playing quantum games with superconducting circuits quantum computing zwiesel... · 39. edgar...

http://www.wmi.badw.de

Quantum

Computation

Rudolf Gross

Walther-Meißner-Institut, Bayerische Akademie der Wissenschaften

andTechnische Universität München

39. Edgar Lüscher-SeminarGymnasium Zwiesel

24.-26- April 2015

http://www.tum.de/

http://www.tum.de/

http://www.we-heraeus-stiftung.de/index.html

http://www.we-heraeus-stiftung.de/index.html

25.04.2015/RG - 2www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel

Research Campus Garching

Walther-Meißner-Institute

FRM II

Physics-Department

Mechanical Engineering

Informatics

Mathematics

LRZMPQ

ESOAstrophysics

Plasma Physics

Extraterrestr. Physics

ZAE

GRS


Contents

• a brief history of computation... from mechanical to quantum mechanical information processing

• computational complexity

• classical computation

• the weird world of quantum mechanics

• quantum computation

• quantum computers... where we are and where we hope to go

• summary


Science Museum London Science and Society Picture Library

Charles Babbage (1791-1871) conceptualized and invented the first mechanical computer in the early 19th century

in 1837 he conceives thecalculating machineAnalytical Engine

only part of the machine was completed before his death

engine incorporated (i) an arithmetic logic unit, (ii) a control flow, and (iii) integrated memory

first design for a general-purpose computer that could be described in modern terms as Turing-complete

First general-purpose computing device


Turing machine (1936)

• is a hypothetical (mathematical) device that manipulates symbols on a strip of tape according to a table of rules

• can be adapted to simulate the logic of any computer algorithm

Alan Mathison Turing(1912 – 1954) ©RosarioVanTulpe


Konrad Zuse was building the firstbinary digital computer Z1 in 1938

The first programmableelectromechanical computer Z3was completed in 1941

Zuse also developed thefirst algorithmic programminglanguage called „Plankalkül“

The first electromechanical computers


First fully automatic, digital computer

Replica of Zuse's Z3 (German Science Museum, Munich)


John von Neumann was proposing the EDVAC computerin 1945

he was introducing the conceptof a computer that is controlledby a

stored program

Programmable machines


Digital electronic programmable computers… with vacuum tubes

Colossus (1943) was the first electronic digital programmable computing device(Max Newman)

US-built ENIAC (Electronic Numerical Integrator and Computer) was the first electronic programmable computer built in the US (John Mauchly, J. Presper Eckert)

30 tons, 200 kW electric power, over 18,000 vacuum tubes, 1,500 relays, and hundreds of thousands of resistors, capacitors, and inductors, 6 operators, 160 m² space


Semiconductor Integrated Circuits

Intel 2nd generation Core i7 chip: 3.4 GHz, 32nm process technology (1.4 Mio. transistors)


SuperMUC @ LRZ Munich:peak performance: 3.6 PetaFLOPS (=1015 Floating Point Operations Per Second)

phase 2: 74 304 cores, Haswell Xeon processor E5-2697 v3

Modern supercomputers


Intel dual-core 45 nm

(2007)

first transistor (1947)

Bardeen, Brattain, & Shockley

vacuum tubes

ENIAC (1946)

Enigma (1940)

technologyphysics

superconducting Qubit

20 µm

WMI

From mechanical to quantum mechanical IP


Collaborative Research Center 631 Cluster of Excellence NIM

CeNS

F

WSI

F

SemiconductorQuantum Dots

MPQ

Trapped Atoms and Ions

2 µm

Al

WMI

Superconducting Qubits

Development of Hardware Platform for QIP Systems

http://www.mpg.de/bilderBerichteDokumente/multimedial/bilderWissenschaft/2005/10/Rempe0501/Web_Zoom.jpeg

http://www.mpg.de/bilderBerichteDokumente/multimedial/bilderWissenschaft/2005/10/Rempe0501/Web_Zoom.jpeg


Superconducting Quantum Computer

Vesuvius 3, 512 qubits,operated at T = 30 mK


multi

electron, spin, fluxon, photon

devices

single/few


devices

today near future

quantifiable,but not quantum

classicaldescription

65 nm process 2005 single electron transistor

PTB

... Solid State Circuits Go Quantum

Intel

• quantumconfinement

• tunneling• …


multi


devices

single/few


devices

quantum


devices

today near future far future



quantumdescription

65 nm process 2005 superconducting qubitsingle electron transistor

PTB


Intel

• superposition of states• entanglement• quantized em-fields

WMI2 µm


Contents







• summary


What is computation?

.... a procedure that

transforms input information to an output resultby a

sequence of simple elementary operations

algorithm

efficiency of algorithm measured by

computational complexity

if there exists an algorithmto solve a given problem, then it can be run on a universal Turing machine


Computational Complexity

.... is the study of the resources (time, memory, energy,...) required to solvecomputational problems

• addition: 𝒕 = 𝜶 ⋅ 𝒏

• multiplication: 𝒕 = 𝜷 ⋅ 𝒏𝟐

• example: time for adding and multiplying two n-digit integer numbers using primaryschool algorithm

multiplication is more complex than addition

precise result for multiplication by classical computer: 𝑂 𝑛 log𝑛 log (log 𝑛) (Schönhage, 1971)

• main distinction:

problems that can be solved using polynomial resources: P (e.g. 𝑡 ∝ 𝑛𝑘, 𝑘 = 𝑐𝑜𝑛𝑠𝑡.)e.g. multiplication: 𝑂 𝑛 log𝑛 log log 𝑛

problems that can be solved using resources that are superpolynominal: NP (e.g. 𝑡 ∝ 𝑘𝑛)

e.g. factorization of an n-digit integer: exp 𝑂 𝑛1/3 log 𝑛 2/3

(general number field sieve – GNFS – algorithm)


