Quantum Information

Stephen M. BarnettUniversity of Strathclyde

steve@phys.strath.ac.uk

The Wolfson Foundation

0 Motivation1 Digital electronics2 Quantum gates3 Principles of quantum computation4 Quantum algorthims5 Errors and decoherence6 Realizations?

1. Probability and Information

2. Elements of Quantum Theory

3. Quantum Cryptography

4. Generalized Measurements

5. Entanglement

6. Quantum Information Processing

7. Quantum Computation

8. Quantum Information Theory

CMOS Device Performance

Device performance doubles roughly every 5 years!

P - solvable problems (computing time is polynomial in input size)

ClassicalDeterministicAlgorithm

ClassicalProbabilisticAlgorithm

Quantum Computing

Factoring Discrete logarithm Quantum simulations ...

Quantum algorithms: scaling of computing time with N~2n

1. F.T. to determine periodicities

f(x+r) mod N = f(x) mod N

find r Classical: O(N) = O(2n)Quantum: O(log2N) = O(n2)

2. Shor’s factoring algorithm

N = pq find p and q given N

Naïve classical (trial): O(N1/2) = O(2n/2)Best known classical: O(2^[n1/3log2/3n])Shor’s algorithm: O(polynomial[logN]) = O(polynomial n)Exponential sp

eed up!

What happens to RSA? W

hat happens to money??

…101101001…

…000111010…Input

Output“Black Box” orComputer

What is a computation?

Generation of an output number (string of bits) based on an input number.

How does the computer achieve this?

6.1 Digital electronics

Physical bit - electrical voltage +5V = 1 0V = 0

Single bit operation

NOT gate

A AA A

0

1

1

0

Two bit operations

AND gateA B

A A A B

0011

0101

B

B

0001

OR gateA B

A A A B

0011

0101

B

B

0111

Two bit operations

NAND gateA B

A A

0011

0101

B

B

1110

A B

Not all the gates are needed

A small set of gates (e.g. NAND, NOT) is universal in that any logical operation can be made from them.

6.2 Quantum gates

Single qubit operations

H 10

10

1

0

212

1

Hadamard

S1

0

1

0

i Phase

T 1)4/exp(

0

1

0

i / 8

and many more

Two qubit operations - CNOT gate

control bit

target bitTC

TC

TC

TC

TC

TC

TC

TC

01

11

10

00

11

01

10

00

CNOT gate can make entangled states

TCTCTCC

11000102

12

1

We can break up any multi-qubit unitray transformation into a sequence of two-state transformations:

jfc

heb

gda

U †3

†2

†1

123

ˆˆˆˆ

ˆˆˆˆˆ

UUUU

IUUUU

jfc

he

gda

UUba

a

ba

b

ba

b

ba

a

U 0ˆˆ

100

0

0

ˆ1

2222

22

*

22

*

1

**

**3

12

2222

22

*

22

*

2

0

0

001ˆ

0

0

1ˆˆˆ010

0

ˆ

jh

feU

jf

he

gd

UUU

ca

a

ca

c

ca

c

ca

a

U

It follows that we can realise any multi-qubit transformation as a sequence of single-qubit and two-qubit unitary transformations. This is the analogue of the universality of NAND and NOT gates in digital electronics.

The CNOT gate, together with one qubit gates are universal

control bit 1

target bit

TCCTCCabcbacba

2121

control bit 2

Exercise: Construct the Toffoli gate using just CNOT gates and single qubit gates. Try to use as few gates as possible.

V V†V

IVVVVViIe

V xx

iˆˆˆˆˆ,ˆˆˆˆ

2ˆ ††2

4/

6.3 Principles of quantum computation

Encode input onto qubit string

100101101101101001

Quantum evolution = unitary transformation

100101101ˆ100101101 U

Measurement gives output = computed function (hopefully!)

000111010100101101ˆ tmeasuremen U

A quantum computation is a (generalised) measurement

Quantum computation?

