playing quantum games with superconducting circuits quantum computing zwiesel... · 39. edgar...
TRANSCRIPT
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
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
25.04.2015/RG - 3www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
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
25.04.2015/RG - 4www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
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
25.04.2015/RG - 5www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
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
25.04.2015/RG - 6www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
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
25.04.2015/RG - 7www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
First fully automatic, digital computer
Replica of Zuse's Z3 (German Science Museum, Munich)
25.04.2015/RG - 8www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
John von Neumann was proposing the EDVAC computerin 1945
he was introducing the conceptof a computer that is controlledby a
stored program
Programmable machines
25.04.2015/RG - 9www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
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
25.04.2015/RG - 10www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
Semiconductor Integrated Circuits
Intel 2nd generation Core i7 chip: 3.4 GHz, 32nm process technology (1.4 Mio. transistors)
25.04.2015/RG - 11www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
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
25.04.2015/RG - 12www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
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
25.04.2015/RG - 13www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
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
25.04.2015/RG - 14www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
Superconducting Quantum Computer
Vesuvius 3, 512 qubits,operated at T = 30 mK
25.04.2015/RG - 16www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
multi
electron, spin, fluxon, photon
devices
single/few
electron, spin, fluxon, photon
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• …
25.04.2015/RG - 17www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
multi
electron, spin, fluxon, photon
devices
single/few
electron, spin, fluxon, photon
devices
quantum
electron, spin, fluxon, photon
devices
today near future far future
quantifiable,but not quantum
classicaldescription
quantumdescription
65 nm process 2005 superconducting qubitsingle electron transistor
PTB
... Solid State Circuits Go Quantum
Intel
• superposition of states• entanglement• quantized em-fields
WMI2 µm
25.04.2015/RG - 18www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
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
25.04.2015/RG - 19www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
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
25.04.2015/RG - 20www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
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)
25.04.2015/RG - 21www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
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
25.04.2015/RG - 23www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
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
#
25.04.2015/RG - 24www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
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):
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
25.04.2015/RG - 25www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
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):
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
exponential speed-up due to quantum algorthm (Shor)
Integer Factorization 989
25.04.2015/RG - 26www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
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
25.04.2015/RG - 28www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
989 = ??
Integer Factorization
25.04.2015/RG - 29www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
989 = 23 ∙ 43
Integer Factorization
25.04.2015/RG - 30www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
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
25.04.2015/RG - 31www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
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
25.04.2015/RG - 32www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
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)
25.04.2015/RG - 33www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
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)
25.04.2015/RG - 34www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
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
25.04.2015/RG - 35www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
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
25.04.2015/RG - 36www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
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
25.04.2015/RG - 37www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
• 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
25.04.2015/RG - 38www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
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
25.04.2015/RG - 39www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
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
25.04.2015/RG - 40www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
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
25.04.2015/RG - 41www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
»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«
25.04.2015/RG - 42www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
»Quantum«
25.04.2015/RG - 45www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
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
25.04.2015/RG - 46www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
• 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
25.04.2015/RG - 47www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
Weird Quantum World
25.04.2015/RG - 48www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
Weird Quantum World
25.04.2015/RG - 49www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
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
25.04.2015/RG - 50www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
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)
25.04.2015/RG - 51www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
𝚿 = − | ⟩
A B
A B A B
Entangled Dices
25.04.2015/RG - 52www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
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:
25.04.2015/RG - 53www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
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
25.04.2015/RG - 54www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
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
25.04.2015/RG - 55www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
• 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
25.04.2015/RG - 56www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
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 ?
25.04.2015/RG - 57www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
Will quantum effects be part of our everyday life?
Yes, they already are !
interesting applications ⇒ quantum technologies
• quantum computation
• quantum communication
• quantum simulation
• quantum metrology
• ............