Complexity Classes

NPC

P

NP P = NPor

P = polynomial timeNP = superpolynomial time

NPC = NP-complete

believed to be right believed to be wrong

(problem in NP is NPC ifany problem in NP ispolynomially reducibleto it)

.... are still under debate


0 10 20 3010

0

101

102

103

104

105

106

Tim

e -

no o

f opera

tions

no of digits

exp(N)

N2

polynomial (P):

𝒕 ∝ #𝒐𝒑 ∝ 𝒏𝒌

non-polynomial (NP):

𝒕 ∝ #𝒐𝒑 ∝ 𝒌𝒏

(n: # of digits)

time # of operations (#op)

complexity of a problem

integer factorization is NP problem on a classical computer

algorithm is known, but too slow

classical computer

Integer Factorization 989

𝐞𝐱𝐩 𝒏

𝒏𝟐

#


2000 2010 2020 203010

-3

100

103

106

109

1012

1015

2048 bits

1024 bits

512 bits

min

iatu

riza

tion

limit

(

years

)

year of fabrication

polynomial (P):




(n: # of digits)




algorithm is known, but too slow

classical computer


𝐞𝐱𝐩 𝒏


100 1000 1000010

-1

100

101

102

103

4096 bits

2048 bits

1024 bits

512 bits

(

min

ute

s)

no of bits

quantum computer (100 MHz)

polynomial (P):




(n: # of digits)




exponential speed-up due to quantum algorthm (Shor)



Richard Feynman (1981):

“...trying to find a computer simulation of physics, seems to me to be an excellent program to follow out...and I'm not happy with all the

analyses that go with just the classical theory, because nature isn’t classical, dammit, and if you want to make a simulation of nature, you'd better make it quantum mechanical, and by golly it's a wonderful problem because it doesn't look so easy.”

example:

N interacting spins, 𝑺 =𝟏

𝟐, ↑↓

for N = 1000:

dimension of Hilbert space: 21000 > number of atoms in universe

……..

989Interacting Quantum System


989 = ??

Integer Factorization


989 = 23 ∙ 43

Integer Factorization


Contents







• summary


Binary Arithmetics

.... are used because arithmetical rules are simple

a b s c

0 0 0 0

0 1 1 0

1 0 1 0

1 1 0 1

binary addition table

1 1 1 0 1

1 0 1 0 1

1 1 1 0 1

1 1 1 0 1

1 1 1 0 1

1 0 0 1 1 0 0 0 0 1

1 1 1 0 1

1 0 1 0 1

1 1 0 0 1 0

multiplicationaddition

(29)

(21)

(29)

(21)

(50)

(609)

s: sumc: carry over


The Classical “Shannon” Bit

0 or 1 ↑ or ↓ or or

• elementary unit of classical information

copyingmachine

• classical bits can be copied

Claude Elwood Shannon (1916 – 2001)

(important difference to quantum bits)


What is Information?Claude E. Shannon (1948)• information is a general concept

similar to the concept of entropy or energy, appearing in many forms: mechanical, thermal, electrical,...

information content of variable 𝑥appearing with probability 𝑝𝑥:

𝑰 𝒑𝒙 = 𝐥𝐨𝐠𝒂 𝟏/𝒑𝒙 = − 𝐥𝐨𝐠𝒂 𝒑𝒙

example: binary alphabet, 𝑎 = 2,

𝑥1𝑥2 = 00, 01, 10,11; 𝑝𝑥1 = 𝑝𝑥2 = 𝑝 = 1/2

𝑰 𝒑 = −

𝒊=𝟏

𝒌

𝐥𝐨𝐠𝟐𝟏

𝟐= 𝒌 ⋅ 𝑯 𝒑 = 𝒌 = 𝟐

• can be packed into many equivalent forms:

0,1 ↑,↓ ,

good morning, guten Morgen

numberof bits

binaryentropyfunction𝐻2 𝑝

• information is physical (Landauer 1991)

- ink on paper- charge on capacitor- currents in leads- spins- polarization of photons- .....

unit: 1 Shannon (sh)

𝑯𝟐𝒑

𝒑(𝒙 = 𝟏)


What is Information?

• example:

- we consider the character string „Honolulu“ with 𝑘 = 8 characters

- the alphabet ist 𝑍 = 𝐻, 𝑜, 𝑛, 𝑙, 𝑢

- with probabilities 𝑝 𝐻 =1

8, 𝑝 𝑜 =

1

4, 𝑝 𝑛 =

1

8, 𝑝 𝑙 =

1

4, 𝑝 𝑢 =

1

4

𝑰 = −

𝒊=𝟏

𝟖

𝐥𝐨𝐠𝟐 𝒑𝒊 = 𝟐 ⋅ 𝟑 + 𝟔 ⋅ 𝟐 = 𝟏𝟖 𝐛𝐢𝐭

• we calculate the total information by using the log basis 𝑎 = 2 to get the result in the unit of bits:

we need 18 bit to optimally code the word „Honolulu“ in a binary basis


Elementary Logic Gates• in any computation:

𝑛-bit input is tranferred into 𝑙-bit output 𝒇: 𝟎, 𝟏 𝒏 → 𝟎, 𝟏 𝒍

can be decomposed into sequence of elementary logical operations

logical (one-bit and two-bit) gates: AND, OR , XOR, NOT, NAND, NOR, XNOR,FANOUT, SWAP

AND: 𝑨 ∧ 𝑩A B Y0 0 00 1 01 0 01 1 1

OR: 𝑨 ∨ 𝑩A B Y0 0 00 1 11 0 11 1 1

NOT: 𝒀 = 𝑨

A Y

0 1

1 0

I: 𝒀 = 𝑨

A Y

0 0

1 1

one-bit gates two-bit gates

ANSI/IEEE Std 91/91a-1991


Universal Logic Gates

• any logical function 𝒇: 𝟎, 𝟏 𝒏 → 𝟎, 𝟏 𝒍 can be constructed bya universal set of elementary gates:

