introduction to quantum computingintroduction to quantum computing einar pius university of...

Introduction to Quantum Computing

Einar PiusUniversity of Edinburgh

Thursday, 19 April 12

Summary of Quantum Mechanics


From Classical to Quantum Computing

• Boolean circuits

• Reversible computing

• Quantum computing


Goals

• Construct a quantum model for computation

• Relate the quantum model of computation to the classical one

• Point out the differences between classical and quantum computing

• Show that classical computations can be done on a quantum computer


Boolean CircuitsA mathematical model for classical computing


Boolean Circuits

• Any function can be computed using a small fixed set of boolean gates

f : {0, 1}n ! {0, 1}m

ANDab a∧b NAND ¬(a∧b)a

bNOT ¬aa ORab a∨b


Boolean Circuits


• Each gate corresponds to some boolean function

f : {0, 1}n ! {0, 1}m




Boolean Circuits


• Each gate corresponds to some boolean function

• A network of boolean gates is called a boolean circuit

f : {0, 1}n ! {0, 1}m

NANDNAND

NANDNAND

NANDNAND

NAND

NANDNAND

NAND

x0

x1

x3

x4

x6

x7

x5

x2

f(x)


Boolean Circuits


• This fixed set of boolean gates is called a universal set of gates

f : {0, 1}n ! {0, 1}m




Boolean Circuits


• This fixed set of boolean gates is called a universal set of gates

• There exists more than one universal set of gates

f : {0, 1}n ! {0, 1}m




Boolean Circuits

• Boolean circuits are irreversible

• In general, we cannot recover the input to a boolean circuit from its output

• Quantum operators are reversible

• A quantum analog to the boolean circuits should therefore be reversible


Reversible Classical Computing


Reversible Classical Computing

• Unitary evolution is reversible (for every U, there exists U dagger)

• Hence quantum computing is going to be reversible


Universal Reversible Computing

• Two bit boolean gates are not enough

• At least three bit gates are required

• A universal reversible gate is the Toffoli gate

a

b

c

a

b

c ⊕ ( a ∧ b )


Reversible Functions

• We can use the Toffoli gate to simulate a NAND gate

• Hence every boolean circuit can be represented as a reversible circuit consisting of only Toffoli gates

a

b

1

a

b

1 ⊕ ( a ∧ b ) = ¬ ( a ∧ b )


Reversible Functions

• We can use the Toffoli gate to simulate a NAND gate

• Hence every boolean circuit can be represented as a reversible circuit consisting of only Toffoli gates

• It also turns out, that the Toffoli gate acts as a unitary operatora

b

1

a

b

1 ⊕ ( a ∧ b ) = ¬ ( a ∧ b )


Quantum CircuitsA mathematical model for quantum computing


Quantum Circuits

• The quantum analogues of bits are qubits

abcd


Quantum Circuits

• qubits are represented as wires in the quantum circuit model

abc

abc

d dThursday, 19 April 12

Quantum Circuits


• unitary operators acting on the qubits are represented as quantum gates

U7

abc

a'b'c'

U6

U5U1

U2

U3

d d'U8U4

U9

U10


Quantum Circuits



• gates can act on multiple qubits

U7

abc

a'b'c'

U6

U5U1

U2

U3

d d'U8U4

U9

U10


Quantum Circuits



• gates can act on multiple qubits

U7

abc

a'b'c'

U6

U5U1

U2

U3

d d'U8U4

U9U10


The Gates We Need

• We could use any unitary gate in this model

• This would be an unrealistic model

• In practice, not every unitary is implementable

• What is the smallest set of universal gates?


Quantum Toffoli

• The Toffoli gate implements a unitary operator

• As we can use every unitary operator in quantum computing, we can use Toffoli as a quantum gate.

a

b

c

a

b

c ⊕ ( a ∧ b )


Quantum Toffoli



a

b

c

a

b

c ⊕ ( a ∧ b )

• Therefore we can implement any classicalcomputation on a quantum computer


Quantum Toffoli



a

b

c

a

b

c ⊕ ( a ∧ b )

• Therefore we can implement any classicalcomputation on a quantum computer

• What about probabilistic computations?


Quantum Toffoli



• It turns out that Toffoli is “too much” for peerforming reversible classical computing on quantum computers


Quantum Toffoli



• It turns out that Toffoli is “too much” for peerforming reversible classical computing on quantum computers

H T T H

a

b

c

a

b

c ⊕ ( a ∧ b )

S

T

T † T †

T † T †


Decomposing Toffoli

• The more qubits a gate acts on, the harder it is to implement in the real world

• Two qubit gates is hard to implement, three qubit ones is extremely difficult

H T T H

a

b

c

a

b

c ⊕ ( a ∧ b )

S

T

T † T †

T † T †


Universal Set of Quantum Gates

• Theoretically a completely universal set of gates is not needed if we are interested in building a quantum computer for a single algorithm


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