Principles l off quantum computing i Levon Altunyan M Sc Computer Science and M.Sc. Communications Engineering University Duisburg-Essen

Talk Outline Background z What is Quantum Computation? p z Quantum Algorithms z Decoherence D h and d Noise N i z Implementations z Applications z

Background: Classical Computation Input

Computation

2+2

Output

4

Hello World!

C:\Hello.exe What is the essence of computation?

Classical Computation Theory Church-Turing Church Turing Thesis: Computation is anything that can be done by a Turing machine. This definition coincides with our intuitive ideas of computation: addition, multiplication, binary logic, etc… What is a Turing machine?

…0100101101010010110…

Input

Finite State Automaton (control module)

Infinite tape

Computation …0000001011111111100…

Read/Write R d/W it head …0100101101010010110… …1110010110100111101…

Output

Classical Complexity S Some problems bl are more diffi difficult lt th than others. th Polynomial hierarchy NP-complete NP complete

All Turing machine-equivalent computers have an identical hi hierarchy. h

NP P

Require exponential(?) time to solve Require exponential time to solve Require polynomial time to solve

Classical Complexity Some important S i t t problems bl d do nott h have kknown classical l i l polynomial algorithm and or a known place in the hierarchy. Polynomial hierarchy NP-complete NP complete NP

?

Factoring

Best known algorithm to factor N N-digit digit number: Time ~ Exp(N1/3)

P ?

Graph Isomorphism

Best known algorithm to compare two N N-node node graphs: Time ~ Exp(N)

Classical Computation Theory Whatt kind Wh ki d off systems t can perform f universal computation?

Desktop computers

Cellular automata

Billiard balls

DNA

These can all be shown to be equivalent to each other and to a Turing machine!

The Big Question: What next?

Talk Outline Background z What is Quantum Computation? p z Quantum Algorithms z Decoherence D h and d Noise N i z Implementations z Applications z

What Is Quantum Computation?

Conventional computers, no matter how exotic, all obey the laws of classical physics.

O the other hand, a quantum computer obeys the laws off quantum physics. On

The Bit The basic component of a classical computer is the bit, a single binary variable of value 0 or 1. 1

0

0

1

At any given time, the value of a bit is either ‘0’ 0 or ‘1’ 1.

The state Th t t off a classical l i l computer t is i described d ib d by b some long bit string of 0s and 1s. 0001010110110101000100110101110110...

The Qubit A quantum bit bit, or qubit, qubit is a two-state two state system which obeys the laws of quantum mechanics. Valid qubit states: =|1〉

=|0〉

|ψ〉 = |0〉 |ψ〉 = |1〉 |ψ〉 = (|0〉- eiπ/4 |1〉)/√2 |ψ〉 = (2|0〉 (2|0〉- 3ei5π/6 |1〉)/√13

qubit |ψ〉 can be thought g of as a vector in The state of a q a two-dimensional Hilbert Space, H2, spanned by the Basis vectors |0〉 and |1〉.

Computation with Qubits How does the use of qubits affect computation?

Classical Computation

Data unit: qubit

Data unit: bit = ‘1’

= ‘0’

=|1〉

=|0〉

Valid states:

Valid states:

|ψ〉 = c1|0〉 + c2|1〉

x = ‘0’ or ‘1’ x=1

x=0

Quantum Computation

|ψ〉 = |0〉

0

0

1

1

|ψ〉 = |1〉

|ψ〉 = (|0〉 + |1〉)/√2

Computation with Qubits How does the use of qubits affect computation?

Classical Computation

Quantum Computation Operations: unitary Valid operations:

Operations: logical Valid operations: in NOT =

σX =

0

1

1

0

out

1-bit

1-qubit σy =

0 1 1 0 0

i

-i

0

in

in

0 1

0

0

0

1

2 bit 2-bit out

2 qubit 2-qubit

CNOT =

1 0 0 -1 1

Hd = 1 √2 1 0 0 0

0 1

AND =

σz =

0 1 0 0 0 0 0 1 0 0 1 0

1 1 1 -1

Computation with Qubits How does the use of qubits affect computation?

Classical Computation

Quantum Computation

Measurement: deterministic

Measurement: stochastic

State x = ‘0’ x = ‘1’

Result of measurement ‘0’ ‘1’

State |ψ〉 = |0〉 |ψ〉 = |1〉 |ψ〉 = |0〉- |1〉 √2

Result of measurement ‘0’ ‘1’ ‘0’ ‘1’ 1

50% 50%

More than one qubit T Two qubits bit

S Single qubit

|00〉,|01〉,|10〉,|11〉

|0〉,|1〉 Hilbertt Hilb space

Arbitrary state

Operator

H2 =

1 0

,

0 1

c1 |ψ〉 = c1|0〉 + c2|1〉 = c2

U|ψ〉 =

u11 u12 c1 u21 u22 c2

1 0 0 0

H2 ⊗2 = H2⊗H2 =

|Ψ〉 =

,

0 1 0 0

c1|00〉 + c2|01〉 + = c3|10〉 + c4|11〉

U|Ψ〉 =

u11 u21 u31 u41

u12 u22 u32 u42

u13 u23 u33 u43

u14 u24 u34 u44

, c1 c2 c3 c4

c1 c2 c3 c4

0 0 1 0

,

0 0 0 1

Quantum Circuit Model Example Circuit

Two-qubit operation

One-qubit operation

|0〉

σx

|1〉

|0〉

|0〉

1 0 0 0

0 0 1 0

Measurement

CNOT

CNOT =

1 0 0 0

0 1 0 0

|1〉

‘1’ 1

|1〉

‘1’

0 0 0 1

0 0 0 1

0 0 0 1

0 0 1 0

Quantum Circuit Model Example Circuit |0〉 + |1〉 ______ √2 |0〉

σx

50%

50% |0〉 + |1〉 ______ √2 |0〉

1/√2 0 1/√2 0

1/√2 0 1/√2 0

CNOT

? ?

