Quantum Computing

Stephen Bartlett

www.physics.usyd.edu.au/~bartlett

A Puzzle Two rooms:

One room has three light switches These are connected to three bulbs in the other room

You don’t know which bulbs are connected to which switches

A Puzzle Condition: you’re only allowed to go into each room

once

PROBLEM: how do we figure out which bulb is connected to which switch?

The mathematician’s problem As a mathematical problem, there is no solution I.e., there is no configuration for the switches (which you

can only set once) that will give a unique matching of bulbs to switches when you observe the lights

NO SOLUTION?

?? ?

??

The physicist’s solution As a physics problem, there is a solution In the switch room:

Turn on two switches for a few minutes, then turn one off In the bulb room:

See which bulb is on, feel which other bulb is hot

On On, then off

OffOnHotOff

IBM Research

Information is... ... abstract, but its use requires a physical

representation

... encoded in the symbols on a page, the registers of a computer, the neurons of a brain or the base-pairs in DNA

... governed by the laws of physics!!!

No information without representation!

A physicist’s view of computers

Input Output

=?

0101001001001010101001 10101100101010001010101

Energy HeatPHYSICS

=?

Is there a fundamental difference between computers?

What are their limitations, if any?

Information and physicsInformation is physical, and governed by the

laws of physics

Our best framework for physical theories is quantum mechanics

Use quantum mechanics to describe information

Quantum information!

Quantum information investigates the processing, storage, and acquisition of information using quantum physics

Quantum computation We can use quantum physics to solve mathematical

problems

Shor’s quantum algorithm can factor numbers very quickly

Difficulty of factorizing is the basis for modern cryptosystems used on the internet

M. Nielsen, Scientific American, Nov 2002

Best classical algorithm:

1024 steps

Shor’s quantum algorithm:

1010 steps

On classical THz computer:

150,000 years

On quantum THz computer:

<1 second

Example: factor a 300-digit number

Quantum Cryptography Two remote parties can communicate securely by using

the laws of quantum physics

Quantum physics provides a powerful trade-off

Information gain Disturbance

Quantum Algorithms

What is an algorithm? Consider a problem where each instance has a solution

Example of a problem: Is an integer p a prime number? The instance: a particular choice of integer The solution: either yes or no (a decision problem)

Algorithm: a detailed step-by-step method for solving a problem Example algorithm: a program PRIMALITY(p) that runs on a

computer and gives yes or no for any input integer p

Alan Turing

Computer: a universal machine that can implement any algorithm

Example: discrete Fourier transform Problem: for a given vector (xj), j=1,...,N, what is the discrete

Fourier transform (DFT) vector

Algorithm: a detailed step-by-step method to calculate the DFT (yj) for any instance (xj)

With such an algorithm, one could: write a DFT program to run on a computer build a custom chip that calculates the DFT train a team of children to execute the algorithm

Computational complexity Consider an algorithm that solves a given problem Question: how much computing power do I need to

execute this algorithm for a given input (instance) size?

Let N be an integer describing the size of our instance Example: N could be the number of bits needed to write

the input in memory How does the number of steps in our algorithm depend

on N? (Definition of “steps” is a bit arbitrary, but the choice doesn’t affect scaling)

+more?

Computational complexity of DFT For the DFT, N could be the dimension of the vector

To calculate each yj, must sum N terms This sum must be performed for N different yj

Computational complexity of DFT: requires N2 steps DFTs are important ! a lot of work in optical computing

(1950s,1960s) to do fast DFTs 1965: Tukey and Cooley invent the Fast Fourier

Transform (FFT), requires N logN steps FFT much faster ! optical computing almost dies

overnight

Complexity classes - P and NPNaively categorise problems: P: the set of problems with an algorithm that requires

resources that are polynomial in the size of the problem Problems in P are considered “solvable” Not the whole story: an algorithm that scales as N100

is not easy in practice Both DFT and FFT are in P but FFT requires fewer

resources NP: the set of problems for which a “guessed” solution

can be checked using polynomial resources Some problems in NP can be used for cryptography

(data encryption, secure communication, etc.)

P

NP

All problems

P = NP ?

