quantum computer meets blackhole and quantum...

Quantum Computer Meets

Blackhole and Quantum Chaos

Workshop on OTO correlators

藤井啓祐東京大学大学院工学系研究科付属光量子科学研究センター

JST さきがけ

量子情報とOTO

2008年4月～2011年3月京都大学工学研究科原子核工学専攻博士課程日本学術振興会特別研究員 DC1

2011年4月～2013年3月大阪大学基礎工学研究科物質創成専攻井元研特任研究員

2013年4月～2015年3月京都大学白眉センター (情報学研究科通信情報システム専攻岩間研特定助教)

2015年4月～2016年3月京都大学白眉センター (理学研究科物理学宇宙物理学専攻高橋研特定助教)

2016年４月～東京大学工学系研究科光量子科学研究センター小芦研助教 (物理工学専攻兼担)

2016年10月～ JSTさきがけ「量子の状態制御と機能化」さきがけ研究員兼任

自己紹介：藤井啓祐工学

基礎工学

情報

物理

物理工学

Outline• Quantum information perspective of random unitary:

blackhole as a mirror

• Random unitary and OTOC

• Quantum computational supremacy benchmark on a quantum computer

• How to characterize random unitary operation: quantum chaos

• Toward random unitary with a translation invariant system using universality (simulate random unitary on a quantum simulator)

All these are about complex unitary operations, which is now becoming experimentally accessible!

Blackhole as a mirrorHayden-Preskill ’07

Alice’s information

Initial stateof black hole

Scrambling dynamics

=random unitary

Bob (observer with an unlimited computational power)

Hawking radiation

Where is Alice’s information?

Classical toy modelHayden-Preskill ’07


Initial stateof blackhole

random permutationover 2n n-bit

strings Hawking radiation

00101011

00000000(n-k bit string)

(k bit string)

1101…(s bit string)

Bob wants to decode Alice’s information from HR. How many bits in HR is required to do this task?

→Channel capacity



Initial stateof blackhole

random permutation

over 2n n-bit strings HR

00101011

00000000(n-k bit string)

(k bit string)

1101…(s bit string)

source of information

encoding to an error

correction code

channel output

erased

eraser channel

The random permutation is an encoding to achieve the optimal channel capacity! s=k is enough to decode!


source of information

00000 000…0 00001 000…0 00010 000…0

11111 000…0

…

Alice

{2k

Bob has a table of the random permutation,and decode what’s Alice’s information from HR.

10110 110…1 01001 101…0 01010 100…0

11010 010…1…

HR erased

{s bits

pfail = 2k2�s

probability to have the same bit strings in HR for different Alice’s message.

Total number of Alice’s bit strings

If s=k+c, pfail is sufficiently small.

encoding to an error

correction code

random permutation

over 2n bit strings

Entanglement assisted quantum eraser channel capacity


Initial stateof blackhole R

maximally entangled state

BobE

B’

reference

maximally entangled state

NScrambling dynamics

=random unitary

V B

maximally entangled state?

Alice

black hole R

Bob

MES

E

B’

N

VB

MES

⇢NB(VB) = (IN ⌦ VB)⇢NB(IN ⌦ V †B)

Entanglement assisted quantum eraser channel capacity

⇢NB0(VB) = TrR[⇢NB(VB)]

⇢N (VB) = TrB0 [⇢NB0(VB)]

F (VB) = h�|⇢M̃N |�i = 1� k⇢NB0(VB)� ⇢N (VB)⌦ I 0Bk1

s=k+c qubit is enough to decode Alice’s info.

Decoupling theorem [Hayden et al ‘07]:Z

dVBk⇢NB0(VB)� ⇢N (VB)⌦ IB0k1 |NB||R|2 Tr[⇢2NB ] = 22(k�s)

kAk1 = Tr[pAA†]

2

⇢NB = TrE [⇢NBE ]

Random unitaryExact Haar random unitary: not efficient, exponential time is required

Approximated t-design (Dankert et al., ‘06):X

i

piU⌦ti ⇢(U †

i )⌦t '

Z

Haar

U⌦t⇢(U †)⌦tdU

(using operator 1-norm or diamondnorm as a map)

decoupling theorem & approximated 2-design:X

i

pik⇢NB0(Vi)� ⇢N (Vi)⌦ I 0Bk21 22(k�s) + ✏

approximated 2-design is efficiently achievable(Dankert et al., ’06):