(i) AND, OR , NOT, FANOUT

(ii) NAND, FANOUT

AA

A

• the FANOUT gate is acting as a copying machine for classical bits

• we will see later that this gate cannot be realized for a quantum computer: no cloning theorem


• the AND gate is not logically reversible

• therefore, the (non-reversible) AND gatethrows away or erases information

• Landauer showed that erasing a bit ofinformation results in energy dissipation

Landauer principle

Universal Logic Gates

AND: 𝒀 = 𝑨 ∧ 𝑩A B Y0 0 00 1 01 0 01 1 1


Information and Energy

Landauer principle (1961):

each time a single bit of information iserased, the amount of energy dissipatedinto the environment is at least 𝒌𝑩𝑻 𝐥𝐧𝟐

equivalently, we may say that the entropyof the environment is increased by at least 𝒌𝑩 𝐥𝐧 𝟐

Rolf William Landauer(February 4, 1927 – April 28, 1999)

example:

computer with 109 gates operated at 3 GHz clockspeed at 300 K dissipates at least

𝑃 = 1.38 ⋅ 10−23 × 300 × 3 ⋅ 109 × ln 2 ≃ 10 mW

R. Landauer, "Irreversibility and heat generation in the computing process," IBM Journal of Research and Development, vol. 5, pp. 183-191, 1961


Reversible Computation• most of the classical logic gates are irreversible

we cannot recover the input from a given output

the Boolean operators erase a bit of information

energy dissipation is unavoidable (Landauer principle)

ANDA B Y0 0 00 1 01 0 01 1 1• is it possible to do reversible computation without

energy consumption?

Yes, replace irreversible gates by reversible generalization

Identity and NOT gate are reversible

important reversible two-bit gate is CNOT (reversible XOR)

additional three-bit gate required: Toffoli gate (C-CNOT)

𝐴

𝐵

𝐴′ = 𝐴

𝐵′ = 𝐴⊕𝐵

CNOT𝐀 𝐁 𝐀′ 𝐁′

0 0 0 00 1 0 11 0 1 11 1 1 0

C. H. Bennett, "Logical reversibility of computation," IBM Journal of Research and Development, vol. 17, no. 6, pp. 525-532, 1973.

𝐴 + 𝐵 𝑚𝑜𝑑2


Contents







• summary


»Quantum« is an amount of energy that can no longer be subdivided

quantum hypothesis of Max Planck (1900):

𝑬 = 𝒉 ⋅ 𝝂

• quantum physics: theories, models and concepts based on thequantum hypothesis of Max Planck

• quantum jump: transition between twoquantum states

∼ 𝟏𝟎𝟐𝟒 light quanta are required to heat up 1 liter of H2O to 100°C

• quantum mechanics and theory of relativity: foundations of modern physics

microcosm macrocosm

What do we understand by »Quantum«


»Quantum«


quantum objects are wave and particle at the same time

diffraction of lightat double-slit

diffraction of electronsat double-slit

Excursion to the quantum world


• quantumtunneling

• quantumsuperposition

e.g. Schrödinger cat

• uncertaintyrelation

tunneling of a wave packet

a physical system — e.g. an electron — exists partly in all its particular theoretically possible states simultaneously

Excursion to the quantum world


Weird Quantum World


Ruth Bloch, Entanglement II, bronze, 27" (2000)

Quantum Entanglement

quantum correlation:measurements of observables of entangled particles arecorrelated

decoherence:entanglement is broken through the interaction with the environment

entangled quantum states:two or more particles can show nonlocal correlationssuch that their quantum states can no longer bedescribed independently

𝚿𝒆 =𝟏

𝟐(|𝟎𝑨𝟎𝑩⟩ + |𝟏𝑨𝟏𝑩⟩) (entangled state)

Ψ ≠ Ψ 𝐴 ⊗ Ψ 𝐵

Ψ ∈ ℋ𝐴 ⊗ℋ𝐵

Ψ𝑠 =1

2(|0𝐴1𝐵⟩ + |1𝐴1𝐵⟩) (separable state)

=1

2(|0𝐴⟩ + 1𝐴 ) ⊗ |1𝐵⟩ Ψ = Ψ 𝐴 ⊗ Ψ 𝐵


Can the quantum mechanical description ofthe physical reality be considered complete?

Are there hidden variables?

A. Einstein, B. Podolsky, N. Rosen, Phys. Rev. 47, 777 (1935)

Einstein (1935): „spuky action at a distance“

Schrödinger (1935): „entanglement“

Bell (1964): „principle of locality is in conflict with quantum theory“

EPR Paradox

measurement in basis{|𝑷⟩, |𝑸⟩}

𝐏 =𝟏

𝟐(| ⇑⟩ + | ⇓⟩) ; 𝑸 =

𝟏

𝟐(| ⇑⟩ −⇓⟩)

A B A B

𝜳 =𝟏

𝟐(|𝑷𝐀𝑸𝐁⟩ − |𝑸𝐀𝑷𝐁⟩)

J.S. Bell, Physics 1, 195–200 (1964)


𝚿 = − | ⟩

A B

A B A B

Entangled Dices


Bell‘s Inequality (a simple logic exercise)

• we consider a collection of objects with parameters A, B, and C:e.g. 𝐴 = male, 𝐵 = taller than 1.8 m, 𝐶 = blue eyes (classroom example)

# 𝑨, 𝒏𝒐𝒕 𝑩 + # 𝑩, 𝒏𝒐𝒕 𝑪 ≥ #(𝑨, 𝒏𝒐𝒕 𝑪) this relationship is called

Bell's inequality

John Stewart Bell (1928 – 1990)