Constraints of unitarity? Consider the two bit map

BAABA 1,11,10,01,0

0,10,10,00,0

State overlap

U

Problem. Our computation requires

10,00,001,00,00010

0000

Unitary evaluation of the function f

ˆ U f)(afb

aa

b

a = input string …101101001...

b = input string, usually set to “zero” …000000000...

Exercise: Show that the states transformation is an allowed unitary transformation.

We can show this by an explicit construction:

bbafafaaUafba

f)(,0

)(00)(ˆ

II

bbafafaaU

UU

afbaf

ff

ˆˆ

)()(00ˆ

ˆˆ

)(,0

2

†

Parallel quantum computation

ˆ U f

aca

a

b

)(afaca

a

Can input a superposition of many possible bit strings a.Output is an entangled stated with values of f (a) computed for each a.

Deutsch’s algorithm

A black box that computes one of four possible one-bit functions:

We wish to know if the function is constant or balanced. We can do this by performing two computations To give f (0) and f (1) . Can we do it in one step?

A f (A)Black Box

Constant functions:

f (0) 0

f (1) 0or

f (0) 1

f (1) 1

Balanced functions:

f (0) 0

f (1) 1or

f (0) 1

f (1) 0

)0()0(1021 ff ˆ U f

102

1

102

1

A quantum computer allows solution in a single run:

+ for constant for balanced

10&102

12

1

are orthogonal states and so canbe identified without error.

Exponential speed up

Suppose our box computes a one bit function of n bits and that this function is either constant or balanced.

Constant: 0 or 1 independent of input

Balanced: 0 or 1 for exactly half of the possible inputs

Orthogonal states for constant or balanced functions so solution in ONE computation.

Guaranteed classical solution in computations2n 1 1

Quantum?

xxf

x

n

n

)(

2

2/ 12

10102 2/)1( nn

Exponential speed up.

6.4 Quantum algorithms

1. F.T. to determine periodicities

f(x+r) mod N = f(x) mod N

find r

Classical: O(N) = O(2n)Quantum: O(log2N) = O(n2)

2. Shor’s factoring algorithm

N = pq find p and q given N

Naïve classical (trial): O(N1/2) = O(2n/2)Best known classical: O(2^[n1/3log2/3n])Shor’s algorithm: O(polynomial[logN]) = O(polynomial n)

3. Grover’s search algorithm - searching a database

Classical: O(N) Quantum: O(N1/2)

Factorisation algorithm

N: Given big integer to be factorisedm: Small integer chosen at randomn = 0,1,2, …

1. Make the series FN(n) = mn mod N

2. Find the period r : FN(n+r) = FN(n)

3. The greatest common divisor of N and mr/2±1 divides N

Example: N = 15, m = 2 => FN(0) = 1 FN(1) = 2 FN(2) = 4 FN(3) = 8 FN(4) = 1 FN(5) = 2 …

=> r = 4

=> mr/2 – 1 = 3 mr/2 + 1 = 5

Both OK

Example: N = 15, m = 11 => FN(0) = 1 FN(1) = 11 FN(2) = 1 FN(3) = 11 …

=> r = 2

=> mr/2 – 1 = 10 => GCD 5 mr/2 + 1 = 12 => GCD 3

Both OK

Shor’s algorithm to factorise N

1. Find integers q and M such that: q = 2M > N2 and prepare two registers each

containing M qubits.

2. Set the qubits in the first register in the state (|0 + |1)/21/2 and those in the second in the state |0.

010002

001001

000000

where

01

21

1

0

nq

q

n

3. Choose an integer m at random and entangle the two registers so that

21

1

0

mod1

Nmnq

nq

n

This can be achieved by a unitary transformation (on a suitably programmed quantum computer) within polynomial time.