Relevance of Quantum Phenomena
25.04.2015/RG - 58www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
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
25.04.2015/RG - 59www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
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
25.04.2015/RG - 60www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
The Quantum Bit
• Bloch sphere representation (geometrical picture of qubit)
𝚿 = 𝐜𝐨𝐬𝚯
𝟐𝟎 + 𝒆𝒊𝝓 𝐬𝐢𝐧
𝚯
𝟐𝟏
=𝐜𝐨𝐬
𝚯
𝟐
𝒆𝒊𝝓𝐬𝐢𝐧𝚯
𝟐
𝟎 or 𝒈
𝟏 or 𝒆
ground state
excited state
25.04.2015/RG - 61www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
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
25.04.2015/RG - 62www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
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
25.04.2015/RG - 63www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
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⟩
25.04.2015/RG - 64www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
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
25.04.2015/RG - 65www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
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
25.04.2015/RG - 66www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
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
25.04.2015/RG - 67www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
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
25.04.2015/RG - 68www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
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
25.04.2015/RG - 69www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
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
25.04.2015/RG - 70www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
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
25.04.2015/RG - 72www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
Bloch sphere
𝚿 = 𝐜𝐨𝐬𝚯
𝟐𝟎 + 𝒆𝒊𝝓 𝐬𝐢𝐧
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)
25.04.2015/RG - 73www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
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)
25.04.2015/RG - 74www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
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
25.04.2015/RG - 75www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
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
25.04.2015/RG - 76www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
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
25.04.2015/RG - 77www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
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
25.04.2015/RG - 78www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
Quantum Decoherence
𝚿 = 𝐜𝐨𝐬𝚯
𝟐𝟎 + 𝒆𝒊𝝓 𝐬𝐢𝐧
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
25.04.2015/RG - 79www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
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
25.04.2015/RG - 80www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
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
25.04.2015/RG - 81www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
J.-S. Tsai, Proc. Jpn. Acad., Ser. B 86 (2010)
Quantum Processor at Workalgorithm
25.04.2015/RG - 82www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
• 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
25.04.2015/RG - 83www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
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 )
25.04.2015/RG - 84www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
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
25.04.2015/RG - 85www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
• 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)
25.04.2015/RG - 86www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
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
25.04.2015/RG - 87www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
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"
25.04.2015/RG - 88www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
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
25.04.2015/RG - 89www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
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
25.04.2015/RG - 90www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
multi
electron, spin, fluxon, photon
devices
single/few
electron, spin, fluxon, photon
devices
quantum
electron, spin, fluxon, photon
devices
today near future far future
quantifiable,but not quantum
classicaldescription
quantumdescription
65 nm process 2005 superconducting qubitsingle electron transistor
PTB
... Solid State Circuits Go Quantum
Intel WMI2 µm
25.04.2015/RG - 92www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
... 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)
25.04.2015/RG - 97www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
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
25.04.2015/RG - 99www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
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
25.04.2015/RG - 100www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
mK Technology for SC Quantum Circuits
25.04.2015/RG - 101www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
1 GHz ~ 50 mK
ħωr ~ 10-24 J
Optical “table” @ mK temperature
25.04.2015/RG - 102www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
1 GHz ~ 50 mK
ħωr ~ 10-24 J
Optical “table” @ mK temperature
25.04.2015/RG - 103www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
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
25.04.2015/RG - 104www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
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
25.04.2015/RG - 105www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
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
25.04.2015/RG - 106www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
Superconducting Quantum Computer
Martinis group @ UCSB and Google, superconducting quantum circuit with five Xmon qubits
..... towards superconducting quantum circuits, computers, simulators, ....
Photo credit: Erik Lucero
25.04.2015/RG - 107www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
Superconducting Quantum Computer
25.04.2015/RG - 109www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
D-Wave One™ Systems
Superconducting Quantum Computer
25.04.2015/RG - 110www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
IBM three-qubit chip:basis for a much larger quantum computer
Photo: IBM
Superconducting Quantum Computer
transmon qubit
25.04.2015/RG - 111www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
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
25.04.2015/RG - 112www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
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
25.04.2015/RG - 113www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
..... 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, .... ??
25.04.2015/RG - 114www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
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
25.04.2015/RG - 115www.wmi.badw.de 39. Edgar Lüscher-Seminar, Gymnasium Zwiesel
The WMI team
Thank you !