J.S. Bell, Physics 1, 195–200 (1964)

• we proof that the number of objects which have parameter A but not parameter B plus the number of objects which have parameter B but not parameter C is greater than or equal to the number of objects which have parameter A but not parameter C:


Bell‘s Inequality (a simple logic exercise)

Proof (has nothing to do with quantum mechanics):

• we assert that # 𝑨, 𝒏𝒐𝒕 𝑩, 𝑪 + # 𝒏𝒐𝒕 𝑨,𝑩, 𝒏𝒐𝒕 𝑪 ≥ 𝟎

pretty obvious, since either no group members have these combinations or some members do

• we add # 𝑨, 𝒏𝒐𝒕 𝑩, 𝒏𝒐𝒕 𝑪 + # 𝑨,𝑩, 𝒏𝒐𝒕 𝑪 on both sides

# 𝑨,𝒏𝒐𝒕 𝑩, 𝑪 + # 𝑨, 𝒏𝒐𝒕 𝑩, 𝒏𝒐𝒕 𝑪 + # 𝒏𝒐𝒕 𝑨,𝑩, 𝒏𝒐𝒕 𝑪 + # 𝑨,𝑩, 𝒏𝒐𝒕 𝑪 ≥ 𝟎 + # 𝑨, 𝒏𝒐𝒕 𝑩, 𝒏𝒐𝒕 𝑪 + # 𝑨,𝑩, 𝒏𝒐𝒕 𝑪

q.e.d.

• no other assumptions made than(i) logic is a valid way to reason (ii) parameters A, B, C exist whether they are measured or not

(there is a reality separate from its observation)

# 𝑨, 𝒏𝒐𝒕 𝑩 + # 𝑩, 𝒏𝒐𝒕 𝑪 ≥ #(𝑨, 𝒏𝒐𝒕 𝑪)

# 𝑨, 𝒏𝒐𝒕 𝑩 # 𝑨, 𝒏𝒐𝒕 𝑪

since either 𝐵 or 𝑛𝑜𝑡 𝐵must be true

# 𝑩, 𝒏𝒐𝒕 𝑪

since either 𝐶 or 𝑛𝑜𝑡 𝐶must be true

since either 𝐴 or 𝑛𝑜𝑡 𝐴must be true


Bell‘s Inequality (applied to electron spin)

# 𝑨, 𝒏𝒐𝒕 𝑩 + # 𝑩, 𝒏𝒐𝒕 𝑪 ≥ #(𝑨, 𝒏𝒐𝒕 𝑪)

𝐴: 𝜃 = 0° = ↑𝐵: 𝜃 = 45° =𝐶: 𝜃 = 90° = →

spin directions:

# ↑, + # ,← ≥ #(↑,←) 𝑛𝑜𝑡 𝐴: 𝜃 = 180° = ↓𝑛𝑜𝑡 𝐵: 𝜃 = 135° =𝑛𝑜𝑡 𝐶: 𝜃 = 90° = ←

• measurement of # ↑, , # ,← , #(↑,←)

measurement of # of electrons with ↑, ↓,←,→ or , gives 50% for each projection but if we try to measure ↑ and at the same time, we have a problem:

only 15% are (and 85% would be ), if we have measured ↑ before the preceeding measurement of # ↑ irrevocably changes #

in the same way: the measurement of # irrevocably changes # ←

in classroom example this would mean that measuring the gender would changetheir height: pretty weird but true for electron spins

• Bell‘s inequality

e-gun


• measurement of spin directionsarbitrary spin state:

𝚿 = 𝐜𝐨𝐬𝚯

𝟐↑ + 𝒆𝒊𝝓 𝐬𝐢𝐧

𝚯

𝟐↓

↑

↓

Bell‘s Inequality (applied to electron spin)

# ↑, : if we have measured | ↑⟩ in the first measurement,the probability to find | ⟩ in the second is

↑ . 2 = cos135°

2

2

= 0.146

# ,← : if we have measured | ⟩ in the first measurement,the probability to find | ←⟩ in the second is

. ← 2 = cos135°

2

2

= 0.146

#(↑,←): if we have measured | ↑⟩ in the first measurement,the probability to find | ←⟩ in the second is

↑ ← 2 = cos90°

2

2

= 0.5

# ↑, + # ,← = 𝟎. 𝟏𝟒𝟔 + 𝟎. 𝟏𝟒𝟔 ≥ # ↑,← = 𝟎. 𝟓 violation of Bell‘s inequality

first experiments demonstrating violation: Clauser, Horne, Shimony and Holt in 1969 using photon pairs


Violation of Bell‘s Inequality

• assumptions made in deriving Bell’s inequality

(i) logic is a valid way to reason

(ii) parameters A, B, C exist whether they are measured or not≡ electrons have spin in a given direction even if we do not measure it≡ there is a reality separate from its observation≡ hidden variables exist

(iii) no information can travel faster than the speed of light≡ locality≡ hidden variables are local

violation of Bell‘s inequality is in conflict with local realism !!

.....but what if logic is not valid ?


Will quantum effects be part of our everyday life?

Yes, they already are !

interesting applications ⇒ quantum technologies


• quantum communication

• quantum simulation

• quantum metrology

• ............

