THE THREE THEMES IN QUANTUM MECHANICS

DONG PYO CHISeoul National University

INTRODUCTION

The Official birth of Quantum Mechanics

Why introduce quantum mechanics in computation?

Because nature isn't classical, dammit...

... If you want to make a simulation of nature, You’d better make it

Quantum Mechanical…!

Richard Phillips FeynmanSimulating physics with computers, Int. J. Theo. Phys. 21, 467 (1982).

“Quantum mechanics is simple.”

Axioms of Quantum MechanicsQuantum state Bit vs QubitQuantum Parallelism

Axioms of Quantum Mechanics

State A complete description of a physical system. A ray or a unit vector in a Hilbert space over C.

i.e. such that =1. Observable

A property of a physical system that can be measured. A self-adjoint operator, i.e., A = A.

Spectral Representation: A = n an Pnan: an eigenvalue of APn: the orthogonal projection onto the corresponding eigenspace.

Axioms of Quantum Mechanics

Measurement The numerical outcome of a measurement of the

observable A is an eigenvalue of A. The outcome an is obtained with probability

Prob(an) = || Pn||2 = Pn. If the outcome an is attained, the quantum state becomes

Pn-1/2 Pn. Dynamics

Time evolution of a quantum state is unitary. Schrödinger Equation: d(t)/dt = - iH(t)

(H is the Hamiltonian)i.e. (t) = U(t) (0) when H is time-independent,

U(t) = e-iH.

(continued)

Quantum state A quantum state is a mathematical object that

fully describes a quantum system. If the basis states are written and , then the

state vector is

where and are complex numbers with

A mixed quantum state is a statistical ensemble of pure states.

kk

kkp

Bit vs Qubit

Classical bit Qubit

0 or 1

00, 01, 10, or 11

1

,1022

1

1

,111001002222

2

Quantum Parallelism

000 f (000)

f (x)

001 f (001)010 f (010)011 f (011)100 f (100)101 f (101)110 f (110)111 f (111)

Classical processor

Quantum Parallelism

f (x)

000 001010011100101110111

f (000)f (001)f (010)f (011)f (100)f (101)f (110)f (111)

Quantum parallelismReversible operation

Quantum processor

(continued)

“Quantum mechanics is subtle.”

Schrödinger Cat Heisenberg Uncertainty Principle No-Cloning Theorem EPR Paradox Local Hidden-Variable Theory Bell's Inequality Quantum entanglement

Schrödinger Cat

Quantum mechanics predicts that the cat is alive and dead at the same time!

⇒ Superposed state

Heisenberg Uncertainty Principle

For a pair of non-commutative observables(ex. Position / Momentum)Knowing the value of one observable⇒ Makes the value of another observable

more uncertain

Any measurement of the output state that yields information in a classical way

⇒ Destruction of the remaining information

Heisenberg in 1925, at the age of 24

No-cloning Theorem

The polarization of single photon cannot be copied

⇒ Eavesdropper cannot have the same quantum information that Bob has

Alice BobEavesdropper

Proof of No-cloning theorem

Initial state of the copying machine: s

Some unitary evolution U now effects the copying procedure:

)( sUs

Suppose this copying procedure works for two particular pure states:

)(

)(

sU

sU

Taking the inner product of these two equations:

2)(

or orthogonal are and

state quantum pureunknown :

EPR Paradox

Einstein, Podolsky and RosenMaximally entangled state of two qubits

Faster-than-light communication?

BABA 11002

1

Local Hidden-Variable Theory

Einstein Locality: Suppose that A and B are spacelike separated systems. Then in a complete description of physical reality an action performed on system A must not modify the description of system B.

Measurement is actually fundamentally deterministic, but appears to be probabilistic because some degrees of freedom are not precisely known.

Let a, b, and c be any three observables such that a, b, c = ±1, i.e., they are functions of the hidden variables that take only the two values +1 and 1.

Bell's inequality:where denotes the expectation value.

