sequence of gates has the following sequence of effects on a computational basis state
|a, bi,
|a, bi ! |a, a
! |a
! |b, (a
bi
(a
b), a
b)
bi = |b, a
bi = |b, ai ,
bi
(1.20)
where all additions are done modulo 2. The effect of the circuit, therefore, is to inter-
126
CHAPTER III. QUANTUM COMPUTATION
change the state of the two qubits.
Figure 1.7. Circuit swapping two qubits, and an equivalent schematic symbol notation for this common and useful
circuit.
Figure III.11: Diagram for swap (from NC).
There are a few features allowed in classical circuits that are not usually present in
quantum circuits. First of all, we don’t allow ‘loops’, that is, feedback from one part of the
C.3
Quantum
circuits
quantum
circuit to another;
we say the circuit is acyclic. Second, classical circuits allow
, with the resulting single wire
wires to be ‘joined’ together, an operation known as
¶1. Quantum circuit: A quantum circuit is a sequential series of quantum
of the inputs. Obviously this operation is not reversible and
containing the bitwise
transformations
ondon’t
a quantum
register.
in our quantum circuits. Third, the inverse
therefore
not unitary, so we
allow
, whereby several copies of a bit are produced is also not allowed in
operation,
¶2. The inputs are usually computational basis states (all |0i unless stated
quantum circuits. In fact, it turns out that quantum mechanics forbids the copying of a
operation impossible! We’ll see an example of this in the next
qubit,otherwise).
making the
section when we attempt to design a circuit to copy a qubit.
¶3. Quantum circuit diagrams are drawn with time going from left to right,
As we proceed we’ll introduce new quantum gates as needed. It’s convenient to inwith the quantum gates crossing one or more “wires” (qubits) as aptroduce another convention about quantum circuits at this point. This convention is
propriate.
illustrated
in Figure 1.8. Suppose U is any unitary matrix acting on some number n of
It
represents
a sequence
of unitary
on Then
a quantum
register
qubits, so U can be regarded
as a quantum
gate operations
on those qubits.
we can define
a
rather than
physical
wires. extension of the controlledgate. Such a gate
controlled-U
gate which
is a natural
has a single control qubit, indicated by the line with the black dot, and n target qubits,
¶4.
Unique
arenothing
not allowed.
indicated
by thefeatures:
boxed U . IfAcyclic:
the controlloops
qubit(feedback)
is set to 0 then
happens to the
target qubits. If the control qubit is set to 1 then the gate U is applied to the target qubits.
¶5.
(equivalent
to OR)
is not gate
allowed,
it it notgate,
reversible
which is or
The Fan-In
prototypical
example of the
controlled-U
is the since
controlledunitary. gate with U = X, as illustrated in Figure 1.9.
a controlled-U
Another important operation is measurement, which we represent by a ‘meter’ symbol,
¶6. Fan-Out is not allowed, because it would violate the No-cloning Theorem.
(N.B.: This does not contradict the universality of the To↵oli or Fredkin gates, which are universal only with respect to classical states.)
¶7. CNOT: Fig. III.9 (right) shows the symbol for CNOT and its e↵ect.
¶8. Swap: The swap operation is defined |xyi 7! |yxi, or explicitly
X
|yxihxy|.
x,y22
¶9. We can put three CNOTs in series to swap two qubits (Exer. III.27).
It has a special symbol as shown in Fig. III.11.
C.24 QUANTUM
INFORMATION
Introduction
and overview
127
Figure 1.8. Controlled-U gate.
Figure III.12: Diagram for controlled-U (from NC).
¶10. Controlled-U: In general, any unitary operator (on any number of
qubits) canFigure
be controlled
(see
Fig. III.12).
If the control
bit is 0, it
1.9. Two different
representations
for the controlled.
does nothing, otherwise it does U .
as shown in Figure 1.10. As previously described, this operation converts a single qubit
¶11.
This is implemented by |0ih0| ⌦ I + |1ih1| ⌦ U .
state | i = |0i + |1i into a probabilistic classical bit M (distinguished from a qubit by
E↵ectively, the operators are entangled.
2
drawing it as a double-line wire), which is 0 with probability | | , or 1 with probability
| |2 .
¶12. Example: Suppose the control bit is in superposition, | i = a|0i+b|1i.