O(log(n) log(1/✏))time O(

pn log(n) log(1/✏))

localitycircuit depth circuit depth

Random unitary & OTOCRandom unitary → delocalization of quantum information:

2-norm of commutator

k[A(0), B(t)]k22kWk2 = (Tr[WW †])1/2

OTOC:

[A,B] = 0 $ hABABi = 1

{A,B} = 0 $ hABABi = �1

hABABi = 1

dTr[ABAB]

OTOC & 2-norm of commutator:

( A2 = B2 = I is assumed, e.g. Pauli operators)

k[A,B]k22 = 2Tr[I �ABAB]= 2d(1� hABABi)

2-design & 4-point OTOCScrambling (unitary 2-design) implies decay of OTOC

Unitary 2-design and Haar random unitary both result in the same 4-point OTOC!

4-point (2k-point) OTOC is a good witness (related to frame potential) of random unitary of 2-design (k-design).(Roberts-Yoshida ’16)

Wswap ⌘X

ij

|ii|jihj|hi|

Tr[AB(t)AB(t)⇢]

= Tr[(A⌦A)(Ut ⌦ Ut)(B ⌦B)(Ut ⌦ Ut)†Wswap]

Tr[AB(t)AB(t)]

Quantum “computational” supremacy

Google’s Milestone Chip Can Achieve Quantum Supremacy By The End Of 2017https://fossbytes.com/google-quantum-chip-50-qubit/

IBM’s First Commercial “Universal Quantum Computer” Can Be The New Future Of AIhttps://fossbytes.com/ibm-q-quantum-computing-50-qubit-computer/

22qubits-> 50qubits 16,17qubits

50qubits, d=250=1015 → O(100 peta bit)

What is quantum computer?“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.” R. Feynman, Simulating Physics with Computers, Int. J. Theor. Phys. 21, 467 (1982).

→quantum computer, quantum simulator

W. S. Bakr et al., Nature 462, 74 (2009)

Kelly et al., Nature 519, 66 (2015)

Is there a quantum machine that can simulate physics universally?

What is quantum computer?

・・・

time evolution

unitary operator (dim = 2n)

measurement・・・ single-qubit

unitary operator

2qubit unitary operator

quantum circuit

|0i|0i|0i

|0i

initial state

・・・{N

qubi

t

qubit

Universal quantum computer|0i|0i|0i

|0i

initial state time evolution 測定

・・・

・・・

unitary operator

・・・{N qu

bit

Is there a quantum machine that can simulate physics universally?

H =1p2

✓1 11 �1

◆

Hadamard

e�i(⇡/8)Z =

✓e�i⇡/8 0

0 ei⇡/8

◆

π/8 (T gate)

UCNOT = |0ih0|⌦ I + |1ih1|⌦X

=

0

BB@

1 0 0 00 1 0 00 0 0 10 0 1 0

1

CCA

CNOT

Universal gate set

universal quantum computer=can implement an arbitrary unitary operator

R. P. Feynman, Quantum Mechanical Computers, Optics News (1985)

How to characterize the quantum machine

Boixo et al., arXiv:1608.00263

random unitary circuit

APS march meeting 20179 qubit, depth 10, error per gate 0.3%,

total fidelity 98%

How can we characterize complex many-body quantum dynamics universally?

Neutron scattering and quantum chaos (random unitary)

resonance level spacing: Wigner-Dyson distributionresonance strength: Porter-Thomas distribution

“Handbook of Nuclear Engineering - Chapter3”Evaluated Nuclear Data

P. Oblozinsky, M. Herman and S.F. Mughabghab

The scattering strength is proportional to which diagonalizes the Hamiltonian H.

|Uij |2



random unitary circuit

}Porter-Thomas distribution

| i =2nX

i=1

ci|ii

p(c1, c2, ..., c2n) / �

1�

X

i

|ci|2!

p(|c|2) / e�|c|2/ ¯|c|2

randomly chosen



probability × dim

prob

abilit

y de

nsity

noise less

uniform distribution PT entropy

42qubit

36qubit

PT entropy

How are OTOC decay and convergence of PT distribution related?