Let where

, , , 3)(

3)(

2)(

1)( nncnbna BABA

bcacab 1

.10

01 ,

00

,0110

321

ii

),,,( 321

Bell's Inequality

Consider a pair of spin-1/2 objects in the state

Let be separated by successive angles.Then quantum mechanics predictsab = bc =1/2, ac = -1/2which violates the Bell's inequality:

(continued)

BABA 11002

1

321 , , nandnn 60

21

211

21

211

Bell's Inequality

Quantum entanglement “Spooky action at a distance”

The result of a measurement on one part of a quantum system can have an instantaneous effect on the result of a measurement performed on another part, regardless of the distance separating the two parts

Known as "non-local behavior" or "quantum weirdness"

Entangled state

…Can not be prepared by any local quantum operation and classical communication

States of many-qubit systems that cannot be broken down into a tensor product

i.e. there exist no s.t. , , ,

BBAA

BA4BA3BA2BA1

1 01 0

11011000

Measuring Entangled States

After measuring an entangled pair for the first time, the outcome of the second measurement is known 100%

First qubit measurement

50% prob with 0A

100% with 0 B

Second qubit measurement

BABA 11002

1

Measuring Entangled States

After measuring an entangled pair for the first time, the outcome of the second measurement is known 100%

First qubit measurement

50% prob with 1A

100% with 1 B

Second qubit measurement

BABA 11002

1

Bell states

In two-qubit system, the four possible entangled states are named Bell states

10012

1 , 10012

1

,11002

1 , 11002

1

“Quantum mechanics is useful.”

Quantum TeleportationDense CodingQuantum AlgorithmsQuantum Cryptography

Quantum Teleportation

Quantum Teleportation

AliceBob

10qubit unknown

Alice wants to send her unknown quantum information to Bob.

A and B do not have a quantum channel: only classical communication is allowed.

Alice cannot tell Bob what the values are,nor can she measure | to see what they are.

TeleportationAlice

Bob

Alice applies a CNOTand a Hadamard to and her EPR half.

She measures the bits in the 0/1 basis and sendsthe information to Bob.

Bob receives the twobits and acts on hisside of the EPR, in a way that recreatesthe unknown qubit | = α|0+β|1.

unknown qubit | = α|0+β|1.

Shared EPR pair

Danube River

Experimental Quantum Teleportation

Zeilinger group, 2004. 8. 19. Nature

Dense Coding

Dense Coding

EPRsource

Alice

2-bitmessage

2-bitmessage

Bob

QuantumGate

QuantumGate

Classical 1 bitQuantum 1 bit

Dense Coding

Alice and Bob share an EPR pair On her member of the entangled pair, Alice

performs one of four transformations and sends her qubit:

Bob performs an orthogonal collective measurement on the pair that projects onto the maximally entangled basis

0110

1100

21

21

BABA

BABA

ABAB

ABAB

ABAB

ABAB

A

A

A

A

:

:

:

:

)(

3

)(

2

)(

1

)(1

AB

,

Quantum Algorithms

Quantum Factoring Algorithms Data Search Algorithm Quantum algorithms without Initialization Hidden Subgroup Problems

Quantum Factoring Algorithms

Factoring problem Find a nontrivial factor of a given integer N Best known classical algorithm

O (exp[(64/9) 1/3 (log N) 1/3 (log log N) 2/3 ]) RSA-129 Challenge (1977)

Factoring 129 digits (429 bits) number into 64 & 65 digits primes⇒ 600 volunteers + 1500 workstations + 8 months (1994) 1000 digits : 1025 years

(1016 years = 106 * the age of the universe) Shor’s quantum factoring algorithm (1994)

O((log N)2) 129 digits: 1 hour 1000 digits: 24 hours

Fourier Transform DFT(Discrete Fourier Transform)

Linear Operator defined on C N : O(N 2) FFT(Fast Fourier Transform)

Cooley and Turkey (1965) : O(N log N) QFT(Quantum Fourier Transform)

P. Shor (1994) D. Coppersmith (1994)

Exact QFT: O ((log N )2) Approximate QFT: error bound

Size: O(log N log(log N /)), Depth: O(log N) A. Yu. Kitaev (1995)

Approximate QFT: error bound Size: poly(log(N/))

Factoring as Finding Period

Period-Finding Problem f : Z2n Z2m , a periodic function with unknown period

T, f(x) = f(x+kT) for 0 k [ 2n /T ] Find T efficiently!

