Dal gatto di Schrödinger ai computer quantistici

Irene Marzoli

Scuola di Scienze e TecnologieUniversità di Camerino

irene.marzoli@unicam.it

What is quantum computing?

Quantum computing is

essentially harnessing and

exploiting the amazing laws

of quantum mechanics to

process information

A universal quantum simulator

“Nature isn't classical,

dammit, and if you want to

make a simulation of

nature, you'd better make it

quantum mechanical, and

by golly it's a wonderful

problem, because it doesn't

look so easy.”

(Richard Feynman, 1982)

Peter Shor’s algorithm (1994)

• A quantum computer could

factor large numbers (500-

digit) in an efficient way

• Factorization is a hard

problem for classical

computers

• RSA encryption relies on the

factoring problem

Overview

• Dal bit al qubit

• Porte logiche quantistiche• Un algoritmo quantistico

• Entanglement• Il gatto di Schrödinger

• Tecnologie quantistiche

Turing machine

Alan Turing (1912-1954)

From the bit to the qubit

1sine0cos i θθ ϕ+=Ψ

Benjamin Schumacher, Phys. Rev. A 51, 2738 (1995)

A qubit or quantum bit is the unit ofquantum information

with

πϕπθ 20 and 2

0 ≤≤≤≤

Superposition of states

=+++=Ψ

11

10

01

00

11100100 11100100

αααα

αααα

12

1

2

0 =+ αα

=+=Ψ

1

010 10

αα

αα

with

12

11

2

10

2

01

2

00 =+++ ααααwith

One qubit

Two qubits

Quantum register

120

2 =∑<≤ nx

xα

∑<≤

=Ψnx

x xt20

)( α

with

The general state of n qubits can be any superposition of 2n states

Example: the computational basis of 3 qubits is

{ }111,110,101,100,011,010,001,000

Quantum logic gates

• Building blocks of quantum circuits;

• Linear unitary transformations

• Reversible;

• Act on few qubits (one or two);

• Represented as 2x2 or 4x4 matrices.

IUUUU == ++

Single-qubit gates

Hadamard gate

−=

11

11

2

1H

( ) 1,0 with 11 =−+−→ xxxx xH

Hadamard gate: examples

2

101

2

100

−→

+→

H

H

Let’s apply the Hadamard gate to one qubit

If we apply the Hadamard gate to a two-qubit register

( )111001002

1

2

10

2

1000

+++=

=++

→H

Pauli-X gate

=

01

10X

01 and 10 →→ XX

Quantum equivalent of a NOT gate. Rotation of π about the x-axis of the Bloch sphere

Pauli-Y gate

−=

0

0

i

iY

01 and 10 ii YY −→→

Rotation of π about the y-axis of the Bloch sphere

Phase-shift gates

= θθ iR

e0

01

1e1 and 00 θθθ iRR →→

They modify the phase of the quantum state. Equivalent to tracing a horizontal circle (line of latitude) on the Bloch sphere by θ radians.

Pauli-Z gate

−=

10

01Z

11 and 00 −→→ ZZ

It is a special case of a phase-shift gate with θ = π.Equivalent to a rotation about the z-axis of the Bloch sphere by π.

Single-qubit operations

( )

( )

( )[ ]1sin0cos

1sin0cos10102

1

102

1

102

10

2/

2

2

2

θθ

θθ

ϕ

θ

θ

πϕ

θ

iR

iH

iR

H

e

ie

e

+ →

−∝−++→

+→

+→

+

Any single-qubit state can be generated by applying a sequence ofHadamard and phase-shift gates.

Two-qubit gates

SWAP gate

=

1000

0010

0100

0001

SWAP

yxxy SWAP →

It swaps two qubits

Controlled NOT gate (CNOT)

=

0100

1000

0010

0001

CNOT

yxxyx CNOT ⊕ →

It performs the NOT operation on the target qubit when thecontrol qubit is in state |1>.

Conditional phase shift

=

θ

θ

ie

RC

000

0100

0010

0001

)(

)( yxeyx ixyRC θθ →

It operates a phase shift on the target qubit when the control qubit is in state |1>.

Universal quantum gates

Hadarmard, phase shift and controlled-phase shift form a set of universal quantum gates.

Example: realization of CNOT with a sequence of Hadamard and controlled-phase shift.

H HRπ

Testa o croce?

The Deutsch algorithmD. Deutsch, Proc. R. Soc. Lond. A 400, 97 (1985); R. Cleveet al., Proc. R. Soc. Lond. A 454, 339 (1998)

Let’s consider a Boolean functionf which maps{0,1} {0,1} .There are four functions of this type:

Two constant

• f1(0) = f1(1) = 0

• f2(0) = f2(1) = 1

Two balanced

• f3(0) = 0; f3(1) = 1

• f4(0) = 1; f4(1) = 0

How to find out whetherf is constant or balancedwith just one function evaluation?

Solution

0

0

The Boolean function evaluation amounts to x y → x f (x) ⊕ y

H H

H1 f

Measure the first qubit:• |0⟩ constant• |1⟩ balancedThe 2nd qubits always ends up in state |1⟩.

State after information

acquired

Information acquired

To get information from

bits

Can state be learnt from

Bits?

Reversible operations on

states

Subsets of n Bits

States of n Bits

QubitsBitsClassical vs. quantum

Bits

1 with ,2 =∑∑ xx x ααnxx 20 with , <≤

State after information

acquired

Information acquired

To get information from

bits

Can state be learnt from

Bits?