Relevance of Quantum Phenomena


Contents







• summary


The Quantum Bit

𝚿 = 𝜶 + 𝜷| ⟩

0 or 1 ↑ or ↓ or or

• classical bit (two distinct states)

• quantum bit (arbitrary superposition of two quantum states – computational basis)

𝚿 = 𝜶 𝟎 + 𝜷|𝟏⟩ with 𝜶 𝟐 + 𝜷 𝟐 = 𝟏

decisive

indecisive

• measurement („collapse“ of wave function)

Ψ = 𝛼 + 𝛽| ⟩

observer| ⟩

with probability 𝜶 𝟐


The Quantum Bit

• Bloch sphere representation (geometrical picture of qubit)


𝟐𝟎 + 𝒆𝒊𝝓 𝐬𝐢𝐧

𝚯

𝟐𝟏

=𝐜𝐨𝐬

𝚯

𝟐

𝒆𝒊𝝓𝐬𝐢𝐧𝚯

𝟐

𝟎 or 𝒈

𝟏 or 𝒆

ground state

excited state


The Quantum Bit

• quantum bit (mathematical picture of qubit):

- two-level quantum system whose state is represented by a ket | ⟩ lying in a 2D Hilbert space 𝓗, which has the orthonormal basis 𝟎 , |𝟏⟩

- each ket can be thought of as a column vector

𝟎 =𝟏𝟎

and 𝟏 =𝟎𝟏

𝚿 = 𝜶 𝟎 + 𝜷 𝟏 = 𝜶𝟏𝟎

+ 𝜷𝟎𝟏

=𝜶𝜷

• tensor product of Hilbert spaces: 𝓗 =𝓗𝟏 ⊗𝓗𝟐 ⊗⋯⊗𝓗𝒏

𝟎 𝟏 = 𝟎𝟏 = 𝟎 ⊗ 𝟏 =𝟏𝟎

⊗𝟎𝟏

=𝟏 ⋅

𝟎𝟏

𝟎 ⋅𝟎𝟏

=

𝟎𝟏𝟎𝟎

state in 2n-dimenisonal Hilbert space: 𝚿 = 𝚿𝟏 𝚿𝟐 … 𝚿𝒏 = 𝚿𝟏𝚿𝟐…𝚿𝒏

• example: 𝓗 =𝓗𝟏 ⊗𝓗𝟐


Measuring the State of a Qubit

• example: consider a 2-dim. quantum system in state 𝚿 = 𝜶 𝟎 + 𝜷|𝟏⟩

• what happens if we measure Ψ in the basis ± =1

20 ± |1⟩ ?

- we first express Ψ in the basis ± : 𝚿 =𝜶+𝜷

𝟐

𝟎 +|𝟏⟩

𝟐+

𝜶−𝜷

𝟐

𝟎 −|𝟏⟩

𝟐

thus measurement of Ψ in the basis ± yield two possible results:

𝚿 →𝟎 + |𝟏⟩

𝟐𝚿 →

𝟎 − |𝟏⟩

𝟐

probability =𝜶+𝜷 𝟐

𝟐probability =

𝜶−𝜷 𝟐

𝟐

note: we cannot completely control the outcome of the measurement


Representing Integers

• let ℋ2 be a 2-dim. Hilbert space with orthonormal basis 0 , |1⟩

• then, ℋ =⊗0𝑛−1 ℋ2 is a 2𝑛-dim. Hilbert space with induced orthonormal basis

𝟎…𝟎𝟎 , |𝟎…𝟎𝟏⟩ , 𝟎…𝟏𝟎 , 𝟎…𝟏𝟏 ,… , |𝟏…𝟏𝟏⟩

• we represent the integer 𝑚 with binary expansion

𝒎 = 𝒋=𝟎𝒏−𝟏𝒎𝒋𝟐

𝒋 , 𝒎𝒋 = 𝟎 𝒐𝒓 𝟏, ∀𝒋

as the ket

𝒎 = |𝒎𝒏−𝟏𝒎𝒏−𝟐 … 𝒎𝟏𝒎𝟎⟩

• example

𝟐𝟑 = |𝟎𝟏𝟎𝟏𝟏𝟏⟩


The No Cloning Theorem

• quantum bit cannot be copied copyingmachine

proof: - assume that there is a unitary operator 𝑈 producing copies of |𝐴⟩ and |𝐵⟩

remark: - cloning is inherently nonlinear

- quantum mechanics is inherently linear

- however, the quantum copying machine fails in copying state 𝐶 =1

2(|𝐴⟩ + |𝐵⟩)

𝑼 𝑨 𝒃𝒍𝒂𝒏𝒌 ] = |𝑨𝑨⟩ and 𝑼 𝑩 𝒃𝒍𝒂𝒏𝒌 ] = |𝑩𝑩⟩

𝑼 𝑪 𝒃𝒍𝒂𝒏𝒌 ] =𝟏

𝟐(|𝑨𝑨⟩ + |𝑩𝑩⟩) ≠ |𝑪𝑪⟩

quantum replicators do not exist


Massive Parallelism

• example: 𝚿𝐢 =𝟏

𝟐(|𝟎⟩ + |𝟏⟩) for 𝑖 = 1,2,3,… , 𝑛

𝚿𝟎𝚿𝟐…𝚿𝒏−𝟏 =

𝒊=𝟎

𝒏−𝟏𝟏

𝟐(|𝟎⟩ + |𝟏⟩)⊗

=𝟏

𝟐

𝒏

(|𝟎⟩ + |𝟏⟩) (|𝟎⟩ + |𝟏⟩) … (|𝟎⟩ + |𝟏⟩)

=𝟏

𝟐

𝒏

(|𝟎𝟎…𝟎⟩ + 𝟎𝟎… 𝟏 +⋯ |𝟏𝟏…𝟏⟩)

=𝟏

𝟐

𝒏

𝒂=𝟎

𝟐𝒏−𝟏

|𝒂⟩

then

the 𝒏-qubit register contains all 𝒏-bit binary numbers simultaneously !!!