❴✤✤ ❴ ❴ ❴ ❴ ❴ ! ❴ ❴ ✤✤
(|0ih0| ⌦ I + |1ih1| ⌦ U )| , ✤✤✤✤ i ✙ ✙ ✙ ✙ ✤✤✤✤
✤❴ ❴ ❴ ❴ ❴✙ ✙ ❴ ❴ ❴ ✤
= (|0ih0| ⌦ I + |1ih1| ⌦ U
)(a|0i + b|1i) ⌦ | i
Figure 1.10. Quantum circuit symbol for measurement.
= |0ih0|(a|0i + b|1i) ⌦ I| i + |1ih1|(a|0i + b|1i) ⌦ U | i
We shall find=quantum
a|0i ⌦circuits
| i + useful
b|1i ⌦asUmodels
| i of all quantum processes, including
but not limited =
to computation,
communication,
and even quantum noise. Several simple
a|0, i + b|1, U i.
examples illustrate this below.
We have a superposition of entangled outputs.
1.3.5 Qubit copying circuit?
The Recallgate
useful for=demonstrating
one particularly fundamental property of
¶13.
thatis CNOT
controlled X.
quantum information. Consider the task of copying a classical bit. This may be done
gate,
which takes transformation:
in the bit to copy (in some
x)
usingConditional
a classical
¶14.
or
controlled
If U0unknown
and U1 state
are uniand atary
‘scratchpad’
bit
initialized
to
zero,
as
illustrated
in
Figure
1.11.
The
output
is
two
operators, then we can make the choice between them conditional
bits, on
botha ofcontrol
which are
same state x.
bitinasthefollows:
Suppose we try to copy a qubit in the unknown state | i = a |0i + b |1i in the same
manner by using a
gate. The input state of the two qubits may be written as
|0ih0|i⌦ U0 + |1ih1| ⌦ U1 .
h
a |0i + b |1i |0i = a |00i + b |10i,
(1.21)
¶15. For example,
is to negate the second qubit when the first qubit is 1, and thus
The function of
= |0iwe
h0| successfully
⌦ I + |1ih1|copied
⌦ X.| i? That is, have
(III.17)
the output is simply a |00i +CNOT
b |11i. Have
we
created the state | i| i? In the case where | i = |0i or | i = |1i that is indeed what this
circuit does; it is possible to use quantum circuits to copy classical information encoded
as a |0i or a |1i. However, for a general state | i we see that
| i| i = a2 |00i + ab|01i + ab|10i + b2 |11i.
(1.22)
26
Introduction and overview
understood via the equations
|
xy i
|0, yi + ( 1)x |1, ȳi
p
,
2
Figure
1.8. Controlled-U
CHAPTER
III. gate.
QUANTUM
128ȳ is the negation of y.
where
(1.27)
COMPUTATION
In
|00i
Out
p
(|00i + |11i)/ 2 | 00 i
p
|01i (|01i + |10i)/ 2 | 01 i
p
Figure 1.9.
Two different
for the controlled.
|10i (|00i
|11i)/
2 |representations
10 i
p
|11i (|01i |10i)/ 2 | 11 i
as shown in Figure 1.10. As previously described, this operation converts a single qubit
state | i = |0i + |1i into a probabilistic classical bit M (distinguished from a qubit by
Figure 1.12. Quantum circuit to create Bell states, and its input–ouput quantum2‘truth table’.
1 withNC
probability
drawing
it as a double-line
which
is 0 with probability
| | , or[from
Figure III.13:
Quantum wire),
circuit
for generating
Bell states.
fig. 1.12]
| |2 .
1.3.7 Example: quantum
❴✤✤ ❴ ❴ teleportation
❴ ❴ ❴ ! ❴ ❴ ✤✤
✙
✤✤
We will now apply the techniques of the last few✤✤✤ pages
✙✙
✤✤ to understand something non✤✤
✙✙ ✙
trivial, surprising, and a lot of fun – quantum teleportation!
Quantum teleportation is a
❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ✤
technique for moving quantum
states
around,circuit
even symbol
in theforabsence
of a quantum commuFigure
1.10. Quantum
measurement.
nications
channel
linking
the sender
the quantum state
the recipient.
Figure
III.14:
Symbol
for ofmeasurement
of atoquantum
state (from NC).
Here’s
teleportation
and Bob
long ago
but nowincluding
live
Wehow
shallquantum
find quantum
circuitsworks.
useful Alice
as models
of allmet
quantum
processes,
far apart.
