Upgrade to Pro
— share decks privately, control downloads, hide ads and more …
Speaker Deck
Features
Speaker Deck
PRO
Sign in
Sign up for free
Search
Search
Qubism: self-similar visualization of a many-bo...
Search
Piotr Migdał
January 10, 2013
Science
1
280
Qubism: self-similar visualization of a many-body wavefunction
Article, code and more:
http://qubism.wikidot.com/
Piotr Migdał
January 10, 2013
Tweet
Share
More Decks by Piotr Migdał
See All by Piotr Migdał
Detecting trypophobia triggers (with deep learning)
pmigdal
1
170
Teaching Machine Learning
pmigdal
7
1.5k
A game needs to framework
pmigdal
1
140
Visualizing word coincidences
pmigdal
1
65
Dreams, Drugs and ConvNets
pmigdal
1
810
{Machine, Deep} Learning for software engineers
pmigdal
1
2k
Lightning talk - Teaching machine learning
pmigdal
0
1.7k
Interaktywna wizualizacja danych w d3.js
pmigdal
2
610
Gry naukowe, moja gra kwantowa
pmigdal
0
210
Other Decks in Science
See All in Science
プロダクト開発を通して学んだナレッジマネジメントの哲学
sonod
0
130
Machine Learning for Materials (Lecture 7)
aronwalsh
0
800
ベイズ最適化をゼロから
brainpadpr
2
580
深層学習を利用して 大豆の外部欠陥を判別した研究事例の紹介
kentaitakura
0
190
構造設計のための3D生成AI-最新の取り組みと今後の展開-
kojinishiguchi
0
420
解説!データ基盤の進化を後押しする手順とタイミング
shomaekawa
1
310
学術講演会中央大学学員会八王子支部
tagtag
0
210
HAS Dark Site Orientation
astronomyhouston
0
5.2k
Raccoon Roundworm
uni_of_nomi
0
140
最新のAI技術を使った材料シミュレーションで材料研究現場に変革を
matlantis
0
720
ultraArmをモニター提供してもらった話
miura55
0
170
General Parasitology
uni_of_nomi
0
110
Featured
See All Featured
How to name files
jennybc
75
98k
Debugging Ruby Performance
tmm1
72
12k
"I'm Feeling Lucky" - Building Great Search Experiences for Today's Users (#IAC19)
danielanewman
225
22k
Practical Orchestrator
shlominoach
185
10k
What's in a price? How to price your products and services
michaelherold
242
11k
Designing on Purpose - Digital PM Summit 2013
jponch
114
6.8k
No one is an island. Learnings from fostering a developers community.
thoeni
18
2.9k
Imperfection Machines: The Place of Print at Facebook
scottboms
263
13k
How to Think Like a Performance Engineer
csswizardry
16
960
Web development in the modern age
philhawksworth
205
10k
Git: the NoSQL Database
bkeepers
PRO
425
64k
Facilitating Awesome Meetings
lara
49
5.9k
Transcript
self-similar visualization of many-body wavefunctions QUBISM: presented by: Piotr Migdał
(ICFO, Barcelona)
Don’t take plots for granted!