Universal quantum computer

|0i|0i|0i

|0i

initial state time evolution measurement

・・・

・・・unitary operator

・・・{n qu

bit

H1

H2

H1,2

on

off

quantum circuit = spatiotemporal modulationof Hamiltonian

Kelly et al., Nature 519, 66 (2015)

Can we achieve universality with more natural dynamics？

Feynman quantum computer R. Feynman, Quantum mechanical computers, Opt. News, vol. 11, pp. 11–46, 1985

H =TX

t=1

Ut ⌦ |tiht� 1|+ U†t ⌦ |t� 1iht|

working space clock

initial state

working space clock| ini|0i

time evolution TX

t=0

ct(⌧)UtUt�1 · · ·U1| ini|tie�iH⌧

(history state)

Hamiltonian dynamics with time-independent Hamiltonian can implement universal quantum computation!

様々な万能量子計算モデル

量子回路型

万能計算を埋め込むために利用する自由度

時間・空間的にハミルトニアンを変動

Feynman型定常ハミルトニアン

断熱型Aharonov et al, ‘04

(QMA-hardness経由)

定常ハミルトニアン＋断熱操作(横磁場イジングはダメ Bravyi et al ‘06)

オートマトン型定常並進対称ハミルトニアン＋初期状態

[4-local: Feynman’85, 2-local: Nagaj ’10&12]

[5-local: Kitaev ‘02, 3-local: Kempe-Kitaev-Regev’06; 2D 2-local Oliveira-Terhal ’08; 2D fermions: Schuch-Verstraete ’09, 1D 2-local: Aharonov et al ‘09; 2D XY: Cubitt-Montanaro ’13, and many more]

量子ウォーク型定常ハミルトニアン[adjacency matrix: Childs’09,bosons&fermions: Childs-Gosset-Webb’13, and many more]

定常並進対称ハミルトニアン＋時間的に変動 [2D local: Janzig-Wocjan ‘04]

[1D local: Raussendorf ‘05]

Scrambling (random unitary)with translation invariant system

ZZ X Z ZZ X Z

{

…

mea

sure

simul

tane

ously

Hzz

= g

N�1X

i=1

�z

i

�z

i+1, Hz

= hz

NX

i=1

�z

i

, Hx

= hx

NX

i=1

�x

i

{ { { { {

Time intervals are chosen randomly.

Convergence to PT distribution#

of c

ount

s

probability × dim

8 qubit, depth 300

0.01

0.1

1

10

100

0 1 2 3 4 5 6 7 8 9 10

depth=3depth=15depth=30

depth=300

Convergence to PT distribution

depth

entro

py

4 qubit

5 qubit

6 qubit

7 qubit

1.5 2

2.5 3

3.5 4

4.5 5

5.5 6

0 50 100 150 200 250 300

4 qubit5 qubit6 qubit7 qubit8 qubit9 qubit

PT entropy

8 qubit

9 qubit

S = log(2

n)� 1 + �

PT entropy:

Euler’s constant

Convergence of PT entropy and decay of OTO correlator

entro

pyO

TOC

More systematic understanding

would be required!-0.2

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100 120 140

1 2 3 4 5 6 7 8 9

0 20 40 60 80 100 120 140

entropyPT entropy

Higher moment of PT distribution?

9qubit

Sensitivity of PT distribution against decoherence

0.01

0.1

1

10

100

1000

0 1 2 3 4 5 6 7 8 9 10 0.01

0.1

1

10

100

1000

0 1 2 3 4 5 6 7 8 9 10 0.01

0.1

1

10

100

1000

0 1 2 3 4 5 6 7 8 9 10

0.01

0.1

1

10

100

1000

0 1 2 3 4 5 6 7 8 9 10 0.01

0.1

1

10

100

1000

0 1 2 3 4 5 6 7 8 9 10 0.01

0.1

1

10

100

1000

0 1 2 3 4 5 6 7 8 9 10

0.01

0.1

1

10

100

1000

0 1 2 3 4 5 6 7 8 9 10 0.01

0.1

1

10

100

1000

0 1 2 3 4 5 6 7 8 9 10

p=0.001 p=0.002 p=0.005

p=0.01 p=0.02 p=0.03

p=0.05 p=0.1 Depolarizing error with prob. p.

PT distribution rapidly converges to uniform distribution.

Summary & Discussion• Random unitary (universal computation) connects quantum

computer, quantum chaos, and blackhole.

• OTOC is a good witness of “randomness” of unitary, which seems to be sensitive to delocalization.

• PT distribution is directly measurable from the output of random unitary.

• PT distribution is sensitive to decoherence.

• Experimental realization with cold atom quantum simulator.

• Integrable v.s. quantum chaotic system (random free-fermionic dynamics→ random matchgate)?

quantum computer meets blackhole and quantum...

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