Order finding can be reduced to Period finding Finding the order r s.t. x r =1(modN )

⇒ Finding the period of f s.t. f (t ) = x t (mod N )

Many other similar reductions Discrete logarithm Solution of Pell equation PID problem over quadratic number field

Data Search Algorithm

Review of Search Algorithms

Grover (1996) Single query when t(# of solutions) = N/4

Boyer, Brassard, Hoyer and Tapp (1996) Multiple solutions

Brassard and Hoyer (1997) Single query when t = N/2

Biron et al. (1998) Arbitrary initial amplitude

Chi and Kim (1999) Single query with certainty when N/4 ≤ t ≤ N

Grover Algorithm

Given F : ZN Z,find an index j ZN s.t F( j )1

Grover’s algorithm can search a database by queries

Grover operator : GF D SF Conditional phase transform:

Diffusion transform: D W S0 W* I 2 P( P (Pjk))

O N

otherwise

1)( if

j

jFjjFS

Several iterations of Conditional phase transform and Diffusion Operator

Measurement

Take a n-qubit register, where Nn 2

After n-dimensionHadamard Gate:

12

021 n

in

i

Quantum algorithms without Initialization

Initialization Free

Control register Biron et al. (1998) : Search algorithm Carlini & Hosoya (1999)

Generalized search algorithm, Counting algorithms

Auxiliary register Parker & Plenio (2000)

Factoring algorithm D. P. Chi, J. Kim & S. Lee (2001, 2002)

Initialization-free generalized Deutsch-Jozsa algorithm Quantum functional oracles

D. P. Chi, J. S. Kim & S. Lee (2005) Simon algorithm, period-finding algorithm

GDJ-EB Problem

f : ZN ZM is evenly balanced if exactly one half of the output values of f have parity 0. A function f : ZN ZM is said to be evenly distributed

iff has evenly spaced D values and | f -1(a + jL)| = | f -1(a + kL)| for j, k ZD (L M / D).

GDJ-EB problem is to determine whether f : ZN ZM is constant or evenly balanced. Cleve et al. (1997) Dong Pyo Chi, Jinsoo Kim & Soojoon Lee (2001)

Simon Problem

Given : f : Z2n Z2n two-to-one Promise : sZ2n s.t. f(x) = f(y) iff x = y n s

Find s efficiently.

Given : a group G = (Z2n, n), f : G GPromise : H = {0, s} G s.t. f(x) = f(y) iff x n H = y n H

Find H efficiently.

Circuit for Simon Algorithm

W : Walsh-Hadamard transformUf

: functional evaluation operator

{

W

W

WUf

W

W

W

n0

n0

nn 00

Circuit for Simon Algorithm

{{

W

W

WUf

W

W

W

n0

n0

0 0

0

0

0

2 1( )

00/

01/

1 ( 1) ( 1) ( )2

1 ( 1) ( )2

n

nx y x s yn

yx G H

x yn

x G H y H

y f x

y f x

first register measure0 0 1

1 for some with probability .2ny y H

Without Initialization

Original algorithm

Algorithm without Initialization

0

0

( ) ( )01

/

first register measure0 0 1

10 0 ( 1) ( )2

1 for some with probability 2

n f n x yn nn

x G H y H

n

y f x

y y H

W I U W I

Algorithm ( )

/

first register measure0

20 ( 1) ( 1)G

y while is not touched

n w f x x y

x G Hy H

y

Circuit for the Algorithm without Initialization(1)

{{

Uf

Sw

WW

W

Sw

WW

WUf

k

k k

n0

n0

Circuit for the Algorithm without Initialization(2)

