introduction to quantum computation neil shenvi department of chemistry yale university

Introduction to Quantum Computation

Neil Shenvi

Department of Chemistry

Yale University

Talk Outline

BackgroundWhat is Quantum Computation?Quantum AlgorithmsDecoherence and NoiseImplementationsApplications

Quantum Random Walks

O

Noise in Grover’sAlgorithm

Decoherence in Spin Systems

Background: Classical Computation

C:\Hello.exe Hello World!

Input Computation Output

What is the essence of computation?

2 + 2 4

Classical Computation Theory

Church-Turing Thesis: Computation is anything that can be done by a Turing machine. This definition coincides with our intuitive ideas of computation: addition, multiplication, binary logic, etc…

What is a Turing machine?

…0100101101010010110…

Infinite tape

Read/Write head

Finite State Automaton (control module)

…0000001011111111100…

Computation

…1110010110100111101… Output

…0100101101010010110… Input

Classical Computation Theory

What kind of systems can perform universal computation?

Desktop computers Billiard balls DNA

Cellular automata

These can all be shown to be equivalent to each other and to a Turing machine!

The Big Question: What next?

Talk Outline


What Is Quantum Computation?

Conventional computers, no matter how exotic, all obey the laws of classical physics.

On the other hand, a quantum computer obeys the laws of quantum physics.

The Bit

The basic component of a classical computer is the bit, a single binary variable of value 0 or 1.

1

0

0

1

The state of a classical computer is described by some long bit string of 0s and 1s.

0001010110110101000100110101110110...

At any given time, the valueof a bit is either ‘0’ or ‘1’.

The Qubit

A quantum bit, or qubit, is a two-state system which obeys the laws of quantum mechanics.

=|1 =|0

Valid qubit states:

| = |0 | = |1| = (|0- ei/4 |1)/2 | = (2|0- 3ei5/6 |1)/13

Spin-½ particle

The state of a qubit | can be thought of as a vector in a two-dimensional Hilbert Space, H2, spanned by theBasis vectors |0 and |1.

Computation with Qubits

How does the use of qubits affect computation?

Classical Computation

Data unit: bit

x = 0 x = 1

0

1

0

1

Valid states:x = ‘0’ or ‘1’ | = c1|0 + c2|1

Quantum Computation

Data unit: qubit

Valid states:

| = |0 | = |1 | = (|0 + |1)/√2

=|1 =|0= ‘1’ = ‘0’


0 1

1 0



Operations: logicalValid operations:

AND =

0 i

-i 0

1 0

0 -1

1 1

1 -1

0 1

0

1

0 0

0 1

NOT =0 1

1 0

in

out

out

in

in

1 0 0 0

0 1 0 0

0 0 0 1

0 0 1 0

1-bit

2-bit

Quantum Computation

Operations: unitary

Valid operations:

σX =

σy =

σz =

Hd =

CNOT =

√21

1-qubit

2-qubit




Measurement: deterministic

x = ‘0’

State Result of measurement

‘0’

x = ‘1’ ‘1’

Quantum Computation

Measurement: stochastic

| = |0

| = |0- |1

State Result of measurement

| = |1

2

‘0’

‘1’

‘0’ 50%

‘1’ 50%

More than one qubit

1000

u11 u12

u21 u22

Single qubit

c1

c2

c1

c2

Two qubits

H2 = 10

01,

|0,|1

H2 2 = H2H2 = ,

|00,|01,|10,|110100

,

0010

,

0001

c1

c2

c3

c4

c1

c2

c3

c4

u11 u12 u13 u14

u21 u22 u23 u24

u31 u32 u33 u34

u41 u42 u43 u44

Hilbertspace

U| = U| =Operator

| = c1|0 + c2|1 = | c1|00 + c2|01 +c3|10 + c4|11

==Arbitrarystate

Quantum Circuit Model

1000

0 0 1 00 0 0 11 0 0 00 1 0 0

σx I =

0010

1 0 0 00 1 0 00 0 0 10 0 1 0

CNOT =

0001

0001

|0

|0

|1

|0

|1

|1

‘1’

‘1’

Example Circuit

σx

One-qubit operation

CNOT

Two-qubit operation Measurement

Quantum Circuit Model

1/√2 01/√2 0

1000

σx CNOT

|0 + |1

|0

Example Circuit

√2______

1/√2 01/√2 0

1/√2 0 01/√2

0001

|0 + |1

|0√2

______ ‘0’

‘0’or

‘1’

‘1’

or

50% 50%

Separable state:can be written astensor product

| = | |

Entangled state:cannot be written as tensor product

| ≠ | |

??

Some Interesting Consequences

Quantum SuperordinacyAll classical quantum computations can be performed by a quantumcomputer. U

No cloning theoremIt is impossible to exactly copy an unknown quantum state

||0

||

ReversibilitySince quantum mechanics is reversible (dynamics are unitary),quantum computation is reversible.

|00000000 | |00000000

Talk Outline


Quantum Algorithms: What can quantum computers do?

Grover’s search algorithmQuantum random walk search algorithmShor’s Factoring Algorithm

Grover’s Search Algorithm

Imagine we are looking for the solution to a problem withN possible solutions. We have a black box (or ``oracle”) that can check whether a given answer is correct.

78

Question: I’m thinking of a number between 1 and 100. What is it?

Oracle No

3 Oracle Yes


The best a classical computer can do on average is N/2 queries.

1 Oracle No

...

2 Oracle No

3 Oracle Yes

Classical computer

Oracle1+2+3+... No+No+Yes+No+...

