Quantum Network Coding Harumichi Nishimura 西村治道 Graduate School of Science, Osaka Prefecture...

Quantum Network Coding

Harumichi Nishimura西村治道

Graduate School of Science, Osaka Prefecture University大阪府立大学理学系研究科

March 30, 2011, Institute of Network Coding

Why Quantum Network Coding?

Power of quantum information processing– Fast algorithms

• Factoring large integers • Database search

– Secure cryptosystems• Unconditionally secure key distribution• Public key cryptosystems

– Communication-efficient multi-party protocols• Communication complexity • Leader election

But quantum channel is "expensive"

Q. Can we reduce the amount of quantum communication using the idea of network coding?

Outline of This Talk

• Basic of quantum information – States– Measurements– Transformations

• Quantum network coding– Negative results– Positive results

Basic of Quantum Information

Mathematical Representation of Quantum Mechanics

Quantum mechanics Representations

quantum state vector

vector space

projection

unitary

state space

measurement

evolution

Qubit

• Bit ：＝ Basic unit of information and computation

• Quantum bit (qubit) ：＝ Basic unit of "quantum" information and computation– allows a superposed state of the 2 basis

states which corresponds "0'' and "1"

|0>+|1>

Implemented by various micro systems:

• Nuclear spin of an atom ("up" or "down")

• Polarization of the light （ "vertical" or "horizontal" ）　

QubitRepresented by a unit vector of the two dim. complex-valued vector space

0

1:0

1

0:1

ket

b

aba :10

(probability) amplitude

ie

basis stateorthogonal

a

b

0

1If a, b are real, a state is represented in 2D plane

If a, b are complex, it

is identified as a vector

on the unit ball, called

"Bloch sphere"

102

1

102

1i

and represent the same state

To Get Classical Information

10 ba Measurement

After the measurement

)(:2

0

2 Ea 0

1

prob.

prob.

The state becomes 　　　

0

1

a

b

Measurement0

a

b

Measurement is represented by a set of projections

The state becomes 　　　

10 ,EE

projection on 0span)(:2

1

2 Eb

Multiple Qubits A m-qubit state is in the 2m dim. vector space that is the tensor product of

m 2-dim. spaces.

0

0

0

1

0

1

0

1:00

1

0

0

1:01

0

1

1

0:10

1

0

1

0:11

11100100

:

dcba

2-qubit state

nx

x x1,0

Measurement

After the measurement 2

xprob. nxxE 1,0

x

The state becomes

x

x

E

Ex

01:01

normalization

Entanglement

Entanglement is one of important key words in the quantum world,

which is a quantum correlation between two (or more) qubit states.

Mathematically, a two-qubit state is called entangled if it cannot be "represented" by any tensor product of two single qubits

101011002

1dcba

Ex.

QPPQ

A quantum state on two quantum registers (=quantum systems) R and Q are called entangled if

R Q

Partial Measurement

)111000(3

1

2-qubit state

113

1

0)10(3

1

)( baba

State after getting 0: ＝ 0)10(2

1

3

2

0)10(3

1

)(

)(

0

0

EI

EI

Measure 2nd qubit

11

)1000(2

1

Result 0 (prob. 2/3)

Result 1 (prob. 1/3)

To Transform Quantum StatesA measurement collapses a superposition

10 ba

Measurement

0

On the contrary, the quantum mechanics allows us to transform a superposed state into another state, which is mathematically described as a unitary transformation.

Quantum gate

10 ,EE

U U

Important quantum gates

• Hadamard transformation

• Controlled NOT

• (Controlled) Phase-shift

Hadamard Transformation

1102

1 xxH

11

11

2

1H

Hadamard gate

0

1:0

1

1

2

1:)10(

2

1H

matrix representation

1

0:1

1

1

2

1:)10(

2

1H

Hadamard gate is essential to prepare the uniform superposed state of all n classical bits

000 Apply H for each qubit

102

1 10

2

110

2

1

nx

nx

}1,0{2

1n qubits

Controlled NOT

yxxyxCN

0100

1000

0010

0001

CN

Controlled NOT gatematrix representation

control part target part

Notice: CN is essentially "classical" transformation since it flips the target part if the control part is 1. But the fact that CN can apply to any superposed state is very important in quantum information processing.

Toffoli gate (Controlled-Controlled NOT)

zxyyxzyxT

Basic Fact Any function f computed efficiently by a classical computer can be computed

efficiently using only Toffoli gates, which means that the transformation

),(0 yxfyxyx

can be efficiently implemented by quantum computing

Phase-ShiftPhase-shift gate

xxZ x1

It does not change classical states but change superposed states!