While
together
they
generated
an
EPR
pair,
each
taking
one
qubit
of
the
EPR
but not limited to computation, communication, and even quantum noise. Several simple
pair when
theyillustrate
separated.
Many
years later, Bob is in hiding, and Alice’s mission, should
examples
this
below.
gates:
The| symbol
gatestate
is of
show in
she¶16.
chooseOther
to acceptspecial
it, is to deliver
a qubit
i to Bob.for
Shethe
doesCCNOT
not know the
Fig.
III.10,
the qubit, and moreover can only1.3.5
send classical
information
to Bob. Should Alice accept
Qubit copying
circuit?
or
with
•
for
top
two
connections
and
forfundamental
bottom, representing
the mission?
The
gate is useful for demonstrating one particularly
property of
CCNOT|x,
y, zi
= |x,
xytask
zi,
Intuitively,
things look
pretty
bady,the
for
Alice.
doesn’t
know the
i of be
thedone
quantum
information.
Consider
ofShe
copying
a classical
bit.state
This| may
qubitusing
she or
has
to send
to Bob,
laws of
prevent
her from
put
“CCNot”
inwhich
athe
box.
gate,and
takes
in quantum
the bit to mechanics
copy (in some
unknown
state x)
a classical
determining
the
state
when
she
only
has
a
single
copy
of
|
i
in
her
possession.
What’s
and a ‘scratchpad’ bit initialized to zero, as illustrated in Figure 1.11. The output is two
¶17.
Other
operations
may
shown
letter
symbol
in a box,
worse,
even
if she
did know
thethe
state
|bei,state
describing
preciselyatakes
an or
infinite
amount
bits,
both
of which
are in
same
x. by itputting
of classical
information
since
| for
a continuous
if she
didsame
for
example
“H”
the values
Hadamard
gate.
Suppose
we try to
copy
ai takes
qubit
in
the in
unknown
state | space.
i = aSo
|0ieven
+ b |1i
in the
knowmanner
| i, it by
would
forThe
Aliceinput
to describe
to Bob.
It’sbenot
looking
usingtake
a forever
gate.
state of the state
two qubits
may
written
as
¶18.
H canFortunately
be used to
Bell
states
(Exer.is III.26):
good
for Alice.
for
Alice, quantum
teleportation
a way of utilizing the
h generate
i
|0i +| bi|1i
|0i =with
a |00i
+ ba|10i,
entangled EPR pair in order to asend
to Bob,
only
small overhead of classical(1.21)
communication.
CNOT(H ⌦ I)|xyi = | xy i.
(III.18)
is solution
to negateare
theassecond
qubit
the first
and thus
function
of of the
InThe
outline,
the steps
follows:
Alicewhen
interacts
the qubit
qubit is| 1,
i with
the of
output
is simply
a |00i
+ bmeasures
|11i. Have
wetwo
successfully
copied
| i? That
is, have we
her half
the EPR
pair, and
then
qubits in her
possession,
obtaining
¶19.
The
circuit
fori?generating
Bellthe
states
(Eq.
III.18)
isthat
shown
in Fig.
III.13.
created
the
state
|
i|
In
the
case
where
|
i
=
|0i
or
|
i
=
|1i
is
indeed
what
one of four possible classical results, 00, 01, 10, and 11. She sends this information
to this
does;on
it is
possible
to usemessage,
quantumBob
circuits
to copy
information
Bob. circuit
Depending
Alice’s
classical
performs
one classical
of four operations
onencoded
his
as
a
|0i
or
a
|1i.
However,
for
a
general
state
|
i
we
see
that
half of the EPR pair. Amazingly, by doing this he can recover the original state | i!
¶20. Measurement: It’s also
convenient to have a symbol for quantum
2 1.13 gives a more precise2description of quantum
The quantum circuit shown
ini Figure
|00i
+
ab|01i
+ ab|10i + b |11i.
(1.22)
|
i|
=
a
state measurement, such as Fig.
III.14.
teleportation. The state to be teleported is | i = |0i+ |1i, where and are unknown
amplitudes. The state input into the circuit | 0 i is
|
0i
= | i|
00 i
(1.28)
be seen from the fact that it can be used to construct the AND and NOT operators in the
following way:
T |1, 1, xi = |1, 1, ¬xi
T |x, y, 0i = |x, y, x ^ yi
The T gate is sufficient to construct arbitrary combinatorial circuits.
