rutherford appleton laboratory particle physics department g. villani qc seminar ral 2008 1 notes on...

1

Rutherford Appleton LaboratoryParticle Physics Department

G. Villani QC seminar RAL 2008

Notes on Quantum Computation

E.G. VillaniSTFC

Rutherford Appleton Laboratory

2



OutlineOutline

• Quantum Computation introduction

• Algorithms examples

• Quantum Computation Technology

• Conclusions

3



Introduction notesIntroduction notes

Topics:Quantum information and computation theoryImplementation using different technologies

Topological Quantum computingQuantum magnetism

… 25th Winter School in Theoretical Physics

Institute for Advanced StudiesJerusalem 2007

4



Jerusalem notesJerusalem notes

Unified (east and west) 1967 Capital city of Israel from 1980Around 800,00 inhabitants : approx 70% Jewish, 29% Muslim, 1% ChristianMostly religious: ‘secular’ population in constant decline Old city around 1km2 :one of the oldest cities in the world

Christian

ArmenianJewish

Moslem

5



Jerusalem notesJerusalem notes

Dome of the ‘Rock’ 7th century AC

Holy SepulchreInitially built by Constantine 4th

Century BC

Western WallRetaining wall for the 2nd

Temple , 1st century BC

6



The general process of computation can be described as an operation performed on initial information and the reading out of the results:

Computation –introduction -

Input OutputOp

Computation can be performed in classical domain using analogue or digital blocks

7



Classical Computation – analogue -

Analogue computation example: PID controller

dtVdt

dVVV in

ininout

sKGsH

sKGsRsC

1

8



Analogue Computation – analogue -

Analogue Sallen Key 5th order low pass filter

9



Classical computation – digital -

Digital Sallen Key 5th order low pass filter IIR implementation

jnyjdknxkcny

knxkcny

10



Quantum limit

Moore’s Law (1965): Exponential size shrinking of electronic devices -> atomic limit will be approached around 2015

11



To simulate a probabilistic system (QM system) with a computer requires an exponential increase of resources (i.e. gates, time)

The simulation of a probabilistic system (QM system) could be done more efficiently with a probabilistic (QM) machine: Quantum Computer

1...111...1...0000...000221 Nccct

Quantum Computing –introduction -

12



computation in the Quantum domain

10 10

10 10

10 10

operatorinput output

A composite system of N Quantum bits (Qubits) is the input

An unitary operator is applied

A non unitary operator is applied to perform measurement

Quantum Computing –introduction -

13



Can be depicted as a point on the surface of a Bloch sphere:

In theory a Qubit can store infinite amount of information,

conserved during evolution. Measurement yields only one of the

two values.

1,102

1

2

010

A Quantum bit (qubit) describes the states of each individual two-level systems. In the computational bases:

12

sin02

cos

ie

Quantum Information –introduction -

Quantum information: qubits

14



An input state vector of non interacting qubits can be written in the form:

NN ...21

E.g.:

nnin 2121

ii

i

iHiHH 1,0, 22

In a Quantum computer a unit vector in a Hilbert space H describes the initial state of the

system

221214 ,,11002

1 with

Quantum Information

A state vector of interacting qubits consists of entangled states

15



In a more general case, a state of n qubits can be found in a

mixed state:

ii

k

iip

1

The density operator used to describe the state of a subsystem, by tracing out the unwanted system

UUUUp ii

k

ii

1

12212121 bbaabbaaTrTr BAB

BA

Quantum Information

16



Quantum computing –introduction -

To perform a Quantum computation an unitary operation is performed on the qubits system:

00exp UHdti

out

nnn fU 212121

The generator Hamiltonian H has to generate this evolution according to Schrodingers’ equation:

tHtdt

di

The Hamiltonian has to be found for a specific operation U:

Quantum processing: Evolution

17



Quantum computing

If U unitary a solution for H always exists.