Quantum computer

Using Grover’s algorithm, a quantum computer can find the answer in N queries!

Superposition over all N possible inputs.


Pros:Can be used on any unstructured search problem, evenNP-complete problems.Cons:Only a quadratic speed-up over classical search.

The circuit is not complicated, but it doesn’t provide an immediatelyintuitive picture of how the algorithm works. Are there any moreintuitive models for quantum search?

O

σz

O

σz

…

……

…

|0|0

|0

O(N) iterations

Hd

Hd

Hd

…

Hd

Hd

Hd…

Hd

Hd

Hd

…Hd

Hd

Hd

…

Hd

Hd

Hd

Quantum Random Walk Search Algorithm

Idea: extend classical random walk formalism to quantum mechanics

A

tp

1tp

Classical random walk:

C S

| t 1| t

Quantum random walk:

1| |t tU

U S C Moves walkers based on coin

Flips coin

Pr( )ijA j i 1t tp A p


To obtain a search algorithm, we use our “black box” to apply a differenttype of coin operator, C1, at the marked node

C0

C1

1 -1-1 -1-1 1 -1 -1-1 -1 1 -1-1 -1-1 1

C0=12 C1=

-1 0 0 0 0 -1 0 0 0 0 -1 0 0 0 0 -1


Pros:As general as Grover’s search algorithm.

Cons:Same complexity as Grover’s search algorithm.Slightly more complicated in implementationSlightly more memory used

Interesting Feature: Search algorithm flows naturallyout of random walk formalism. Motivation for new QRW-based algorithms?

Shor’s Factoring Algorithm

Find the factors of: 57

3 x 19

Find the factors of: 1623847601650176238761076269172261217123987210397462187618712073623846129873982634897121861102379691863198276319276121

whimper

All known algorithms for factoring an n-bit number on a classical computer take time proportional to O(n!).

But Shor’s algorithm for factoring on a quantum computer takes time proportional to O(n2 log n).

Makes use of quantum Fourier Transform, which is exponentiallyfaster than classical FFT.

# bits 1024 2048 4096factoring in 2006 105 years 5x1015 years 3x1029 yearsfactoring in 2024 38 years 1012 years 7x1025 yearsfactoring in 2042 3 days 3x108 years 2x1022 years

with a classical computer

# bits 1024 2048 4096# qubits 5124 10244 20484# gates 3x109 2X1011 X1012

factoring time 4.5 min 36 min 4.8 hours

with potential quantum computer (e.g., clock speed 100 MHz)

R. J. Hughes, LA-UR-97-4986

Shor’s Factoring Algorithm

The details of Shor’s factoring algorithm are more complicated thanGrover’s search algorithm, but the results are clear:

Talk Outline


Decoherence and Noise

What happens to a qubit when it interacts with an environment?

0

0 1,

1

z

j jj

H H V

H B

V A

Quantum computer Environment

V

Quantum information is lost through decoherence.

σ1σ2 σ3

σN…

Types of Decoherence

T1 processes: longitudinal relaxation, energy is lost to the environment

V

T2 processes: transverse relaxation, system becomes entangled with the environment

V+

+

What are the effects of decoherence?

Effects of Environment on Quantum Memory

Fidelity of stored information decays with time.

T1 – timescale oflongitudinal relaxation

T2 – timescale oftransverse relaxation

Effects of Environment on Quantum Algorithms

Errors accumulate, lowering success rate of algorithm

Gro

ver’

s a

lgo

rith

m s

ucc

ess

rat

e

n = # of qubits

O

O

Idealoracle

Noisyoracle

Suppressing Decoherence

1. Remove or reduce V, i.e. build a better computer

System isolated from environment

2. Increase B, i.e. increase level splitting

B

E

|0

|1 When E >> V, decoherenceis smallE

3. Use decoherence free subspace (DFS)

4. Use pulse sequence to remove decoherence

Talk Outline


Some Proposed Implementations for QC

NMR

B

Ion trap

Optical Lattice

Kane Proposal

The Loss-Divincenzo Proposal

D. Loss and D.P. DiVincenzo, Phys. Rev. A 57, 120 (1998); G. Burkhard, H.A. Engel, and D. Loss, Fortschr. der Physik 48, 965 (2000).

Solid State Electron Spin Qubit

Silicon lattice

Phosphorus impurity

Electron wavefunction

Si28 (no spin)

Si29 (spin ½)

External MagneticField, B

Hyperfine couplingDipolar coupling

System Hamiltonian

Electronspin

N nuclearspins

( , )S z I jz j j jk j k

j j j k

H BS BI A S I b I I

Hyperfine coupling Dipolar coupling

~105 Hz ~102 Hz~107 Hz / T~1011 Hz / T

Hyperfine-Induced Longitudinal Decay

21( ) 82

cz

BS t

B

For B > Bc, T1 is infinite

jjc

S I

AB

Critical field for electronspin relaxation:

Hyperfine-Induced Transverse Decay

Free evolution Spin echo pulse sequence

Spin echo pulse sequence removes nearly all dephasing!

Talk Outline


Applications

Factoring – RSA encryptionQuantum simulationSpin-off technology – spintronics, quantum

cryptographySpin-off theory – complexity theory,

DMRG theory, N-representability theory

Acknowledgements

Dr. Julia Kempe, Dr. Ken Brown, Sabrina Leslie, Dr. Rogerio de Sousa

Dr. K. Birgitta WhaleyDr. Christina ShenviDr. John Tully and the Tully Group

introduction to quantum computation neil shenvi department of chemistry yale university

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