None
None
bar chart - William Playfair (1786) scatter plot - Francis
Galton (a century later)
Dmitri Mendeleev | Periodic Table of Elements (1869) periodic table
- Dimitri Mendeleev (1869)
Back to the quantum world
↵|"i + |#i
↵|"i + |#i ⇠ = ↵| i + |•i
↵|"i + |#i ⇠ = ↵| i + |•i ⇠
= ↵|0i + |1i
↵|"i + |#i ⇠ = ↵| i + |•i ⇠
= ↵|0i + |1i ↵00 |00i + ↵01 |01i + ↵10 |10i + ↵11 |11i
↵|"i + |#i ⇠ = ↵| i + |•i ⇠
= ↵|0i + |1i ↵00 |00i + ↵01 |01i + ↵10 |10i + ↵11 |11i ↵000 |000i + ↵001 |001i + ↵010 |010i + ↵011 |011i + ↵100 |100i + ↵101 |101i + ↵110 |110i + ↵111 |111i
↵|"i + |#i ⇠ = ↵| i + |•i ⇠
= ↵|0i + |1i 2n complex parameters ↵00 |00i + ↵01 |01i + ↵10 |10i + ↵11 |11i ↵000 |000i + ↵001 |001i + ↵010 |010i + ↵011 |011i + ↵100 |100i + ↵101 |101i + ↵110 |110i + ↵111 |111i
None
None
00 01 10 11
00 01 10 11 00 01 00 01 10 11
10 11 00 01 00 01 10 11 10 11
00 01 10 11 00 01 10 11 00 01
10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 00 01 10 11 10 11 00 01 00 01 10 11 10 11
00 01 10 11 00 01 10 11 00 01
10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 00 01 10 11 10 11 00 01 00 01 10 11 10 11 |101000i
00 01 10 11 00 01 10 11 00 01
10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 00 01 10 11 10 11 00 01 00 01 10 11 10 11 |101000i
00 01 10 11 00 01 10 11 00 01
10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 00 01 10 11 10 11 00 01 00 01 10 11 10 11 |101000i
00 01 10 11 00 01 10 11 00 01
10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 00 01 10 11 10 11 00 01 00 01 10 11 10 11 |101000i
00 01 10 11 00 01 10 11 00 01
10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 00 01 10 11 10 11 00 01 00 01 10 11 10 11 |101000i
00 01 10 11 00 01 10 11 00 01
10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 00 01 10 11 10 11 00 01 00 01 10 11 10 11 FM: 000000... FM: 111111...
00 01 10 11 00 01 10 11 00 01
10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 00 01 00 01 10 11 10 11 00 01 00 01 10 11 10 11 FM: 000000... FM: 111111... AFM: 010101... AFM: 101010...
Examples
Dicke state |01i + |10i p 2
Dicke state |01i + |10i p 2 00 10 01
11
Dicke state (|0011i + |0101i +|0110i + |1001i +|1010i +
|1100i) / p 6
Dicke state particles zeros ones 6 3 3
Dicke state particles zeros ones 8 4 4
Dicke state particles zeros ones 10 5 5
Dicke state particles zeros ones 12 6 6
Dicke state particles zeros ones 14 7 7
Product state (↵|0i + |1i)n
Heisenberg AFM X ~ Si · ~ Si+1 (periodic boundary
cond.)
Heisenberg AFM (1,2) (3,4) (5,6) (7,8) ... X ~ Si
· ~ Si+1 (periodic boundary cond.)
Heisenberg AFM (1,2) (3,4) (5,6) (7,8) ... (n,1) (2,3) (4,5)
(6,7) ... X ~ Si · ~ Si+1 (periodic boundary cond.)
Heisenberg AFM (1,2) (3,4) (5,6) (7,8) ... (n,1) (2,3) (4,5)
(6,7) ... X ~ Si · ~ Si+1 (open boundary cond.)
It works for any qudit 1D spin chains
-- -0 -+ 0- 00 0+ +- +0 ++ +
qutrits (spin-1) 0 -
AKLT state Affleck, Lieb, Kennedy and Tasaki (| +i +
|00i + | + i)/ p 3 + 1 3 ⇣ ~ Si · ~ Si+1 ⌘2 X ~ Si · ~ Si+1
AKLT state particles 4 Affleck, Lieb, Kennedy and Tasaki +
1 3 ⇣ ~ Si · ~ Si+1 ⌘2 X ~ Si · ~ Si+1
AKLT state Affleck, Lieb, Kennedy and Tasaki + 1 3
⇣ ~ Si · ~ Si+1 ⌘2 X ~ Si · ~ Si+1 particles 6
AKLT state Affleck, Lieb, Kennedy and Tasaki + 1 3
⇣ ~ Si · ~ Si+1 ⌘2 X ~ Si · ~ Si+1 particles 8
AKLT state Affleck, Lieb, Kennedy and Tasaki + 1 3
⇣ ~ Si · ~ Si+1 ⌘2 X ~ Si · ~ Si+1 particles 10
Alternative qubistic schemes
00 01 11 10 anti-ferromagnetic ferromagnetic
Heisenberg AFM X ~ Si · ~ Si+1
X z i z i+1 X x i Ising transverse
field
X z i z i+1 X x i Ising transverse
field = 1
X z i z i+1 X x i Ising transverse
field
X z i z i+1 X x i Ising transverse
field = 1
None
Product state
Product state Dicke half-filled
Product state Dicke half-filled Ising transverse field (ground state)
Product state Dicke half-filled Ising transverse field (ground state) Heisenberg
(ground state)
You can see entanglement
entanglement: (1,2) vs (3,4,5,6,7,8,9,...)
