is called the
π
8
or Tgate
,
CNOT =
⎛
⎜
⎜
⎝
1
0
0
0
0
1
0
0
0
0
0
1
0
0
1
0
⎞
⎟
⎟
⎠
in the twoqubit basis

00
,

01
,

10
,

11
.
204
ERIC C. ROWELL AND ZHENGHAN WANG
The CNOT gate is called controlledNOT because the first qubit is the control bit,
so that when it is

0 , nothing is done to the second qubit, but when it is

1 , the
NOT gate is applied to the second qubit.
Definition 3.3.
(1) An
n
qubit quantum circuit over a gate set
S
is a map
U
L
: (
C
2
)
⊗
n
→
(
C
2
)
⊗
n
composed of finitely many matrices of the form Id
p
⊗
g
⊗
Id
q
, where
g
∈ S
and
p, q
can be 0.
(2) A gate set is universal if the collection of all
n
qubit circuits forms a dense
subset of SU(2
n
) for any
n
.
The gate set
S
=
{
H, σ
1
/
4
z
,
CNOT
}
will be called the standard gate set, which
we will use unless stated otherwise.
Theorem 3.2.
(1)
The standard gate set is universal.
(2)
Every matrix in
U
(2
n
)
can be eﬃciently approximated up to an overall phase
by a circuit over
S
.
Note that (2) means that
eﬃcient
approximations of unitary matrices follows
from approximations for free due to the Solovay–Kitaev theorem; see [105].
3.4.
Complexity class BQP.
Let
C
T
be the class of computable functions
C
T
,
and let
P
be the subclass of functions eﬃciently computable by a Turing machine.
20
Defining a computing model
X
is the same as selecting a class of computable func
tions from
C
T
, denoted
X
P
.
The class
X
P
of eﬃciently computable functions
codifies the computational power of computing machines in
X
.
Quantum com
puting selects a new class BQP–boundederror quantum polynomial time.
The
class BQP consists of those problems that can be solved eﬃciently by a quantum
computer. In theoretical computer science, separation of complexity classes is ex
tremely hard, as the millennium problem
P
vs.
NP
problem shows. It is generally
believed that the class BQP does not contain NPcomplete problems. Therefore,
good target problems will be those NP problems which are not known to be NP
complete. Three candidates are factoring integers, graph isomorphism, and finding
the shortest vector in lattices.
Definition 3.4.
Let
S
be any finite universal gate set with eﬃciently computable
matrix entries. A problem
f
:
{
0
,
1
}
∗
→ {
0
,
1
}
∗
(represented by
{
f
n
}
:
Z
n
2
→
Z
m
(
n
)
2
)
is in BQP (i.e., can be solved eﬃciently by a quantum computer) if there exist
polynomials
a
(
n
)
, g
(
n
) :
N
→
N
satisfying
n
+
a
(
n
) =
m
(
n
) +
g
(
n
) and a classical
eﬃcient algorithm to output a map
δ
(
n
) :
N
→ {
0
,
1
}
∗
describing a quantum circuit
U
δ
(
n
)
over
S
of size
O
(poly(
n
)) such that
U
δ
(
n
)

x,
0
a
(
n
)
=
I
a
I

I ,

I
=

f
n
(
x
)
,z

a
I

2
≥
3
4
,
where
z
∈
Z
g
(
n
)
2
.
20
More precisely, this should be denoted as FP since P usually denotes the subset of decision
problems.
MATHEMATICS OF TOPOLOGICAL QUANTUM COMPUTING
205
The
a
(
n
) qubits are an ancillary working space, so we initialize an input

x
by appending
a
(
n
) zeros and identify the resulting bit string as a basis vector in
(
C
2
)
⊗
(
n
+
a
(
n
))
. The
g
(
n
) qubits are garbage. The classical algorithm takes as input
the length
n
and returns a description of the quantum circuit
U
δ
(
n
)
. For a given

x
, the probability that the first
m
(
n
) bits of the output equal
f
n
(
x
) is
≥
3
4
.
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 Fall '08
 BLOOMFIELD,A
 Quantum Field Theory, Sula, Passing, Quantum entanglement, MATHEMATICS OF TOPOLOGICAL QUANTUM COMPUTING