Introduction to Quantum Information and Computation

IPQI-2010-Anu Venugopalan 1

Introduction to Quantum Information and Computation

Anu Venugopalan

Guru Gobind Singh Indraprastha UniveristyDelhi

_______________________________________________ INTERNATIONAL PROGRAM ON QUANTUM INFORMATION (IPQI-

2010) Institute of Physics (IOP), Bhubaneswar

January 2010


Introduction

Quantum computation and quantum information is the study of the information processing tasks that can be accomplished using quantum mechanical systemsQuantum

MechanicsComputer ScienceInformation Theory CryptographyMathematics

Quantum information &computation


Introduction

Real computers are physical systems

Stonehenge Pebbles and beads Tokens, abacus Mechanical computers Difference Engine I (Charles Babbage) Analytical engine Electromechanical, electronic (vacuum tubes) Electronic (semiconductors)


Introduction

Real computers are physical systems

Computer technology in the last fifty years- dramatic miniaturization

Faster and smaller –

- the memory capacity of a chip approximately

doubles every 18 months – clock speeds and transistor density are rising exponentially...what is their ultimate fate????


Moore’s law [www.intel.com]


Miniaturization and computers

In spite of the dramatic miniaturization in

computer technology in the past five decades, our basic understanding of how a computer functions – or what it can do – has not changed. The tiny components inside all computers today still behave and are understood according to classical physics.


Extrapolating Moore’s law

If Moore’s law is extrapolated, by the year 2020 the basic memory component of the chip would be of the size of an atom – what will be space, time and energy considerations at these scales (heat dissipation…)?

At such scales, the laws of quantum physics would come into play - the laws of quantum physics are very different from the laws of

classical physics - everything would change!


Quantum computation

In anticipation of ultimately hitting atomic scales in computer technology, the field of quantum computation was first envisaged

Quantum physics offers something new and spectacular. By exploiting delicate quantum phenomena that have no classical analogues, it is possible to do certain computational tasks much more efficiently than can be done by any classical computer – even a supercomputer


Quantum computation

Quantum information and computing offers a new paradigm and possibilities for computing and will change the way in which scientists think about fundamental operations in computing and the capabilities and ultimate limits to computing

Offers powerful techniques for storage and manipulation of information

New phenomena - Quantum teleportation - Quantum cryptography


Quantum mechanics

At the turn of the last century, several experimental observations could not be explained by the established laws of classical physics and called for a radically different way of thinking --this led to the development of Quantum Mechanics which is today regarded as the fundamental theory of Nature

The price to be paid for this powerful tool is that some of the predictions that Quantum Mechanics makes are highly counterintuitive and compel us to reshape our classical (‘common sense’) notions.........


Introduction to theoretical computer science

What are the capabilities of a computer? What are the limits of a computer? What are the problems that can be solved

efficiently on a computer and what are the ones that cannot?

How are these questions related to the actual physical make up of the computer?

Does it matter if the computer is made up of gears and columns, vacuum tubes or integrated chips?



- Developed a ‘classical’ mathematical models for computation which was supposedly ‘free’ of any assumptions pertaining to the actual physical mechanism involved in a computer

- on closer examination these models revealed subtle assumptions that might well break down when we encounter a new regime of Nature…the quantum domain.

Alonso Church, Kurt Gödel, Emil Post

Alan Turing(1936)



Why should we spend time investigating classical computer science (TCS) if we are to study quantum computation?

• TCS has a vast body of concepts and techniques that can be applied to and reused in QI and QC- many of the triumphs of QI and QC have come by combining existing ideas from computer science with novel ideas from quantum mechanics



One learns to ‘think like a computer scientist’-

Computer scientists think in a very different style than does a physicist – anybody wanting a deeper understanding of QI&QC must learn to think like a computer scientist (at least some times!)

- very useful for studying QI and QC



Key concept of computer science – Algorithm

An algorithm is a precise recipe for performing some task – e.g. adding two numbers

The fundamental model for algorithms is

the Turing Machine


The Turing Machine

The most influential computer model due to Alan Turing (1936) – captures in a mathematical definition, what we mean when we use the intuitive concept of an algorithm

It is said to have been Turing’s response to David Hilbert’s challenge (‘Entscheidungsproblem’) and is also regarded as a computational analog of Gödel’s Incompleteness Theorem in Logic.


The Turing Machine

The proof process

If one were to look over a mathematician's shoulders during a proof derivation, what would one see in his/her notes?