Example: Factoring Factoring: given a number, what are its prime factors? Considered a “hard” problem in general, especially for numbers

that are products of 2 large primes

Best factoring algorithm requires resources that grow exponentially in the size of the number (RSA-129 took 17 years)

Example: factor a 300-digit number Best algorithm: takes 1024 steps On computer at THz speed: 150,000 years

Difficulty of factoring is the basis of security for the RSA encryption scheme used, e.g., on the internet

Information security of interest to private and public sectors

Example: 4633 = 41 x 1131143816257578888676692357799761466120102182 96721242362562561842935706935245733897830597123563958705058989075147599290026879543541 = 3490529510847650949147849619903898133417764638493387843990820577 x 32769132993266709549961988190834461413177642967992942539798288533

RSA-129

Quantum algorithms Feynman (1982): there may be quantum systems

that cannot be simulated efficiently on a “classical” computer

Deutsch (1985): proposed that machines using quantum processes might be able to perform computations that “classical” computers can only perform very poorly

Concept of quantum computer emerged as a universal device to execute such quantum algorithms

PProblems a quantum system can solve

?

David Deutsch

Richard Feynman

Factoring with quantum systems Shor (1995): quantum factoring algorithm

To implement Shor’s algorithm, one could: run it as a program on a “universal quantum computer” design a custom quantum chip with hard-wired algorithm find a quantum system that does it naturally! (?)

Best classical algorithm:

1024 steps

Shor’s quantum algorithm:

1010 steps

On classical THz computer:

150,000 years

On quantum THz computer:

<1 second

Example: factor a 300-digit number

Scientific American, Nov 2002

Implications Information security and e-commerce are based on the

use of NP problems that are not in P must be “hard” (not in P) so that security is unbreakable requires knowledge/assumptions about the algorithmic

and computational power of your adversaries Quantum algorithms (e.g., Shor’s factoring algorithm)

require us to reassess the security of such systems Lessons to be learned:

algorithms and complexity classes can change! information security is based on assumptions of what is

hard and what is possible ! better be convinced of their validity

How do quantum algorithms work? What makes a quantum algorithm potentially faster than

any classical one? Quantum parallelism: by using superpositions of quantum

states, the computer is executing the algorithm on all possible inputs at once

Dimension of quantum Hilbert space: the “size” of the state space for the quantum system is exponentially larger than the corresponding classical system

Entanglement capability: different subsystems (qubits) in a quantum computer become entangled, exhibiting nonclassical correlations

We don’t really know what makes quantum systems more powerful than a classical computer

Quantum algorithms are helping us understand the computational power of quantum vs classical systems

Implementations of Quantum Computing

Experimental QIP Realising quantum information processing in a lab is

extremely difficult Requires two almost mutually-exclusive conditions:

Experimental effort: to gain strong, precise control over quantum systems that maintain their quantum nature

Low noise

i.e., an isolated, closed system

Strong control

i.e., strongly coupled to user

Example 1: spin of electrons The spin of an electron gives a quantum system We have strong control over this spin using electric

and magnetic fields

But through spin-spin interactions, a single electron spin interacts with every other electron nearby!

U

Example 2: polarised photons The polarisation of a photon gives a quantum system Photons in free space do not interact with each other

(i.e., with electric or magnetic fields)

But how can we entangle two photons if we can’t interact them?

U?

DiVincenzo criteriaDavid DiVincenzo (IBM) – requirements for aquantum computer:1. The machine must have a scalable collection of bits

2. It must be possible to initiate all of the bits to zero3. The error rate should be sufficiently low

4. It must be possible to perform elementary logical operations between pairs of bits

5. Reliable readout of the final result must be possible

Each bit must be individually addressable, and it must be possible to scale up to a large number of bits

Decoherence times must be much longer than the gate operation times

Physical implementations

Liquid-state NMR NMR spin lattices Linear ion-trap

spectroscopy Neutral-atom optical

lattices Cavity QED + atom Linear optics Nitrogen vacancies in

diamond

Electrons in liquid He Superconducting Josephson

junctions charge qubits flux qubits phase qubits

Quantum Hall qubits Coupled quantum dots

spin, charge, excitons Spin spectroscopies, impurities

in semiconductors

Many sub-fields of physics have proposals for QC

Ion traps Qubit: internal electronic state of

atomic ion in a trap (ground and excited)

Coupling: use quantised vibrational mode along linear axis (phonons)

Single qubit gates: using laser

Cirac and Zoller, Phys. Rev. Lett. (1995)

The latest:Monroe group – UMich

“T-Junction trap”