Reversible operations on

states

Generally have no statesAlways have statesSubsets of n Bits

States of n Bits

QubitsBitsClassical vs. quantum

Bits

1 with ,2 =∑∑ xx x ααnxx 20 with , <≤

State after information

acquired

Information acquired

To get information from

bits

Can state be learnt from

Bits?

Unitary transformationsPermutationsReversible operations on

states

Generally have no statesAlways have statesSubsets of n Bits

States of n Bits

QubitsBitsClassical vs. quantum

Bits

1 with ,2 =∑∑ xx x ααnxx 20 with , <≤

State after information

acquired

Information acquired

To get information from

bits

NoYesCan state be learnt from

Bits?

Unitary transformationsPermutationsReversible operations on

states

Generally have no statesAlways have statesSubsets of n Bits

States of n Bits

QubitsBitsClassical vs. quantum

Bits

1 with ,2 =∑∑ xx x ααnxx 20 with , <≤

State after information

acquired

Information acquired

MeasureJust lookTo get information from

bits

NoYesCan state be learnt from

Bits?

Unitary transformationsPermutationsReversible operations on

states

Generally have no statesAlways have statesSubsets of n Bits

States of n Bits

QubitsBitsClassical vs. quantum

Bits

1 with ,2 =∑∑ xx x ααnxx 20 with , <≤

State after information

acquired

x, with probability |αx|2xInformation acquired

MeasureJust lookTo get information from

bits

NoYesCan state be learnt from

Bits?

Unitary transformationsPermutationsReversible operations on

states

Generally have no statesAlways have statesSubsets of n Bits

States of n Bits

QubitsBitsClassical vs. quantum

Bits

1 with ,2 =∑∑ xx x ααnxx 20 with , <≤

Different: nowSame: stillState after information

acquired

x, with probability |αx|2xInformation acquired

MeasureJust lookTo get information from

bits

NoYesCan state be learnt from

Bits?

Unitary transformationsPermutationsReversible operations on

states

Generally have no statesAlways have statesSubsets of n Bits

States of n Bits

QubitsBitsClassical vs. quantum

Bits

1 with ,2 =∑∑ xx x ααnxx 20 with , <≤

x x

Conditional dynamics and

entanglement

( ) ( )

=

+++=

+⊗+=⊗=Ψ

11

01

10

00

11011000

1010

11100100

1010

βαβαβαβα

βαβαβαβαββααφψ

Suppose we have two qubits: one in state

10 10 ααψ +=

and the other in state

10 10 ββϕ +=

The pair state is

( ) ( )

=

+++=

+⊗+=⊗=Ψ

11

01

10

00

11011000

1010

11100100

1010

βαβαβαβα

βαβαβαβαββααφψ

=+++=Ψ

11

10

01

00

11100100 11100100)(

αααα

ααααt

10011100

11

01

10

00

11

10

01

00

αααα

βαβαβαβα

αααα

=⇔

=

A general two-qubit state is a product state if and only if

Nonproduct states of multi-qubit systems are called entangled states.

The CNOT gate generates entangled states

( ) 1100010 1010 αααα + →+ CNOT

Entanglement

• Il termine entanglement fu coniato da Erwin Schrödinger nel 1935, in una recensione del famosoarticolo sul paradosso EPR

• L'entanglement o correlazione quantistica è un fenomeno puramente quantistico, in cui ogni stato quantico di un insieme di due o più sistemi fisici dipende dallo stato di ciascun sistema, anche se essi sono spazialmente separati.

• Esso implica la presenza di correlazioni a distanza tra le quantità fisiche osservabili dei sistemi coinvolti, determinando il carattere non locale della teoria.

Schrödinger’s cat

Schrödinger’s cat

• The cat is entangled with

the radioactive particle

• Measurement output are

correlated

deadalive φϕ +=Ψ

Decoherence and the classical world

From Gedankenexperiment …

“We never experiment with

just one electron or atom or

(small) molecule. In thought

experiments, we sometimes

assume that we do; this

invariably entails ridiculous

consequences …”

(Schrödinger, 1952)

Erwin Schrödinger (1887-1961)

… to the Nobel prize in Physics 2012

“for ground-breaking experimental methods that enable measuring

and manipulation of individual quantum systems”

Serge Haroche David J. Wineland

Present and future quantum technologies

• Quantum key distribution is commercially available

• Atomic clocks and standard technologies

• Quantum sensors and actuators to navigate the

nanoscale world with remarkable precision and

sensitivity

• A fully functioning quantum computer is a longer-

term goal …

The DiVincenzo’s criteria D. P. DiVincenzo, The Physical Implementation of Quantum Computation,

Fortschritte der Physik 48, 771 (2000).

• A scalable physical system with well defined qubits;

• The ability to initialize the state of the qubits;

• A universal set of quantum gates;

• Decoherence times much longer than the gate operation time;

• A qubit-specific measurement;

• Interconverting stationary and flying qubits;

• Transmitting qubits over long distances.

Systems for quantum computation

• Nuclear magnetic resonance (NMR);

• Trapped ions;

• Cold atoms in optical lattices;

• Photonic systems;

• Solid state devices (quantum dots, superconducting

circuits, …);

• Electrons on a surface of liquid He.

Niels Bohr (1885-1962)

“Everything we call real is

made of things that cannot

be regarded as real.

If quantum mechanics

hasn’t profoundly

shocked you, you haven’t

understood it yet.”

.

Grazie per l’attenzione!