
v
= [
v
0
v
1
···
v
N

1
]
T
and
w
= [
w
0
w
1
···
w
N

1
]
T
(here
T
denotes transpose so we are writing
the vectors as “column” vectors) is given by
h
w,v
i
=
N

1
X
i
=0
w
*
i
v
i
(1)
Here I’d like to introduce you to the Dirac braket notation. Generally this notation can be used form any complex
Hilbert space (and really in even more general settings, but we certainly won’t need to worry about this) but since
we will be dealing with the vector space
C
N
and the above inner product, it is useful to introduce this notation and
show what it explicitly corresponds to in this complex Hilbert space.
Kets
: Vectors in a complex Hilbert space are denoted in the braket notation by kets. Let’s call our Hilbert space
H
. Then we denote a vector in this space as a ket

v
i ∈ H
. When the vector space we are dealing with is
C
N
, then

v
i
is nothing more than an ordered ntuple of complex numbers. In particular we will ﬁnd it useful to think about

v
i
as a column vector of
N
complex numbers:

v
i ↔
v
0
v
1
.
.
.
v
N

1
(2)
v
i
∈
C
. We can do everything with kets that we can do with vectors: we can add them

v
i
+

w
i
, multiply them by
a scalar
α

v
i
, etc. Now there is a special vector in a vector space, the zero vector. For the zero vector we will never
write it as

0
i
(you’ll see why soon.) Instead we will always just write it as 0, so

v
i
+ 0 =

v
i
.
Bras
: Recall that for a vector space
V
we can deﬁne it a dual vector space,
V
*
. This is the space of linear functionals
on
V
: i.e. scalarvalued linear transformations on
V
. What does this mean? Well it means that an element of the
dual space takes a vector and turns it into a complex number (functional on
V
). Further this transform is linear,
meaning we can add these transforms and multiply them by a scalar. Elements of the dual vector space for a Hilbert
space
H
are written as “bra”s:
h
w
 ∈ H
*
. When we are dealing with
C
N
and the above inner product, then bras are
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 Fall '08
 Staff
 Computer Science

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