The following quantum
circuit, for example, implements a 1 bit full adder using Toffoli129
C. QUANTUM
INFORMATION
and controlled-NOT gates:
|ci
|ci
|xi
|xi
|yi
|yi
⇥
|0i
|0i
⇥
⇥
⇥
⇥
⇥
|si
|c i
Figure
circuit
for 1-bit
full adder
IQC].carry
“x and
y are
whereIII.15:
x and y Quantum
are the data bits,
s is their
sum (modulo
2), c is[from
the incoming
bit, and
is the bits,
new carry
Vedral,
Barenco
and Ekert
et al.
1996] define
morebit,
complex
thec data
s isbit.
their
sum
(modulo
2), [Vedral
c is the
incoming
carry
and c0
circuits
include
in-place addition and modular addition.
is the
newthatcarry
bit.”
The Fredkin gate is a “controlled swap” and can be defined as
F = |0ih0| ⌦ I ⌦ I + |1ih1| ⌦ S
where S is the swap operation
x
0
x
S = |00ih00| + |01ih10| + |10ih01| + |11ih11|.
f(x)
f(x)
x
-1
Φ
The0reader canΦverify that F , like T , is complete for combinatorial circuits.
y
g(x)
g(x)
CNOT
y
y⊕f(x)
0
0
y⊕f(x)
Uf
Figure III.16: Quantum gate array for reversible quantum computation.
C.4
Quantum gate arrays
¶1. Full adder: Fig. III.15 shows a quantum circuit for a 1-bit full adder.
¶2. As we will see (Sec. C.7), it is possible to construct reversible quantum
gates for any classically computable function. In particular the Fredkin
and To↵oli gates are universal.
¶3. Reversibility: Because quantum computation is a unitary operator,
it must be reversible.
You know that an irreversible computation x 7! f (x) can be embedded
in a reversible computation (x, c) 7! (f (x), g(x)), where c are suitable
constants and g(x) represents the garbage bits.
130
CHAPTER III. QUANTUM COMPUTATION
¶4. Note that throwing away the garbage bits (dumping them in the environment) will collapse the state (equivalent to measurement) by entangling them in the many degrees of freedom of the environment.
¶5. Since NOT is reversible, each 1 bit in c can be replaced by a 0 bit
followed by a NOT, so we need only consider (x, 0) 7! (f (x), g(x)).
See Fig. III.16.
¶6. The garbage must be produced in a standard state independent of x,
“because garbage bits whose value depends upon x will in general destroy the interference properties crucial to quantum computation.”
¶7. Uncomputation: This is accomplished by uncomputing.
Specifically, perform the computation on four registers (data, result,
workspace, target):
(x, 0, 0, y) 7! (x, f (x), g(x), y).
Notice that x and y (data and target) are passed through.
¶8. Now use CNOTs to compute y f (x), where
exclusive-or, in the fourth register:
(x, 0, 0, y) 7! (x, f (x), g(x), y
represents bitwise
f (x)).
¶9. Now we uncompute f , but since the data and target registers are passed
through, we get (x, 0, 0, y f (x)).
Ignoring the result and workspace registers, we write
(x, y) 7! (x, y
f (x)).
¶10. Quantum gate array: Therefore, for any computable f : 2m ! 2n ,
there is a reversible quantum gate array Uf : Hm+n ! Hm+n such that
for x 2 2m and y 2 2n ,
See Fig. III.17.
Uf |x, yi = |x, y
f (x)i,
¶11. The first m qubits are called the data register and the last n are called
the target register.
¶12. In particular, Uf |x, 0i = |x, f (x)i.
gates for any classically computable function. In fact, it is possible to conceive of a universal quantum Turing machine [Bernstein and Vazirani 1997]. In this construction we must
assume a sufficient supply of bits that correspond to the tape of a Turing machine.
Knowing that an arbitrary classical function f with m input and k output bits can be implemented on quantum computer, we assume the existence of a quantum gatearray Uf that
implements f . Uf is a m + k bit transformation of the form Uf : |x, yi ! |x, y f (x)i
where denotes the bitwise exclusive-OR6. Quantum gate arrays Uf , defined in this way,
are unitary for any function f . To compute f (x) we apply Uf to |xi tensored with k
C. QUANTUM
INFORMATION
zores
|x, 0i. Since f (x)
f (x) = 0 we have Uf Uf = I. Graphically the transformation
Uf : |x, yi ! |x, y f (x)i is depicted as
|xi
131
|xi
Uf
|yi
|y
f (x)i.
