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
280
Teaching Machine Learning
pmigdal
7
1.6k
A game needs to framework
pmigdal
1
200
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
680
Gry naukowe, moja gra kwantowa
pmigdal
0
230
Other Decks in Science
See All in Science
データベース08: 実体関連モデルとは?
trycycle
PRO
0
700
データベース04: SQL (1/3) 単純質問 & 集約演算
trycycle
PRO
0
870
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
510
地表面抽出の方法であるSMRFについて紹介
kentaitakura
1
750
MoveItを使った産業用ロボット向け動作作成方法の紹介 / Introduction to creating motion for industrial robots using MoveIt
ry0_ka
0
500
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
180
機械学習 - 決定木からはじめる機械学習
trycycle
PRO
0
990
学術講演会中央大学学員会府中支部
tagtag
0
270
データベース09: 実体関連モデル上の一貫性制約
trycycle
PRO
0
710
統計学入門講座 第2回スライド
techmathproject
0
140
生成AIと学ぶPythonデータ分析再入門-Pythonによるクラスタリング・可視化をサクサク実施-
datascientistsociety
PRO
4
1.6k
07_浮世満理子_アイディア高等学院学院長_一般社団法人全国心理業連合会代表理事_紹介資料.pdf
sip3ristex
0
490
Featured
See All Featured
Testing 201, or: Great Expectations
jmmastey
42
7.6k
Docker and Python
trallard
44
3.5k
JavaScript: Past, Present, and Future - NDC Porto 2020
reverentgeek
48
5.4k
Evolution of real-time – Irina Nazarova, EuRuKo, 2024
irinanazarova
8
810
Templates, Plugins, & Blocks: Oh My! Creating the theme that thinks of everything
marktimemedia
31
2.4k
Creating an realtime collaboration tool: Agile Flush - .NET Oxford
marcduiker
30
2.1k
Put a Button on it: Removing Barriers to Going Fast.
kastner
60
3.9k
[Rails World 2023 - Day 1 Closing Keynote] - The Magic of Rails
eileencodes
35
2.4k
The Myth of the Modular Monolith - Day 2 Keynote - Rails World 2024
eileencodes
26
2.9k
The Success of Rails: Ensuring Growth for the Next 100 Years
eileencodes
45
7.5k
GraphQLとの向き合い方2022年版
quramy
49
14k
How to Create Impact in a Changing Tech Landscape [PerfNow 2023]
tammyeverts
53
2.9k
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