𝒏 classical bits can store a single integer 𝑰,the 𝒏-qubit quantum register can be prepared in the corresponding state |𝑰⟩ ofthe computational basis, but also in a superposition


Massive Parallelism: Deutsch’s Problem

• classical machine:𝒙 → 𝒇(𝒙)

0

1

𝑓(0) = 0 𝑜𝑟 1

𝑓(1) = 0 𝑜𝑟 1

we are satisfied to know whether 𝑓 𝑥 = const (𝑓(0) = 𝑓 1 ) or balanced (𝑓 0 ≠ 𝑓 1 ) we have to run the machine twice to find this out

• quantum machine:𝒙 𝒚 → 𝒇(|𝒙⟩ 𝒚⊕ 𝒇|𝒙 )

|0⟩

|1⟩

machine flips the second qubit if 𝑓 acting on the first qubit is 1, and does not do anything if 𝑓 acting on the first qubit is 0

we can determine if 𝑓 𝑥 is constant or balanced by using the quantum black box twice.

Can we get the answer by running the quantum black box just once ?(“Deutsch’s problem”)

choose the input state to be a superposition of 0 and 1

𝑥1

2|0⟩ − |1⟩ → 𝑥

1

2|𝑓(𝑥)⟩ − |1 ⊕ 𝑓(𝑥)⟩ = 𝑥 −1 𝑓(𝑥)

1

2|0⟩ − |1⟩

we have isolated the function 𝑓 in an 𝑥-dependent phase


Massive Parallelism: Deutsch’s Problem

• we also prepare the first qubit 𝑥 in superposition state 1

2|0⟩ − |1⟩

1

20 + |1⟩

1

2|0⟩ − |1⟩ →

1

2−1 𝑓(0) 0 + −1 𝑓(1) 1

1

2|0⟩ − |1⟩

• we perform a measurement that projects the first onto the basis ± =1

20 ± |1⟩

we will always obtain |+⟩ if the function is balanced and |−⟩ if it is constant

the classical computer has to run the black box twice to distinguish a balancedfunction from a constant function, but a quantum computer does the job inone go!

• suppose we are interested in global properties of a function that acts on 𝑁 bits, a function with 2𝑁 possible arguments

to compute a complete table of values of 𝑓(𝑥) we have to calculate 𝑓 exactly 2𝑁 times (completely infeasible for 𝑁 ≫ 1)

with the quantum machine we can choose the input register to be in a state1

20 + |1⟩

𝑁which requires to compute 𝑓 𝑥 only once !!

speedup by massive quantum parallelism


Entangled Qubits

⊗

𝟎 𝑨 ⊗ 𝟏 𝑩

𝟎 𝑨 ⊗ 𝟏 𝑩 − 𝟏 𝑨 ⊗ 𝟎 𝑩

𝟐

𝑼

• entangled• not separable

• not entangled• separable

unitary transformation

definition:if a pure state Ψ ∈ ℋ𝐴 ⊗ℋ𝐵 can be written in the form Ψ = Ψ 𝐴 ⊗ Ψ 𝐵

where Ψ 𝑖 is a pure state of the 𝑖𝑡ℎ subsystem, it is said to be separable


Observing Entangled Qubits

𝟎 𝑨 ⊗ 𝟏 𝑩 − 𝟏 𝑨 ⊗ 𝟎 𝑩

𝟐

observes onlythe blue qubit

with probability 1/2

whoosh !!

𝟎 𝑨 ⊗ 𝟏 𝑩 𝟏 𝑨 ⊗ 𝟎 𝑩

with probability 1/2

• no longer entangled• separable


Elementary Logic Gates

• classical computer: - 𝒏 classical bits form a register of size 𝒏- sequence of elementary operations (e.g. AND, NOT) produce a

given logic function

• quantum computer: - 𝒏 quantum bits form a quantum register of size 𝒏- sequence of elementary operations (QUANTUM GATES) produce

a given logic function

• typical sequence for quantum computation:

initialization: prepare the quantum computer in a well-definded initial state

manipulation: apply elementary quantum gates to manipulate quantum state

readout: perform a quantum measurement at the end of the algorithm


Bloch sphere



𝚯

𝟐𝟏

𝟎

𝟏

e

g

Qubit

1-Qubit-Gate

U1

Quantum Processor: Principle of Operation

M.A. Nielsen, I.L. Chuang, Quantum Computation andQuantum Information (Cambridge Univ. Press, 2000)


2-Qubit-Gate (C-NOT)

e

g

U1

g e

readout

e

g

U1

e

g

Qubit

1-Qubit-Gate

U1

Quantum Processor: Principle of Operation

M.A. Nielsen, I.L. Chuang, Quantum Computation andQuantum Information (Cambridge Univ. Press, 2000)


Single Qubit Gates

• Hadamard gate:

- maps the basis state |0⟩ to 1

2(|0⟩ + |1⟩) and |1⟩ to

1

2(|0⟩ − |1⟩)

- represents a rotation of 𝜋/2 about the axis 1

2( 𝑥 + 𝑧)

|𝐴⟩ B|𝐵⟩𝑯 =𝟏

𝟐

𝟏 𝟏𝟏 −𝟏

• phase shift gate:

- leaves the basis state |0⟩ unchanged, maps |1⟩ to e𝑖𝜙|1⟩- equivalent to tracing a horizontal circle on the Bloch sphere by 𝜙 radians

𝑹𝝓 =𝟏 𝟎𝟎 𝒆𝒊𝝓

• single qubit gates: - rotate the state vector on the Bloch sphere- are represented by unitary matrices: 𝑈𝑈⋆ = 𝐼