Quantum computers can simulate any classical deterministic (Toffoli gate) and probabilistic (Hadamard gate) functions (i.e. they can perform any computations that

a classical computer does)

If classical f not reversible (like universal Boolean classical gates) it can be made reversible by adding extra information. An unitary quantum equivalent can then be built

E.g. Toffoli gate can be used to make any irreversible classical function reversible. Reversible classical gate (in principle) heatless.

18



2, iiii

i ttt

For a given orthonormal bases of HA it is possible to perform Von Neumann measurement:N distinguishable states input to an apparatus that perform a non unitary operation.

Quantum result: Measurement

2

jjjTr

ii i i

Quantum computing

19



Quantum computing – general process -

General Quantum circuit model

iini i AA

opinput output

0

0

0

U

in

UUTr inBoutin 0...0000...000

iini

iin AA

Superoperator: Quantum operation acts on an inputdensity operator plus ancillary register

If unchanged dimension of H

20



Quantum gates

A Quantum gate U rotates the Bloch vector on the Bloch sphere

01

10X

1

0

X

11

11

2

1

H

1

0

An unitary 1-qubit Quantum gate U can be written in terms of rotations around non parallel axes of the

Bloch sphere using Pauli gates :

lml

i RRReU

2

01

10 Xi

x eRX

2

0

0 Yi

y eRi

iY

2

10

01 Zi

z eRZ

Quantum computing

21



THG ,

Universal Quantum gates

An universal set of Quantum gates allows description with arbitrary accuracy of n-qubit unitary operator

(i.e. equivalent to classical NAND/NOR)

An universal set of Quantum gate for 1-qubit operator

11

11

2

1

H

8

8

1

1

i

i

e

eT

An universal set of Quantum gate for n-qubits operator is obtained by any 2-qubit entangling gate with an universal set for 1-qubit

nTHCNOTGn ,,,

0100

1000

0010

0001

CNOT

Efficiency of approximation ( number of gates) using G1(includes inverses)

1log,max, cOVUVU

Quantum computing

22



Quantum measurement

V. Neumann measurement

2

jjjTr

V. Neumann measurement with respect to any orthonormal basis

jj j jU

iij

H

Xj

Example: Computational basis to Bell basis

112

100

2

100

112

100

2

110

102

101

2

101

102

101

2

111

Quantum computing

23



Quantum computing

Quantum communication - teleportation

Bell

ZX

Using two classical bits , it is possible to send the state of a qubit: quantum state transmitted using classical channels!

112

100

2

100

Alice

Bob XZZX 1110010000 2

1

2

1

2

1

2

1

a

b

abab ZXU

Quantum teleportation useful to implement 2-Qubit gates First Quantum teleportation experimentally achieved using photons 1998

24



Quantum computing – algorithms -

First Quantum algorithm- Deutsch Jozsa Oracle

jj j jU

jj j jΣ ∆

Quantum superposition and interference

25



Deutsch Quantum algorithm

Problem :

1,01,0: f

To determine if f is constant or balanced.Classically: 2 queries

Deutsch algorithm: 1 query

xfyxyxU f :

Define the reversible mapping :

2

10

10 ff

Input an eigenstate to the target qubit of an operator and associate the eigenvalue with the control register

2

101

xf

xfU

x x

x

xfU

x

y xfyx

xfU

y

x

xfyx

x


26



xfU

0

2

10

H H

0 1 2 3

2

1000

2

101

2

1

2

100

2

11

2

101

2

1

2

100

2

1 10

2

ff

2

10101 0

3 fff

Simultaneous computation

result

Deutsch Quantum algorithm


27



1,01,0: nf

To determine if f is constant or balanced.Classically: 2n-1+1queries

Deutsch algorithm: 1 queryExponential increase in efficiency

xfU

0

2

10

H H

0 1 2 3

0 H H

0 H H

2

1000

n

nx

nx

1,01

2

1

2

101

2

1

1,02

nx

xf

nx

2

101

2

1

1,0 1,03 z

n nz x

zxxf

n

Problem :

Deutsch Jozsa Quantum algorithm


28



Quantum algorithm – Quantum Fourier TransformProblem: integer factorization

Split odd- non prime power N

Orders r of integers A co-prime with N

Sampling estimates to random integer multiple 1/n

yexQFTn

n

y

yx

i

n

12

0

22

2

1:

Order of random element in ZN

NNNO loglogloglogloglog 2Quantum complexity

NNOe logloglog Classical complexity

Shor’s algorithm for N factoring could compute 100s digits in seconds

Factorization believed to be NP problem but not demonstrated

xaU

n0 QFT QFT-1

1

NsasU a mod:


29



R(ivest)S(hamir)A(dleman) cryptosystem

Alice BobN,E

P,Q : N = PQ

E: GCD(E,(P-1)(Q-1)=1M

MEmod N

(ME)E-1mod N =M

E-1mod (P-1)(Q-1)

Difficult to factor large numbers: classically ~ weeks for 100 digits N :doubling the digits implies a factorization ~ 106 years!


30



Quantum algorithms remarks

Acyclic quantum gate arrays can compute in polynomial time any function computable in polynomial time by CTM .

10 10

10 10

10 10

opinput output

It is not known yet if many NP classical problems can become P using Quantum Computing

QFT (Shor’s algortithm)Deutsch Jozsa algorithm

Grover’s search algorithm( N vs sqrt(N))Simon’s algorithm


31



1-qubit cyclic gate represented by U(2) group

Any P problem can be mapped onto a acyclic quantum gateCyclical quantum gate still not investigated:

Compactness Phase delay

2-qubit cyclic gate represented by U(2) group

11,01

11,10

Quantum computing – further algorithms -

32



Perturbation of a 2-qubit cyclic network via C-NOT gate

Simplified form for Quantum Gates (i.e. FIR vs IIR in DF)Phase delay may lead to instability

Quantum oscillator!

Evolution of a cyclic network after n cycles:

In U eigenbases:

Simplified QFT

Quantum computing – further algorithms -

33



Quantum Computing – implementation techniques

Technical issues for building a Quantum Computer (DiVincenzo 2000)

Reliable representation of Quantum Information

(scalable number of Qubits )

Setting of initial state of qubits

Quantum gates reliable (decoherence)

Readout

Quantum decoherence

Interaction with environment ( I.e. partial measurement operated by

the environment)

Errors due to decoherence can be recovered, if error rate is

around 10-3/ 10-4 (Aharonov 1998)

Quantum error correction algorithms are effective if operations

are performed 10+3/ 10+4 faster than decoherence time

Alternative proposed solution to decoherence and error problem is

topological quantum computation

Several proposed solution for quantum

computation

34



Implementation techniques – Quantum Dots

Quantum well of Si-Ge for bi-dimensional confinement of

electrons, top gates for lateral confinement

Qubits : spins of individual electrons in quantum dots

Orbital coherence time << Spin coherence time (>100ns in

2DEG @ T=5K in GaAs)

Self-assembled Quantum dots using strained epitaxial

growth (i.e. Stranski-Krastanov process, growth of

material on substrate not lattice matched)

10’s nm scale

No nanothechnology required (etching, implanting)

No contamination

Non uniformity in size and position

Quantum dots using lithography :

100nm spacing

Good reproducibility

Nanotechnology required

Contamination

InAs/GaAs SA QD grown with MBE

35



Qubits : spins of individual electrons in quantum dots

For universal set G, coupling J of spins (qubits) needed

Quantum operation performed

by acting on gate voltages (2-qubit) to control J

ESR (electron spin resonance) ( 1-qubit phase rotation)

Dv/v<10^-6 only at low frequency…

21 SStJtH s

swsw

t

sUUtU

tHitU 2

10 :exp

RO using Spin to Q technique

Implementation techniques – Quantum Dots

e- e- AC

SEM of double Q-dot device

36



Implementation techniques – Ion traps -

Ions are confined in free UHV space using electromagnetic

field (Paul trap)

Qubits : ground and excited state level or hyperfine levels

Very long decoherence time

Initial state by optical pumping

Measurement using laser pulses coupled to one of the qubit

states: emitted photons read using CCD camera

Not easily scalable

Chip size planar Ions Trap: 6+6 traps of Mg on a flat

alumina surface

Field applied through gold electrodes

Tens of trapped ions feasible

Limitations in minimum trap size (~ 5µm)

