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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
Off
Hot
On
On
On, then
off
Off
IBM Research
Information is...

... abstract, but its use requires a physical
representation
No information without 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!!!
A physicist’s view of computers
=
=
?
?
Is there a fundamental difference between computers?
What are their limitations, if any?
Energy
Input
0101001001001010101001
PHYSICS
Heat
Output
10101100101010001010101
Information and physics
Information 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
Example: factor a 300-digit number
Best classical
algorithm:
1024 steps
Shor’s quantum
algorithm:
1010 steps
On classical THz
computer:
150,000 years
On quantum THz
computer:
<1 second
M. Nielsen, Scientific
American, Nov 2002
Shor’s quantum algorithm can factor numbers very quickly
Difficulty of factorizing is the basis for modern cryptosystems
used on the internet
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
Computer: a universal
machine that can
implement any algorithm
Alan Turing
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?
+more?

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)
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 NP
All
problems
Naively categorise problems:

NP

P
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 ?
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
RSA-129
Example: 4633 = 41 x 113
1143816257578888676692357799761466120102182 96721242362562561842935706935245733897830597123563958705058989075147599290026879543541 =
3490529510847650949147849619903898133417764638493387843990820577 x 32769132993266709549961988190834461413177642967992942539798288533




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
Quantum algorithms


Richard
Feynman
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
P
David 
Deutsch
Problems a
quantum system
can solve
?
Concept of quantum computer emerged as a
universal device to execute such quantum
algorithms
Factoring with quantum systems

Shor (1995): quantum factoring algorithm
Example: factor a 300-digit number
Best classical
algorithm:
1024 steps
Shor’s quantum
algorithm:
1010 steps
On classical THz
computer:
150,000 years
On quantum THz
computer:
<1 second
Scientific American, Nov 2002

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! (?)
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:
Low noise
Strong control
i.e., an isolated,
closed system
i.e., strongly coupled to
user
Experimental effort: to gain strong, precise control
over quantum systems that maintain their quantum
nature
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
U

But through spin-spin interactions, a single electron
spin interacts with every other electron nearby!
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 criteria
David DiVincenzo (IBM) – requirements for a
quantum computer:
1.
The machine must have a scalable collection of bits
Each bit must be individually addressable, and it must
be possible to scale up to a large number of bits
2.
3.
It must be possible to initiate all of the bits to zero
The error rate should be sufficiently low
Decoherence times must be much longer than the
gate operation times
4.
5.
It must be possible to perform elementary logical
operations between pairs of bits
Reliable readout of the final result must be possible
Physical implementations
Many sub-fields of physics have proposals for QC







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





Quantum Hall qubits
Coupled quantum dots


charge qubits
flux qubits
phase qubits
spin, charge, excitons
Spin spectroscopies, impurities
in semiconductors
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
The latest:
Zeilinger group – UVienna
“One-way” quantum
computing with four qubits
Knill, Laflamme,
Milburn, Nature
(2001)
Superconducting Josephson junctions


Qubit: a) 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
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)
Silicon quantum computing



Kane, Nature (1998)
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
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.
D(E(x,e),d) = x

E.g.: one-time pad (e=d, random string as long as x)
00100
+11010
11110
A
11110
No transmitted information!
B
11110
-11010
00100
Problems with private keys

How are the private keys distributed?
 Security rests on private keys being kept secret
01101
10011
Trusted courier?


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
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
Eve receives a qubit that is either in
Measure in
or
basis?
Always gets
right, leaves state in
50% chance will mistake
for
Collapses
Measure in

into
basis
Disturbance!
basis? Similar result
Information gain by Eve causes an uncontrollable
disturbance
BB84 QKD Protocol



1984: Bennett and Brassard
Alice generates two random bits, a1,a2
Alice prepares a qubit as follows:
bits
00
01
10
11

state
a1 determines
which basis
a2 is an
encoded bit
in that basis
Alice then sends the qubit to Bob
BB84 QKD Protocol


a1
b1
0
0
1
0
1
1
0
1
1
1
0
0
?

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 round
This protocol is repeated (4+)n times
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:


Use this property to protect information


Any attempt to gain information about a quantum
system must alter that system in an uncontrollable way
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