Two Qubit Gates

• SWAP gate: - swaps two qubits

𝐒𝐖𝐀𝐏 =

𝟏 𝟎𝟎 𝟎

𝟎 𝟎𝟏 𝟎

𝟎 𝟏𝟎 𝟎

𝟎 𝟎𝟎 𝟏

• CNOT gate: - performs the NOT operation on the second qubit only when the first qubit is 1

and otherwise leaves it unchanged

𝐂𝐍𝐎𝐓 =

𝟏 𝟎𝟎 𝟏

𝟎 𝟎𝟎 𝟎

𝟎 𝟎𝟎 𝟎

𝟎 𝟏𝟏 𝟎

𝟎𝟎 =

𝟏𝟎𝟎𝟎

𝟎𝟏 =

𝟎𝟏𝟎𝟎

𝟏𝟎 =

𝟎𝟎𝟏𝟎

𝟏𝟏 =

𝟎𝟎𝟎𝟏

|𝐴⟩

|𝐵⟩

|𝐴⟩

𝐵′ = |𝐴⟩ ⊕ |𝐵⟩

|𝐴⟩

|𝐵⟩

|𝐵⟩

|𝐴⟩

𝐴 + 𝐵 𝑚𝑜𝑑2


Universal Quantum Logic Gates

• classical computation: any logical function can be constructed by a universal set of elementary gates:

NAND FANOUT

AA

A

• quantum computation:any logical function can be decomposedinto one-qubit and two-qubit CNOT gates

𝑯,𝑹𝝓, … CNOT


Gate Errors

• quantum computation is performed by a sequence of quantum gates applied to someinitial state |Ψ0⟩:

𝚿𝒏 =

𝒊=𝟏

𝒏

𝑼𝒊|𝚿𝟎⟩

• the unitary operations form a continuous set and any realistic implementation involvessome error (operator 𝑉𝑖 slightly differing from perfect 𝑈𝑖):

𝚿𝒊 = 𝑼𝒊|𝚿𝒊−𝟏⟩

𝚿𝒊 = 𝚿𝒊 + 𝑬𝒊 = 𝑽𝒊|𝚿𝒊−𝟏⟩𝑬𝒊 = (𝑽𝒊−𝑼𝒊) |𝚿𝒊−𝟏⟩error

• after 𝑛 iterations:

𝚿𝒏 − 𝚿𝒏 < 𝒏 𝝐 𝑽𝒊 − 𝑼𝒊 𝒔𝒖𝒑 < 𝝐 (sup norm of operator 𝑉𝑖 − 𝑈𝑖 )

unitary errors accumulate at worst linear with length of computation this takes place for systematic errors, for stochastic errors we expect a 𝑛 growth


Quantum Decoherence



𝚯

𝟐𝟏

• qubits are coherent superpositions of two computational basis states:

quantum decoherence is the loss of coherence or ordering of the phase angles between the components in the quantum superposition due to interaction with the environment (unobservable quantum degrees of freedom)

example: two wave packets interfere to form

interference fringes (left pattern) interaction with the fluctuating environment

(wavy orange lines) blurs the interference pattern (right pattern)

decoherence produces a gradual crossover between wave-like phenomena (interference) and particle-like behavior (classical, localized particles with well-defined trajectories)© Uni Erlangen


Quantum Decoherence

• quantum decoherence occurs when a quantum system interacts with its environment in a thermodynamically irreversible way

entanglement with the environment due to finite coupling

• quantum decoherence can be viewed as the loss of information from a quantum system into the environment

the dynamics of the isolated quantum system is non-unitary, although the combined system plus environment evolves in a unitary fashion

the dynamics of the quantum system alone is irreversible

• quantum decoherence represents a challenge for quantum computers, since they rely heavily on the undisturbed evolution of quantum states

decoherence has to be managed, in order to perform quantum computation


A Few Words on Nomenclature • quantum decoherence:

originates from the quantum effect of the environment, entanglement with the environment

• dephasing: describes the effect that coherences, i.e. the off-diagonal elements of the density

matrix, get reduced in a particular basis, namely the energy eigenbasis of the system dephasing may be reversible if it is not due to decoherence, as revealed, e.g. in spin-

echo experiments.

• phase averaging: a classical noise phenomenon entering through the dependence of the unitary

system evolution on external control parameters which fluctuate examples: (i) vibrations of an interferometer grating, (ii) fluctuations of the classical

magnetic field empirically, phase averaging is often hard to distinguish from decoherence

• dissipation: energy exchange with the environment leading to thermalization usually accompanied by decoherence


J.-S. Tsai, Proc. Jpn. Acad., Ser. B 86 (2010)

Quantum Processor at Workalgorithm


• important difference to classical computation:

the algorithm performed by the quantum computer may be a probabilistic algorithm if we run exactly the same program twice we obtain different results

because of the randomness of the quantum measurement process

the quantum algorithm actually generates a probability distribution of possible outputs

Quantum Algorithm

• quantum algorithm:

sequence of simple elementary logic gate operations

• example:

in fact, Shor s factoring algorithm is not guaranteed to succeed in finding the prime factors – it just succeeds with a reasonable probability

that’s okay though because it is easy to verify whether the factors are correct


A Quantum Computation

E. Knill, Nature 463, 441 (2010)

a. initialize qubits in state |00⟩

b. generate superposition state Ψ = 𝑎 00 + 𝑎 01 + 𝑎 10 + 𝑎|11⟩ with 𝑎 = 0.25

c. apply quantum algorithm making use of quantum parallelism

d. exploit interference to concentrate amplitudes on the marked configuration

e. perform quantum measurement(in the shown case the outcome is deterministic and reveals the location of the mark at 10 )


Contents







• summary


• One- and two-qubit gates: techniques to precisely manipulate and control the qubit state without introducingunwanted computational errors

What do we need to build a quantum computer?

• Qubits: physical medium (hardware platform) that can support quantum systems with twodistinguishable states

• Coherence: adequate isolation of the qubits from the environment to avoid decoherence of quantumstates

• Quantum error correction: method to correct for unavoidable computational errors

• Readout process: fast single-shot readout process with high fidelity

• Quantum algorithm: suitable sequence of simple elementary logic gate operations (software)


Hardware Platforms

• trapped ions (may soon be used for quantum simulation)

• atoms in cavities: cavity & circuit QED systems

• superconducting quantum circuits (current runner up)

• nitrogen vacancies in diamond

• nuclear & electron spins

• optical systems: photons

• mechanical systems: phonons

• electrons on superfluid helium

• ........