)1()1(G2

/

)(

Hy HGx

yxxfw y

Uf

Sw

WW

W

Sw

WW

WUf

}

Probability

)1()1(G2

/

)(

Hy HGx

yxxfw y

register1st theoft Measuremen

s.t. )(Prob 0 y Hy0 is2

/

)(20

0)1()1(G4)(

HGx

yxxfww yP

Expected Probability

The expected probability of on randomly chosen w is

; Same probability as original Simon algorithm

)( 0yPw

G2

)1()1(G4

)1()1(G4 )(1

/ ,

)())()((3

2

/

)(30

0

0

HGxx

yxx

Gw

xfxfw

Gw HGx

yxxfw

Gww

nn

yPG

Hidden Subgroup Problems

Hidden Subgroup Problems

G : finitely generated group, X : finitef : G X s.t. f is constant on the cosets of H G

and distinct on each coset ⇒ Find H

Abelian Hidden Subgroup Problems Factoring, DLP, Order of Permutation PID problem over quadratic number field

Non-Abelian Hidden Subgroup Problems Graph isomorphism ⇒ HSP on Symmetry groups Lattice problem ⇒ HSP on Dihedral groups

HSP for Semi-direct Product Groups

Zpkn

Z2 : Santha et. al. (2002) ZN Zp : Moore et. al. (2004) N, p : prime, p = φ( N ) /poly(logN)

Z p k Z p : Inui and Le Gall, (2004) Zp

r Zp : Bacon et. al. (2005)

ZN Zp : D. P. Chi et. al. (2006) N = p1

r1…pnrn and p | pi-1 for all i

Quantum Cryptography

Quantum CryptosystemQuantum Key Distribution Protocol Security of QKD Physical Implementation of QKD

Eve

Quantum KeyDistribution

Key

Message

Alice

Message

Bob

Code

Key

Insecure Channel

Secure Channel

Code

Quantum Cryptosystem

Quantum Key Distribution Protocol

BB84 protocol EPR protocol B92 protocol

BB84 Protocol

Bennett and Brassard (1984) Two incompatible quantum alphabets Circular and linear polarization of photon

Detection of Eve’s intrusion Heisenberg uncertainty principle

BB84 Protocol: Simulation

Alice Bob0 1

random randomrandom

0

1

00

0

1

1

XX

X

XX

X

0

1

11

0

1

1

Physical implementation

The first prototype implementation of quantum cryptography (IBM, 1989)

EPR Protocol

EPR protocol Ekert (1991)

Detection of Eavesdropping Bell’s inequality

EPR Protocol: Simulation

Alice Bob

EPR

EPR

EPR

EPR

EPR

EPR

1 1

0 0

1 1

0 0

1 1

0 11 0EPR

random

Z Z

X X

Z Z

X X

X Z

Z X

X Z

random

X

XX

X

XX

A B A B(spin 1/2)

Physical realization of EPR

Quantum Optics P.G. Kwiat, K. Mattle, H. Weinfurter, A.

Zeilinger, A.V. Sergienko & Y.H. Shih (PRL, Vol 75, p. 4337, 1995)

B92 Protocol

Bennett (1992)Non-orthogonal basis (photon) Alice: Bob: P0 = 1 P = 1

Detection of eavesdropping No-cloning principle

Security of QKD

Heisenberg Uncertainty Principle

No-cloning Theorem

Heisenberg Uncertainty Principle

For a pair of non-commutative observables(ex. Position / Momentum)Knowing the value of one observable⇒ Makes the value of another observable

more uncertain

Any measurement of the output state that yields information in a classical way

⇒ Destruction of the remaining information

Heisenberg in 1925, at the age of 24

No-cloning Theorem

The polarization of single photon cannot be copied

⇒ Eavesdropper cannot have the same quantum information that Bob has

Alice BobEavesdropper

Physical Implementation of QKD

Optical fiber

Free space QKD

Optical fiber (148.7km)

Free space QKD (144km)

Companies offering commercial quantum cryptography systems

id Quantique (Geneva)

MagiQ Technologies (New York)

SmartQuantum (France)

Summary

Although axioms of quantum mechanics are mathematically simple, its subtleproperties provide us with very usefulapplications such as efficient quantum algorithms and perfectly secure quantum cryptographic protocols.