entanglement: (1,2) vs (3,4,5,6,7,8,9,...) Schmidt rank: A A A A
1 (not entangled)
entanglement: (1,2) vs (3,4,5,6,7,8,9,...) Schmidt rank: A A A A
1 (not entangled) A B B C 3 (entangled!)
entanglement: (1,2,3,4) vs (5,6,7,8,9,...) Schmidt rank:
entanglement: (1,2,3,4) vs (5,6,7,8,9,...) Schmidt rank: A A A A
A A A A A A A A A A A A 1 (not entangled)
entanglement: (1,2,3,4) vs (5,6,7,8,9,...) Schmidt rank: A A A A
A A A A A A A A A A A A 1 (not entangled) A
A B B B B entanglement: (1,2,3,4) vs (5,6,7,8,9,...) Schmidt
rank: A A A A A A A A A A A A A A A A 1 (not entangled) A
A B B B B entanglement: (1,2,3,4) vs (5,6,7,8,9,...) Schmidt
rank: A A A A A A A A A A A A A A A A 1 (not entangled) A B B C B C C B C C C A
A B B B B entanglement: (1,2,3,4) vs (5,6,7,8,9,...) Schmidt
rank: A A A A A A A A A A A A A A A A 1 (not entangled) A B B C B C C D B C C D C D D A B B C B C C B C C C A
A B B B B entanglement: (1,2,3,4) vs (5,6,7,8,9,...) Schmidt
rank: A A A A A A A A A A A A A A A A 1 (not entangled) A B B C B C C D B C C D C D D A B B C B C C B C C C A 5 (entangled!) A B B C B C C D B C C D C D D E
{|0i, |1i}⌦4 {|+i, | i}⌦4 ⌦4 x ⌦4 z Schmidt
number: 1 2 2 3 4 |0000i |GHZi |Wi Dicke half-filling
Renyi fractal dimension (and box counting)
AKLT ground state also works for qutrits (e.g. spin-1) log(4)
log(3) ⇡ 1 . 26 and its fractal dimension
0 0.5 1 1.5 2 0 0.2 0.4 0.6 0.8
1 1.2 1.4 1.6 dq arctan(K) q=0 q=0.5 q=1 q=2 q =104 X ( i ) z ( i +1) z ( i ) x Ising transverse field surface-like line-like point-like
0 0.5 1 1.5 2 0 0.2 0.4 0.6 0.8
1 1.2 1.4 1.6 dq arctan(K) q=0 q=0.5 q=1 q=2 q =104 X ( i ) z ( i +1) z ( i ) x Ising transverse field = 1 surface-like line-like point-like
And how about going the other way?
Jose I. Latorre, arXiv:quant-ph/0510031 (2005) QPEG! matrix product states for
image compression JPEG?
Javier Rodriguez-Laguna Piotr Migdał Miguel Ibanez Berganza Maciej Lewenstein German
Sierra
http://qubism.wikidot.com/ Thanks! paper, code, etc: J.Rodriguez-Laguna, P. Migdał, M. Ibánez
Berganza, M. Lewenstein and G. Sierra. Qubism: self-similar visualization of many-body wavefunctions. New J. Phys. 14, 053028 (2012), arXiv:1112.3560.
None