..... there are many physical systems participating in the game

..... experimental techniques for manipulation and control areoften demanding


The Nobel Prize in Physics 2012

David J. WinelandSerge Haroche

The Nobel Prize in Physics 2012 was awarded jointly to Serge Haroche and David J. Wineland "for ground-breaking experimental methods that enable measuring and manipulation of individual quantum systems"


study of light – matter interaction on a fundamental quantum level

Light-Matter Interaction

• consequences of the quantum nature of light on light-matter interaction

• quantum mechanical control and manipulation of light and matter

basis for quantum information technology & metrology

Cavity QED


e.g. Kimble and Mabuchi groups at CaltechRempe group at MPQ Garching, ….

cavity QED natural atom in optical cavity

Cavity & Circuit QED

Rempe group

circuit QED solid state circuit = “artificial atom”

in µ-wave cavity

e.g. Wallraff (ETH), Martinis (UCSB), Schoelkopf (Yale), Nakamura (Tokyo), Gross (Garching), ….

WMI

advantages of solid state systems: - design flexibility- tunability & manipulation- strong & ultrastrong light-matter interaction achievable- scalability


multi


devices

single/few


devices

quantum


devices

today near future far future



quantumdescription

65 nm process 2005 superconducting qubitsingle electron transistor

PTB


Intel WMI2 µm


... Towards Quantum Electronics

Kondensator

Spule

Widerstand

Diode

- superposition states- entanglement

quantum electronic circuits

superconductingflux quantum bit

conventional electronic circuits

(superposition of clockwise and anticlockwise circulating persistent currents)


Superconducting Quantum Switch

M. Mariantoni et al. Phys. Rev. B 78, 104508 (2008)A. Baust et al., Phys. Rev. B 91, 014515 (2015); arXiv:1412.7372


Circuit-QED: Energy Scales

resonator solid state atom

wr wge

𝝎𝒓/𝟐𝝅 ≈ 𝝎𝐠𝐞/𝟐𝝅 ≈ 𝟏𝟎𝟗 − 𝟏𝟎𝟏𝟎 Hz

1 GHz ≈ 50 mK

ℏ𝝎𝒓 ≈ 𝟏𝟎−𝟐𝟒 J

ultra-low temperature experiments

ultra-sensitive µ-wave experiments

superconducting circuit QED

𝝎𝒓/𝟐𝝅 ≈ 𝟏𝟎𝟔 − 𝟏𝟎𝟗 Hznanomechanical systems


mK Technology for SC Quantum Circuits


1 GHz ~ 50 mK

ħωr ~ 10-24 J

Optical “table” @ mK temperature


One- and Two-Qubit Gates

• have sufficiently accurate quantum gates been demonstrated?

no, and this is one of the key as-yet-unmet challenges

• present consensus/believe:

for practical scalability, the probability of error introducedby the application of quantum gates must be less than 10-4

requirements for qubit-state initialization andmeasurement are more relaxed: 10-2


Moore‘s Law for Qubit Lifetime

M. H. Devoret and R. J. Schoelkopf, Science 339, 1169 (2013)

superconductingqubits

coherence time ofsuperconducting qubits

has been improveddramatically within only

a decade


Quantum Error Correction

• does the analog nature of configuration amplitudes (as opposed to classical digital computers) cause problems?

no

• do the quantum gates have to be increasingly accurate as the number of gates isgrowing?

no

• why?

it is possible to digitize computations arbitrarily accuratelyby applying quantum error correction strategies

error correction removes effects of computational errors anddecoherence processes

this requires relatively limited resources, provided that enoughrequirements for building a quantum computer are met



Martinis group @ UCSB and Google, superconducting quantum circuit with five Xmon qubits

..... towards superconducting quantum circuits, computers, simulators, ....

Photo credit: Erik Lucero


D-Wave One™ Systems



IBM three-qubit chip:basis for a much larger quantum computer

Photo: IBM


transmon qubit


State preservation by repetitive error detection in a superconducting quantum circuit,J. Kelly et al., Nature 519, 66-69 (2015)

Superconducting Quantum Circuit

UCSB&

chip with9 X-mon qubits


Quantum Information Processing – for what?

Optimization problems

Graph theory problems

Material science

Pharmaceuticals

Quantum chemistry

Climate modeling

Bioinformatics

Weather predictions

Risk modeling

Trading strategies

Financial forecasting

Image and pattern recognition

Machine Learning

Communication

Advanced Search

Research

Web

FinancesGoogle, IBM, D-Wave, Microsoft,

Lockheed Martin, NASA, ....

Credit: iStockPhoto


..... are very likely to be wrong !!

“I think there is a world market for maybe five computers.”

Thomas J. Watson, chairman of IBM, 1943

“Whereas a calculator on the Eniac is equipped with 18000 vacuum tubesand weighs 30 tons, computers in the future may have only 1000 tubesand weigh only 1½ tons“

Popular Mechanics, March 1949

“There is no reason anyone would want a computer in their home.”

Ken Olson, president, chairman and founder of DEC, 1977

long term predictions .....

QIP – Perspectives

When will we have a quantum computers, when will they outperform classicalcomputers, which will be the best hardware platform, .... ??


Summary

rapid progress in quantum informationtechnology

quantum computation is close to become reality

quantum communication is at the horizon

quantum simulation and quantum metrology are attracting growinginterest

quantum technology important for fundamental physics experiments

...the future looks bright


The WMI team

Thank you !

playing quantum games with superconducting circuits quantum computing zwiesel... · 39. edgar...

Documents