Shuttling ions around corners

Linear optics Qubit: polarisation of a single photon Coupling: via measurement Single-qubit gates: polarisation rotation

= 1

= 0

Knill, Laflamme, Milburn, Nature (2001)

The latest:Zeilinger group – UVienna

“One-way” quantum computing with four qubits

Superconducting Josephson junctionsa) Magnetic flux trapped in loop

b) Cooper pair charge on metal box

c) Charge-phase Coupling: capacitive/inductive Single-qubit gates: flux bias, charge on

gate, current through junction

Qubit:

Nakamura, Pashkin, Tsai, Nature (1999)The latest:

Schoelkopf group – Yale

Coherent coupling of a single photon to a superconducting qubit (Cooper pair box)

Nuclear magnetic resonance (NMR) Qubit: nuclear spins of atoms in

a designer molecule Coupling and single-qubit gates:

RF pulses tuned to NMR frequency

Gershenfeld and Chuang, Science (1997)

Qubit: Nuclear spin of single P donor Electron spin of single donor

Coupling: gate-controlled electron-electron interaction

Single-qubit gates: NMR pulse; gate bias in magnetic material

Kane, Nature (1998)

Silicon quantum computing

Summary Quantum computation requires precise control over isolated systems Many possible physical realisations may lead to discoveries and

advances in quantum computation Are we at the turning point?

Recent theoretical results strongly suggest QC is feasible Recent experimental developments suggest we might be there soon

Australia is a major player

UNSW, Melbourne and Queensland: experiment

Queensland, Sydney, Macquarie, Griffith: theory

Quantum Cryptography

Cryptography

Alice wants to send a message to Bob, without an eavesdropper Eve intercepting the message

Public key cryptography (e.g., RSA): security rests on assumptions about comp. complexity vulnerable to attacks by a quantum computer!

Quantum mechanics provides a secure solution with quantum key distribution (QKD)

Private Key Cryptography

Private key cryptography can be provably secure Alice has secret encoding key e, Bob has decoding key d Protocol: message x, functions E(x,e) and D(y,d) s.t.

E.g.: one-time pad (e=d, random string as long as x)00100

A B

00100

+11010

11110

11110 11110-11010No transmitted information!

D(E(x,e),d) = x

Problems with private keys How are the private keys distributed?

Security rests on private keys being kept secret

Ideally, A and B wish to generate strings of random numbers secretly and nonlocally

Privacy amplification and information reconciliation can be applied to make near-perfect private keys

Trusted courier?

0110110011

Using quantum mechanics Information gain implies disturbance:

Any attempt to gain information about a quantum system must alter that system in an uncontrollable way

Example: non-orthogonal states of a qubit

Information gain by Eve causes an uncontrollable disturbance

Eve receives a qubit that is either in or

Measure in basis?

50% chance will mistake for

Measure in basis? Similar result

Always gets right, leaves state in

Collapses into basis Disturbance!

BB84 QKD Protocol 1984: Bennett and Brassard Alice generates two random bits, a1,a2

Alice prepares a qubit as follows:

Alice then sends the qubit to Bob

bits state

00

01

10

11

a1 determines which basis

a2 is an encoded bit in that basis

BB84 QKD Protocol Bob receives the qubit Bob chooses a random bit b1 and measures

the qubit as follows: if b1=0, Bob measures in the basis

if b1=1, Bob measures in the basis

obtaining a bit b2

Alice and Bob publicly compare a1 and b1

if they are the same (Bob measured in the same basis that Alice prepared) then a2=b2

if they disagree, they discard that roundThis protocol is repeated (4+)n times

a1 b1 ?

0 0

1 0

1 1

0 1

1 1

0 0

BB84 QKD Protocol

With high probability, Alice and Bob have 2n successes To check for Eve’s interference:

Alice chooses n bits randomly and informs Bob Alice and Bob compare their results for these n bits If more than an acceptable number disagree, they abort

! evidence of Eve’s tampering (or a noisy channel) Alice and Bob use the remaining n bits as a private key!

Summary of quantum crypto Information is physical Information gain implies disturbance:

Any attempt to gain information about a quantum system must alter that system in an uncontrollable way

Use this property to protect information An eavesdropper’s attempt to gain information will alter

the system and thus may be detected! Future attempts to communicate securely or to protect

private information in the midst of public decision may rely on quantum physics