4. Fourier transform for register 1:

rqk

r

nq

k

q

n

nq

n

qkniNmkNmn/for1

period

21

1

0

1

021

1

0

/2expmodmod

5. Measurement on register 1:

=> k = multiple of q/r is obtained with high probability

=> r = q/k

6.5 Errors and decoherence

Interaction with the environment introduces noise and causes errors

Phase error

1

0

1

0

Bit flip error

0

1

1

0

ˆ U f 10

21

)0()0(0121 ff

102

1

Deutsch’s algorithm

Phase error

10

10

10

10

In this case

Bit flip error

01

01

10

10

In this case

Scaling

Probability that a given qubit has no error in time t exp( t)

Probability that none of n qubits has an error in time t )exp( tn

Let t be the time taken to perform a gate operation. For an efficient algorithm we might need n2 operations.

)exp( 3 tn

The number of required gate operations tends to grow at least logarithmically in the n )logexp( 2 tnn

1090requires about 300 qubits. This gives )107exp( 5 t

Decoherence is a real problem. We need efficient error correction!

Quantum error-correction

An error can make any change to a state so it is not obvious that error-correction is possible.

The key idea, of course, is redundancy!

11100010

11110000

33

33

This is a simultaneous eigenstate of

zzzz IIZZIZZI

with eigenvalue +1 in both cases.

If a single spin-flip error occurs 10

110001

101010

011100

111000

IZZ

ZZI

IZZ

ZZI

IZZ

ZZI

110001

110001

101010

101010

011100

011100

110001

101010

011100

111000

II

II

II

x

x

x

111000

111000

111000

We can, in fact correct any single-qubit error using the 7-qubit Steane code:

11010010111100

101101000011111100110

01100111010101000000020 2/37

00101101000011

010010111100000011001

10011000101010111111121 2/37

All the states differ in least four qubits – they are also common eigenstates of 6 operators with eigenvalue +1.

ZIZIZIZ

IZZIIZZ

IIIZZZZ

XIXIXIX

IXXIIXX

IIIXXXX

7777 10110

Any single-qubit error is detectable from a unique pattern of changes to these.“O

K, I’m co

nvinced. W

here ca

n I buy one?”

Ion-trap implementation - Cirac & Zoller, Wineland et al, Blatt et al.

Single ion qubits coupled by their centre of mass motion

0,g

0,e

1,g

1,e

2,g

2,e

Centre of mass motion acts as a ‘bus’ We can entangle the ionic qubits using thecentre of mass motion.

CofM210ee

2CofM1CofM12

1 10 ege

0,1g

0,1e1,1e

0,2g

0,2e1,2e

1,2g

1,1g

CofM21212

1 0ggee

Blatt et al Innsbruck

( Vandersypen, Steffen, Breyta, Yannoni, Cleve, Chuang, July 2000 Physical Rev. Lett. )

Nuclear spins Nuclear spins Nuclear spins Nuclear spins

• 5-spin molecule synthesized

• Pathway to 7-9 qubits

• First demonstration of a fast 5-qubit algorithm

Quantum-dot array proposal

• Well defined extendible qubit array - stable memory

• Preparable in the “000…” state• Long decoherence time (>104 operation time)• Universal set of gate operations• Single-quantum measurements

D. P. DiVincenzo, in Mesoscopic Electron Transport, eds. Sohn, Kowenhoven, Schoen (Kluwer 1997), p. 657, cond-mat/9612126; “The Physical Implementation of Quantum Computation,” quant-ph/0002077.

DiVincenzo’s criteria for implementing a quantum computer

Summary

• Quantum information is radically different to its classical counterpart. This is because the superposition principle allows for many possible states.

• Our inability to measure every property we might like leads to information security, but generalised measurements allow more possibilities than the more familiar von Neumann measurements.

• Entanglement is the quintessential quantum property. It allows us to teleport quantum information AND it underlies the speed-up of quantum algorithms.

• Quantum information technology will radically change all information processing and much else besides!