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
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
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
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
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