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
370
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
270
Teaching Machine Learning
pmigdal
7
1.6k
A game needs to framework
pmigdal
1
190
Visualizing word coincidences
pmigdal
1
73
Dreams, Drugs and ConvNets
pmigdal
1
890
{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
670
Gry naukowe, moja gra kwantowa
pmigdal
0
230
Other Decks in Science
See All in Science
Introd_Img_Process_2_Frequ
hachama
0
560
モンテカルロDCF法による事業価値の算出(モンテカルロ法とベイズモデリング) / Business Valuation Using Monte Carlo DCF Method (Monte Carlo Simulation and Bayesian Modeling)
ikuma_w
0
170
Symfony Console Facelift
chalasr
2
450
KH Coderチュートリアル(スライド版)
koichih
1
41k
機械学習 - ニューラルネットワーク入門
trycycle
PRO
0
800
Hakonwa-Quaternion
hiranabe
1
110
白金鉱業Meetup Vol.16_数理最適化案件のはじめかた・すすめかた
brainpadpr
3
1.8k
MCMCのR-hatは分散分析である
moricup
0
370
サイゼミ用因果推論
lw
1
7.3k
統計学入門講座 第2回スライド
techmathproject
0
140
Agent開発フレームワークのOverviewとW&B Weaveとのインテグレーション
siyoo
0
270
CV_3_Keypoints
hachama
0
190
Featured
See All Featured
Exploring the Power of Turbo Streams & Action Cable | RailsConf2023
kevinliebholz
34
5.9k
Reflections from 52 weeks, 52 projects
jeffersonlam
351
20k
A better future with KSS
kneath
239
17k
VelocityConf: Rendering Performance Case Studies
addyosmani
331
24k
Raft: Consensus for Rubyists
vanstee
140
7k
Practical Tips for Bootstrapping Information Extraction Pipelines
honnibal
PRO
20
1.3k
CSS Pre-Processors: Stylus, Less & Sass
bermonpainter
357
30k
The Success of Rails: Ensuring Growth for the Next 100 Years
eileencodes
45
7.5k
The Cost Of JavaScript in 2023
addyosmani
51
8.5k
Thoughts on Productivity
jonyablonski
69
4.7k
Easily Structure & Communicate Ideas using Wireframe
afnizarnur
194
16k
Building a Scalable Design System with Sketch
lauravandoore
462
33k
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