Turing abstracted the process appearing in

these notes into four principle ingredients


The Turing Machine

1. A set of transformational rules

2. A method for recording each step in the proof

3. A method to go back and forth

4. A mechanism for deciding which rule to apply at a given moment


The Turing Machine

- the four steps listed are simplified so that a machine could be made to implement them –

- translating these steps in terms of symbols (0,1) on a one-dimensional tape with the read/write concept

- In this way Turing translated the mechanistic analogues of the human thought process into a mathematical form

The deterministic Turing machine


The Turing Machine

1 1 1 0 1 0 b b

Infinite tape

Read/write headInternal states/program

--------


The deterministic Turing Machine

- the four main elements of the DTM are

1. Finite State Control

2. Tape

3. A read/write tape head

4. Program


DTM- Finite State Control (FSC)

- Finite State Control

The FSC for a TM can be visualized as a stripped down microprocessor which coordinates the other operations of the machines

A finite set of m internal states: q1 , q2, ……….. qm

qs : starting state

qh: halting state


The Tape

- the tape is a one dimensional strip which stretches off to infinity in one direction – the tape squares are labelled and each contains one symbol drawn from some alphabet

e.g., 0,1 and b (blank)

marks the left hand edge of the tape


The read/write head

- the read/write tape head identifies a single square on the DTM tape as the square that is being currently accessed by the machine


Successor Program[adapted from

www1.cs.columbi.edu/~zeph/3261/L14/L14.ppt]

Sample Rules:

If read 1, write 0, go right, repeatIf read 0, write 1, HALT!If read �, write 1, HALT!

Let’s see how they are carried out on a piece of paper that contains the reverse binary representation of 47:


Successor Program[adapted from

www1.cs.columbi.edu/~zeph/3261/L14/L14.ppt]

Program

1,0,,1, 11 qq

0,1,,,1 hqbq

1,,,, 1 qqs 0,1,,0,1 hqq


Successor Program [adapted from www1.cs.columbi.edu/~zeph/3261/L14/L14.ppt]

1 1 1 1 0 1



0 1 1 1 0 1



0 0 1 1 0 1



0 0 0 1 0 1



0 0 0 0 0 1



So the successor’s output on 111101 was

000011 which is the reverse binary representation of 48.

0 0 0 0 1 1



Similarly, the successor of 127 should be 128, as one can see in the following:

1 1 1 1 1 1 1



0 1 1 1 1 1 1



0 0 1 1 1 1 1



0 0 0 1 1 1 1



0 0 0 0 1 1 1



0 0 0 0 0 1 1



0 0 0 0 0 0 1



0 0 0 0 0 0 0



0 0 0 0 0 0 0 1


The Church-Turing Thesis

.

Church-Turing thesis:

Any algorithmic process can be simulated on a Turing machine

A Turing Machine – an idealized and rigorously defined mathematical model of a computing device.

Many different models of computation are equivalentto the Turing machine (TM).



.

The class of functions computable by a Turing Machine corresponds exactly to the class of functions which we would naturally regard as being computable by an algorithm



.

Despite its lack of adornments, the TM model has proved to be remarkably durable in all the 70 years of its existence. Though computer technology has advanced dramatically, our qualitative understanding of the computation process remains the same

- in the strict theoretical sense, all computers are the same.


Computation

Real computers are finite devices, not infinite, like the Turing machine – they can be understood by a circuit model of computation


Circuits

NOTx 1x

x xNAND

x1 x y y

NOTx0AND

NANDx

10

x

y

x

y x

controlled-not gate (CNOT)


Universality

The NAND gate can be used to simulate the AND, XOR and NOT gates, provided wires, fanout, and ancilla are available.

wire – “memory”

Fanout – e.g Ancilla- bits in pre-prepared

states.

The NAND gate, wires, fanout and ancilla form a universal set of operations for computation.

x

x

xx


Irreversibility

NAND0

11

NAND1

10

From the output of the NAND gate it is impossible todetermine if the input was (0,1), (1,0), or (0,0)

The NAND gate is irreversible- there is no logic gate capable of inverting the NAND.


Irreversibility

Computing machines inevitably involve devices which perform logical functions that do not have a single-valued inverse.

This logical irreversibility is associated with physical irreversibility and requires a minimal heat generation, per machine cycle ~ order of kT for each irreversible function.

Landauer’s principle: Any irreversible operation in a circuit is necessarily accompanied by the dissipation of heat.


Irreversibility

•As the densities and switching speeds of our computational devices continue to increase exponentially, the amount of energy dissipated by these devices must remain at a certain level, otherwise economically impractical cooling apparatus is required.

•Conventional computers perform thermodynamically irreversible logic operations.

•Information, in the form of bits, is erased.

•This bit erasure represents entropy, which is correlated to heat dissipation.


Irreversibility

How can we compute without dissipating heat?

1. Lower the temperature of our computers

2. Develop thermodynamically reversible computers which do not generate entropy and therefore do not dissipate nearly as much heat as conventional, irreversible computers.

Use only reversible circuit elements

quantum gates are the most natural candidates for reversible gates.