‘0’

1/√2 0 0 1/√2

1 0 0 0

Separable state: can be written as tensor product

Entangled state: cannot be written as tensor product

|Ψ〉 = |φ〉 ⊗ |χ〉

|Ψ〉 ≠ |φ〉 ⊗ |χ〉 | 〉

‘0’

or

or

‘1’ ‘1’ 0 0 0 1

Talk Outline Background z What is Quantum Computation? p z Quantum Algorithms z Decoherence D h and d Noise N i z Implementations z Applications z

Quantum Algorithms: What can quantum computers do? Groverâ€™s search algorithm z Shor Shorâ€™ss Factoring Algorithm z

Grover’ss Search Algorithm Grover Imagine we are looking for the solution to a problem with N possible solutions. We have a black box (or ``oracle”) that can check whether a ggiven answer is correct. Question: I’m thinking of a number between 1 and 100. What is it? 78

Oracle

No

3

Oracle

Yes

Grover’ss Search Algorithm Grover Classical computer

Quantum computer

1

Oracle

No

2

Oracle

No

Superposition over all N possible inputs.

3

Oracle

Yes

Using Grover’s algorithm, a quantum computer can find the answer in √N queries!

... The best a classical computer can do on average is N/2 queries.

1+2+3+...

Oracle

No+No+Yes+No+...

Grover’ss Search Algorithm Grover Pros: Can be used on any unstructured search problem, even NP-complete problems problems. Cons: Onlyy a q quadratic speed-up p p over classical search. O(√N) iterations

Hd

…

Hd

O

Hd σ z Hd Hd Hd

…

Hd

…

|0〉

O

Hd σ z Hd Hd Hd

…

Hd Hd

…

|0〉 |0〉

Hd

Hd

… … …

The circuit is not complicated, but it doesn’t provide an immediately i t iti picture intuitive i t off h how th the algorithm l ith works. k

Shorâ€™ss Factoring Algorithm Shor Find the factors of: 57

Find the factors of: 16238476016501762387610762691722612171239872103974621876187 12073623846129873982634897121861102379691863198276319276121

3 x 19

whimper p

All known algorithms for factoring an n-bit number on a classical computer take time proportional to O(n!). But Shorâ€™s Shor s algorithm for factoring on a quantum computer takes time proportional to O(n2 log n).

Shor’ss Factoring Algorithm Shor The details of Shor’s factoring algorithm are more complicated than Grover’s search algorithm, but the results are clear: with a classical computer # bits f t i iin 2006 factoring factoring in 2024 factoring in 2042

1024 105 years 38 years 3 days

2048 5 1015 years 5x10 1012 years 3x108 years

4096 3 1029 years 3x10 7x1025 years 2x1022 years

with potential quantum computer (e.g., clock speed 100 MHz) # bits # qubits # gates f t i ti factoring time

1024 5124 3x109 4 5 min 4.5 i

2048 10244 2X1011 36 min i

4096 20484 X1012 4 8 hours 4.8 h

R. J. Hughes, LA-UR-97-4986

Decoherence and Noise What happens to a qubit when it interacts with an environment? Environment

Quantum computer

V

σ1

σ2

σ3

…

σN

What are the effects of decoherence? Quantum information is lost through g decoherence.

Ideal oracle

O

Noisy oracle

O

G Groverâ€™s algorithm m succe ess rate

Effects of Environment on Quantum Algorithms

n = # of qubits

Errors accumulate, lowering success rate of algorithm

Suppressing Decoherence 1. Remove or reduce V, i.e. build a better computer

System y isolated from environment

2. Increase B, i.e. increase level splitting ||1〉〉

E

When ΔE >> V, V decoherence is small

ΔE |0〉

B

What do QUBITs look like?

Single QUBITs – Superconducting devices ~ 1 μm (0.001mm) compared with a grain of pollen ~ 5 μm (0.005mm)

Quantum computers can be built from different systems: z Ions trapped pp in electrical fields can be manipulated p z Nuclear spins also exhibit quantum properties z Photons from lasers can be used to store quantum i f information ti

Some Proposed Implementations for QC NMR

Ion trap B

Optical Lattice

Kane Proposal

What do quantum computers look like?

D-Wave Systems Inc. uses a computational model known as adiabatic quantum computing (AQC): z z z

These processors exploit quantum effects to solve search and optimization problems in a new way. They look very similar to standard silicon processors â€“ in fact they are currently built using a similar method. Fabricating uses superconducting metals metals, operated at ultra-low temperatures in a magnetic vacuum.

Applications Factoring â€“ RSA encryption z Simulating g Molecules and other systems, y , e.g. g the Human Brain z Optimization/pattern problems z Machine Learning z

Acknowledgements Â…

A special thanks to Dr. Neil Shenvi from the Tully Group Department of Chemistry at Yale University for kindly allowing the use of his presentation materials which can be also found under: p // y / y/ /p http://www.chem.yale.edu/~tully/nashenvi/publications.html

Principles of Quantum Computing

Thank you for listening Any questions?