Low temperature ( ~ -150C)

CNOT operation demonstrated

Trap region

37



Alternative implementation techniques

Nuclear Magnetic Resonance: Qubits are the spin states of the nuclei of the

molecules of the liquid used (demonstrated up to 8 qubits)

Superconductors QC based on Josephson junctions: (~ 1K required), Charge

qubit/Flux qubit

Adiabatic Quantum Computer (D.Aharonov, W. Van Dam et al) based on Adiabatic

Theorem (simulated)

A Quantum System in its ground state remains in it along an adiabatic transformation

in which the Hamiltonian is varied slowly enough from an initial to a final one.

Idea: to vary the Hamiltonian slowly from initial to final state as if an U was performed

on the initial state. The final ground state encodes the solution

jiH 0

jjH ff

1,0,1 10 T

txHxHxxH

38



Topological properties are deformation invariant (i.e. physically unaffected by

perturbations) : this would render quantum computation almost error-free

Topological Quantum computation

using non-abelian anyons(K. Shtengel, UC Riverside)

≡≠

C B A

Idea is to perform Quantum Computation using topological properties of quasi-particles

39



Topological Quantum Computing

Topological differences between 2 and 3 dimensions Quantum systems (Leinaas & Myrheim, 1977)

)i.e. if two particles are confined in 2D, their trajectories involve non-trivial winding if their positions are interchanged twice(

2121 ,, rrerr i

Two identical particles exchange their position anticlockwise:

,0Boson, fermions

,0anyonsIn 2D the phase can take any value:

40



Classes of trajectories taking N anyons from A to B are isomorphic to

BN

Non abelian anyons

A

B

Multiplication of elements of Bn is the

successive execution of the trajectories

Ni 11,

22 jijji

11111 Niiiiiii

12 i

2112

Topological Quantum computing –

41



Topological Quantum Computing

Anyons might arise in some low dimensions

confined many particles systems

Quasiparticles = Localized disturbances of the quanto-

mechanical ground state of the two-dimensional

system

To check if quasiparticles are anyons:

1. Take quasiparticles around each other adiabatically (i.e. intial positions = final positions with interchange)

2. The adiabatic interchange applies a unitary transformation on the ground state (phase):

3. If anyons

dttRE dRRR R

0

Experimental evidence that quasiparticles occurring in fractional Quantum Hall

effect are (non abelian ) anyons is still debated (J. Goldman et al. 2006)

42



Quantum Information – Topological techniques – Pairs of Anyons are brought together:

degeneracy is lifted -> 2 states = qubit

Qubitsin

U

U3 U1U2

Qubitsout

tMapping of unitary operations to braids not trivial

Readout of anyons not trivial

43



Quantum Information – Conclusions and relevance to Particle physics

Quantum Computing promises breakthroughs in solving complex mathematical problems, some hard or insolvable classically (but still investigated)

Computational power is an obvious benefit for all scientific fields, including Particle Physics (e.g. searching through immense databases)

Simulation of Quantum Mechanical systems may be another area of research: essentially, to simulate a quantum mechanical system means really to simulate nature with its laws. This applies to

the world of nanotechnology as Particle Physics too.

Theoretically one could think of modified laws of Quantum Mechanics(e.g. ‘ad’ hoc’ terms,non linearities etc)

Use of Quantum technology for next generation of detectors

44



Quantum Information – Backup slides -

Turing machine:Unbounded tape;Head that can read from the tape and can write on it, with infinite number of states;Instruction table.Given the initial head’s state and initial input the head reads, the table computes:The symbol the head writes on the tape;Where the head moves next on the tape.Church-Turing thesis: any effectively calculable function can be computed by a Turing machine

rutherford appleton laboratory particle physics department g. villani qc seminar ral 2008 1 notes on...

Documents

villani qc seminar ral

quantum information

century bc slide

digital blocks slide

theoretical physics

pid controller slide

general process of computation

probabilistic system