While the T and F gates are complete for combinatorial circuits, they cannot achieve arFigurequantum
III.17:state
Computation
ofInfunction
by quantum
gatetransformations
array [from 7IQC].
bitrary
transformations.
order to realize
arbitrary unitary
,
single bit rotations need to be included. Barenco et. al. [Barenco et al. 1995] show that
Cnot together with all 1-bit quantum gates is a universal gate set. It suffices to include the
following one-bit transformations
C.5 Quantum ✓parallelism
◆ ✓ i
◆
cos
sin
e
0
¶1. Since Uf is linear, if sin
it is cos
applied, to 0a superposition
of bit strings, it will
e i
produce a superposition of the results of applying f to them in parallel
for all 0(i.e.,
 in
 2thetogether
a universal
gates.
As we shall
not to obtain
same with
timetheitCtakes
to compute
it set
onofone
vector):
see, such non-classical transformations are crucial for exploiting the power of quantum
computers. U (c x + c x + · · · + c x ) = c U x + c U x + · · · + c U x
f
1 1
2 2
k k
1
f
1
2
f
2
k
f
k.
5.2 Quantum Parallelism
¶2. For example,
What happens if Uf is applied to input which is in a superposition? The answer is easy
!
papplied to all basis vectors in the
but powerful: since p
Uf is a linear transformation, it is
3
1
3
1
superpositionU
simultaneously
and will
a superposition
results.
way,
|x1 i +
|x2 igenerate
⌦ |0i
=
|x1 , fof(xthe
+ |xIn2 ,this
f (x
f
1 )i
2 )i.
2 f (x) for2n values of x in a single
2 application of Uf2. This effect is
it is possible to compute
called quantum parallelism.
The power
of quantum algorithms
comes
from taking
quantum
parallelism
¶3.
The amplitude
of a result
y will
be theadvantage
sum ofofthe
amplitudes
of all
and entanglement. So most quantum algorithms begin by computing a function of interest
such that y = f (x).
on a superposition of all values as follows. Start with an n-qubit state |00 . . . 0i. Apply the
x
¶4. Quantum parallelism: If we apply Uf to a superposition of all possiis notble
the direct
of vectors.
2m sum
inputs,
it will compute a superposition of all the corresponding
7 More precisely, we mean arbitrary unitary transformations up to a constant phase factor. A constant phase shift
outputs
in parallel
(i.e.,
in the same
time as required for one function
of the state
has no physical,
and therefore
no computational,
significance.
evaluation)!
6
¶5. The Walsh-Hadamard transformation can be used to produce this superposition of all possible inputs:
1
Wm |00 . . . 0i = p (|00 . . . 0i + |00 . . . 1i + · · · + |11 . . . 1i)
2m
1 X
= p
|xi
2m x22m
132
CHAPTER III. QUANTUM COMPUTATION
m
2
1
1 X
= p
|xi.
2m x=0
In the last line we are obviously interpreting the bit strings as natural
numbers.
¶6. Hence,
m
Uf Wm |0i = Uf
2
1
1 X
p
|x, 0i
2m x=0
!
m
m
2
1
2
1
1 X
1 X
=p
Uf |x, 0i = p
|x, f (x)i.
2m x=0
2m x=0
¶7. A single circuit does all 2m computations simultaneously!
¶8. “Note that since n qubits enable working simultaneously with 2n states,
quantum parallelism circumvents the time/space trade-o↵ of classical
parallelism through its ability to provide an exponential amount of
computational space in a linear amount of physical space.” [IQC]
¶9. If we measure the input bits, we will get a random value, and the state
will be projected into a superposition of the outputs for the inputs we
measured.
¶10. If we measure an output bit, we will get a value probabilistically, and a
superposition of all the inputs that can produce the measured output.
¶11. Neither of the above is especially useful, so most quantum algorithms
transform the state in such a way that the values of interest have a
high probability of being measured.
¶12. The other thing we can do is extract common properties of all values
of f (x).
¶13. Both of these require di↵erent programming techniques than classical
computing.