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
390
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
290
Teaching Machine Learning
pmigdal
7
1.6k
A game needs to framework
pmigdal
1
210
Visualizing word coincidences
pmigdal
1
75
Dreams, Drugs and ConvNets
pmigdal
1
910
{Machine, Deep} Learning for software engineers
pmigdal
1
2.1k
Lightning talk - Teaching machine learning
pmigdal
0
1.7k
Interaktywna wizualizacja danych w d3.js
pmigdal
2
700
Gry naukowe, moja gra kwantowa
pmigdal
0
230
Other Decks in Science
See All in Science
データベース14: B+木 & ハッシュ索引
trycycle
PRO
0
470
KH Coderチュートリアル(スライド版)
koichih
1
47k
06_浅井雄一郎_株式会社浅井農園代表取締役社長_紹介資料.pdf
sip3ristex
0
650
02_西村訓弘_プログラムディレクター_人口減少を機にひらく未来社会.pdf
sip3ristex
0
630
03_草原和博_広島大学大学院人間社会科学研究科教授_デジタル_シティズンシップシティで_新たな_学び__をつくる.pdf
sip3ristex
0
620
Gemini Prompt Engineering: Practical Techniques for Tangible AI Outcomes
mfonobong
2
170
データベース12: 正規化(2/2) - データ従属性に基づく正規化
trycycle
PRO
0
980
データベース05: SQL(2/3) 結合質問
trycycle
PRO
0
810
安心・効率的な医療現場の実現へ ~オンプレAI & ノーコードワークフローで進める業務改革~
siyoo
0
350
論文紹介 音源分離:SCNET SPARSE COMPRESSION NETWORK FOR MUSIC SOURCE SEPARATION
kenmatsu4
0
330
My Little Monster
juzishuu
0
110
baseballrによるMLBデータの抽出と階層ベイズモデルによる打率の推定 / TokyoR118
dropout009
2
570
Featured
See All Featured
Intergalactic Javascript Robots from Outer Space
tanoku
273
27k
Cheating the UX When There Is Nothing More to Optimize - PixelPioneers
stephaniewalter
285
14k
Side Projects
sachag
455
43k
Site-Speed That Sticks
csswizardry
11
880
Performance Is Good for Brains [We Love Speed 2024]
tammyeverts
12
1.1k
Building an army of robots
kneath
306
46k
Building Adaptive Systems
keathley
43
2.8k
How to train your dragon (web standard)
notwaldorf
96
6.3k
[RailsConf 2023 Opening Keynote] The Magic of Rails
eileencodes
31
9.7k
Making the Leap to Tech Lead
cromwellryan
135
9.5k
KATA
mclloyd
32
15k
Code Review Best Practice
trishagee
72
19k
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