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Quantum Computation and Quantum Information – Lecture 3
Part 1 of CS406 – Research Directions in Computing
Nick Papanikolaou
Motivation
Quantum computers are built from wires and logic gates, just as classical computers are
The potential of such devices stems from the ability to manipulate superpositions of states
Quantum algorithms solve problems which are not known to be solvable classically!
Lecture 3 Topics
Quantum logic gates Simple quantum circuits Quantum teleportation as a circuit Deutsch’s quantum algorithm
Quantum vs. classical gates
The simplest boolean gate is NOT, with truth table:
Quantum gates have to be defined not only on the equivalents of 0 and 1, but on their superpositions too!
in out
0 1
1 0
Quantum NOT gate: Linearity
Suppose we define a quantum NOT gate as follows:
The action of the quantum NOT gate on a superposition must then be:
All quantum operations are linear
01NOT ,10NOT
01
1NOT0NOT10NOT
The NOT Gate as a Matrix
Because all quantum operations have to be linear, we can represent the action of a quantum gate by a matrix
The quantum NOT, or Pauli-X gate, is written:
01
10X
Quantum State Vectors
Remember that a quantum state is represented by a vector
Notation:
1
01 ,
0
10
10
Quantum NOT
We can express the NOT operation on a general qubit as matrix multiplication:
01
1010X
Other Single Qubit Gates
The Pauli-X gate works on only one qubit Other common single qubit gates are:
– Pauli-Z gate:
– Pauli-Y gate:
– Hadamard gate:
10
01Z
XZY
11
11
2
1NOTH
Z
Y
H
Summary of Simple Gates
X
Z
H
10
10
10
01
10
2
10
2
10
Y10 01 i
Reversibility Requirement
All quantum operations have to be reversible
Boolean operations are not necessarily so A reversible operation is always given by a
unitary matrix, i.e. one for which:
1)*( UUT
The Controlled NOT Gate
The CNOT gate is the standard two-qubit quantum gate
It is defined like this:
1011
1110
0101
0000
CNOT
CNOT
CNOT
CNOT
The Controlled NOT Gate (2)
CNOT is a generalisation of the classical XOR:
The CNOT gate is drawn like this:
ABABA , ,CNOT
“control qubit”
“target qubit”
A
B
A
AB
The Controlled NOT Gate (3)
The matrix corresponding to the CNOT gate is:
The CNOT together with the single qubit gates are universal for quantum computing
0100
1000
0010
0001
CNOT
Quantum Circuits
Using the conventions for control and target qubits, we can build interesting circuits
Example: A Qubit Swap Circuit
A
B A
B
Qubit Swap Circuit
A
B A
BA
BA BA
BBAA )(
ABAB )(
Features of Quantum Circuits
1. No loops are allowed; quantum circuits are acyclic
2. Fan-in is not allowed:
3. Fan-out is not allowed:
Generalised Control Gate
Any quantum gate U can be converted into a controlled gate:
U
One control qubit
n target qubits
If the control qubit is “high,” U is applied to the targets. CNOT is the Controlled-X gate!
Quantum Measurement
Measurement in a quantum circuit is drawn as:
M(classical bit representing outcome of measurement)
10 If then:M = 0 with prob. orM = 1 with prob.
2
2
A Qubit Cloning Circuit?
Using the XOR gate, it is possible to copy a classical bit:
x x
y xy
x
0
x
x
Can we build a quantum circuit that performs does this with qubits?
?
0
A Qubit Cloning Circuit? (2)
0
0
0
0
0
1 1
1
0
10 1100 entangled!!
OK here
A Qubit Cloning Circuit? (3)
11100100
101022
It is impossible to clone a qubit!
Also note that
unwanted terms!
The Bell State Circuit
Hx
y
x y Output
0 0
0
0
1
1
1 1
11002
1
11002
1
10012
1
10012
1
The Bell State Circuit By Example
H0
0
102
1
?
11002
1
10002
1
2
10000
2
10
CNOTCNOT
CNOTCNOT
Quantum Teleportation Circuit
H
XM2 ZM1
11002
1100
M1
M2
Quantum Teleportation Circuit (2)
H
XM2 ZM1
11001110002
11
M1
M2
Quantum Teleportation Circuit (3)
H
XM2 ZM1
01111010
01011000
2
12
M1
M2
Quantum Teleportation Circuit (4)
H
XM2 ZM1
00, 01, 10 or 11
M1
M2
Quantum Teleportation Circuit (5)
If Alice obtains
Then Bob’s qubit is in state
So Bob applies gate
obtaining
00 I
01 X
10 Z
11 Y = ZX
10
01
10
01
10
What have we achieved?
The teleportation process makes it possible to “reproduce” a qubit in a different location
But the original qubit is destroyed!
Next topic: Quantum Parallelism and Deutsch’s quantum algorithm
Quantum Parallelism
Quantum parallelism is that feature of quantum computers which makes it possible to evaluate a function f(x) on many different values of x simultaneously
We will look at an example of quantum parallelism now – how to compute f(0) and f(1) for some function f all in one go!
Quantum Circuits for Boolean Functions
It is known that, for any boolean function
it is possible to construct a quantum circuit Uf
that computes it Specifically, to each binary function f
corresponds a quantum circuit:
1,01,0: f
)(,,: xfyxyxU f
binary addition
Quantum Circuits for Boolean Functions (2)
What can this circuit Uf do? Example:
x x
y yf(x)
0
1
)0(1 ,0
01
10
f
U
U
f
f
fU
Quantum Circuits for Boolean Functions (3)
But what if the input is a superposition?
x x
y yf(x)1fU
102
1
2
)1(,1)0(,0
2
)1(0 ,1)0(0 ,0
2
1000
02
10
ff
ff
U
U
f
f
amazing! we’ve computed f(0) and f(1) at the same time!
Quantum Parallelism Summary
So, a superposition of inputs will give a superposition of outputs!
We can perform many computations simultaneously
This is what makes famous quantum algorithms, such as Shor’s algorithm for factoring, or Grover’s algorithm for searching
Simple q. algorithm: Deutsch’s algorithm
Deutsch’s Algorithm
David Deutsch: famous British physicist Deutsch’s algorithm allows us to compute,
in only one step, the value of
To do this classically, you would have to:1. compute f(0)2. compute f(1)3. add the two results
– Remember:
)1()0( ff
1,01,0: f
Circuit for Deutsch’s Algorithm
x x
y yf(x)
H
H
H0
1
010 2
10
2
101
fU
Circuit for Deutsch’s Algorithm (2)
x x
y yf(x)
H
H
H0
1
2
10
2
10 ),1()0( if
2
10
2
10 ),1()0( if
2
ff
ff
fU
Circuit for Deutsch’s Algorithm (3)
x x
y yf(x)
H
H
H0
1
2
101 ),1()0( if
2
100 ),1()0( if
3
ff
ff
2
10)1()0(
tosimplifies this
3
ff
...and so we have computed
)1()0( ff
fU
End of Lecture 3
Congratulations! If you are still awake, you have learned something about:– quantum gates (X, Y, Z, H, CNOT)– quantum circuits (swapping, no-cloning problem)– teleportation– quantum parallelism– and Deutsch’s algorithm