Some reversible circuit elements

x

y

x

y x

x

y

x

y x

x

y

y

z

y

z x y

x x

The Toffoli gate (or controlled-controlled-not).

y

z

x

y

x

y

x

zz x y


Example: The reversible NAND gate.

x

1

y

1 x y

Computing using reversible circuit elements

x

y


x

0 x

x

Fanout

Computing using reversible circuit elements

quantum gates are natural candidates for reversible gates as any isolated quantum system has a dynamical evolution which is reversible………


The Quantum Turing Machine

• 1973: Charles Bennett – suggested the concept of a reversible Turing Machine

• !980: Paul Benioff - any isolated quantum system had a dynamical evolution which was reversible in the exact sense –and could mimic a reversible TM

• 1982: Richard Feynman - no classical TM could simulate certain quantum phenomena without incurring an exponential slowdown

• 1985: David Deutsch: Described the first Quantum Turing Machine


The Quantum Turing Machine

• In a QTM, all operations – e.g. read, write. shift etc. are accomplished by quantum mechanical interactions

• The tape could exist in states that were highly ‘nonclassical’

• A QTM would encode data not just in bits of 0 and I but in superposition states – qubits

• Quantum parallelism

1

0

Qubit


Complexity

• The fact that a computer can solve a particular kind of problem in principle does not guarantee that it can solve it in practice

• If the running time is too long and the memory requirements are too large, an apparently feasible computation can still lay beyond the reach of any practicable computer

• ‘efficiency’ complexity classes


Complexity classes

• How does the computational cost incurred in solving a problem scale up as the size of the problem increases?

• ‘Measure’ of efficiency – rate of growth of time/memory requirements to solve a problem at the size increases

• ‘Size’: number of bits, L, needed to state the problem to the computer


Complexity classes

• ‘Size’: number of bits, L, needed to state the problem to the computer. E.g. If N is a large integer, then the size of the integer in binary representation is L=Log2N

• Is the ‘cost’ polynomial in L or exponential in L?

• An exponential growth exceeds polynomial growth regardless of the order of the polynomial


Complexity classes

eL

L2

L

Example

The factoring problem

10433 x 16453=--------(polynomial, L2)

---- x ---- =200949083(exponential)


Complexity classes

If you had a very fast (classical) computer which could do, 1010 divisions per second, then to factorize:

L=20 --- one second L=34 ---- one year L=60 ----- 1017 seconds > age of the

Universe!

Multiplying two L=60 numbers would take only a few seconds.


Complexity classes

• Multiplication is ‘easy’ as costs grow only polynomially in size of the problem

• Factoring is ‘hard’ as costs grow exponentially in size of the problem

• The difficulty of factoring large numbers – lies at the heart of public key cryptosystems


RSA-129

• In 1994 Rivest, Shamir and Adleman , using a network of more than 1600 computers solved the most famous cryptography challenge in existence, a challenge that was thought to be unbreakable:

finding the prime factors of a 129 digit number

• Number field sieve (Lenstra 1990)Cost grows as exp[L2 log(L)2/3]

Sub exponential, but super polynomial


Complexity classes

• Problems that can be solved in polynomial time are termed tractable and belong to

• Problems that cannot be solved in polynomial time are termed intractable and may belong to one of several classes, e.g.

Complexity class P e.g. Multiplication

Complexity classes NP, ZPP, BPP


Complexity classes

Nondeterministic polynomial time- problems where the computational costs

incurred is exponential in L but once a candidate solution is found, its correctness can be tested ‘efficiently’, i.e., in polynomial time – this means there is an efficient nondeterministic algorithm for solving the problem

Complexity class NP

e.g. the factoring problem


Complexity classes

Complexity class P e.g. Multiplication

Complexity class NP complete

Complexity class NPe.g. the factoring problem

e.g. the traveling salesman problem

A subset of problems in NP that can be mapped into one another in polynomial time

Is P=NP?


The power of quantum

computing

One of the biggest motivations for studying Quantum Computation is the ability (as shown by Peter Shor in 1994) of a potential quantum computer to perform a computational task like factoring a large number far more efficiently than any conventional computer

– an ability to break 'unbreakable codes' like the most secure public key cryptosystem in the world today- RSA-


The power of quantum

computing

o In public key cryptosystems like the RSA system, the public key is broadcast/published – it contains a number, which if factored would reveal the private/secret key

o Factoring is hard – it is exponentially difficult for

classical (conventional) computers....

But a Quantum algorithm implemented on a quantum

computer could factor a large number very fast (Peter Shor

1994)-A Killer Application for Quantum Computation


Today, it seems technologically infeasible to

build a working quantum computer, but its risky

to think that this will be the position forever

At the pace of technological progress that we

see, a quantum computer might become a

reality sooner than later....

The power of quantum computing

Documents

Introduction to Quantum Information and Computation