Z 102

1 10

2

1

1/√2

1/√2

111001002

1 11100100

2

1

Apply Z at 2nd qubit

10

01Z

matrix representation

How to Process Quantum Information

Prepare quantum bits as much as you need

Choose a few qubits and apply unitary transformation measurement

You can introduce a "fresh" qubit if you want You can use previous measurement results for

choosing your operations The final result may be classical or quantum

(depending on your task)

Ex.1: Controlled Phase-Shift

yxyxCZ xy1

1000

0100

0010

0001

CN

Controlled phase-shift gatematrix representation


Similarly,

yxyx yxf ),(1 is efficiently implementable

CZ can be implemented using Toffoli and Hadamard.

1yx

H on the 3rd qubit

102

111

2

1 yxxyxyyx xy

Toffoli

102

1yx

input

output


Ex.2: Quantum Teleportation

B

pre-shared state (prior "entanglement")

A B

B

output

A

11110

AAAba

)1100(2

122 BABA

input

only local operation

& classical communication

Quantum teleportation is sending an unknown state from A to B under only local operations and classical communication with the assistance of pre-shared state between A and B

Quantum Teleportation: A's Local Transformation

BABABABA

BABAAA

bbaa

ba

110011101000

1100102211

control parttarget part

A applies CN

BABABABAbbaa 111010101000

A applies Hadamard on 1st qubit

BABABABA

BABABABA

bbbb

aaaa

110100011001

111101010000

A B

11110

AAAba

)1100(2

122 BABA

pre-shared

input

A B

B

output

local operation


Quantum Teleportation: A's Measurement

BABABABA

BABABABA

bbbb

aaaa

110100011001

111101010000

BBABBA

BBABBA

baba

baba

01111010

01011000

A measures the two qubits

B

c

BAdbdacd 11

A B

11110

AAAba

)1100(2

122 BABA

pre-shared

input

A B

B

output

local operation


Quantum Teleportation: Classical Communication + B's Local Transformation

B

c

BAdbdacd 11

A sends 2bits c and d

BB

c

BAcddbdacd 11

B applies CN

target part control part

BB

c

BAcdbacd 110

control parttarget part

B applies Controlled phase-shift

BBBA

cdbacd 10

output

A B

11110

AAAba

)1100(2

122 BABA

pre-shared

input

A B

B

output

local operation


Quantum Network Coding

Network Coding Problems in This Talk

We consider the "solvability."We consider the multiple unicast problem.

Instance: – a directed acyclic graph G=(V,E), where each edge

has a unit capacity

– k source-target pairs (s1,t1),...,(sk,tk) where each sj has a message xj

An instance is solvable if there is a network coding protocol which sends xj from sj to tj for every j.

For simplicity, we assume that the alphabet is binary

Revisiting Butterfly

1s 2s

0s

0t

2t 1t

x y

y = x ⊕ x ⊕ y

x

x y

y

x ⊕ y

x ⊕ y

x ⊕ y

x = y ⊕ x ⊕ y

POINT 1Information

can be copied

POINT 2Information can be

encoded

[Ahlswede-Li-Cai-Yeung 2000]

Quantum Butterfly

1s 2s

0s

0t

2t 1t

|Ψ1> |Ψ2>

POINT 1Quantum information

cannot be copied

• Information to be sent is quantum states

• Quantum operation is possible at each node

• Every channel is quantum

• Source nodes may have an entangled state

Q. If an instance is classically solvable, is quantum also?

POINT 2How quantum information is

encoded?

Quantum Information cannot be Copied

(No-cloning theorem: Wootters-Zurek)

An "unknown" quantum state cannot be cloned.

X

111

,000

U

U

a

a

)10)(10(

1100

)10(

Ua

Multiple Unicast is a Natural Target

Multicast Multiple Unicast

|φ1>, |φ2>, |φ3>

|φ1>, |φ2>, |φ3>

|φ1>, |φ2>, |φ3> |φ1>, |φ2>, |φ3>

|φ1> |φ2>

|φ2>|φ1>

c.f. Shi-Soljanin 2006, Kobayashi et al. 2010

Negative Results

|φ1> |φ2>

|φ2>|φ1>

１ qubit １qubit

Unsolvable under a single use

of the network (one shot)

[Hayashi-Iwama-N-Raymond-Yamashita07]

|φ1> |φ2>

|φ2>|φ1>

m qubits m qubits

Unsolvable even asymptotically, i.e., under

the condition m/n goes to 1 asymptotically

[Hayashi07, Leung-Oppenheim-Winter10]

Use of network n timesone shot

Under Additional Resources

• Entanglement– among sources [Hayashi07]

– among neighboring nodes [Leung-Oppenheim-Winter10]

• Classical channel [LOW10, Kobayashi-Le Gall-N-Roetteler09 & 11]

– Much cheaper than quantum: LOCC (Local Operation & Classical Communication) is easier than quantum communication.

Q. If an instance is classically solvable, then is quantum also under free classical communication?

Our Question

sources

sinks

Classical

1x 2x kx

1x 2x kx

classical channel

message is classical

Quantum

sources

sinks

k

kxx

kxx xx,..,

1,..,

1

1,...,

k

kxx

kxx xx,..,

1,..,

1

1,...,

quantum channel

message is quantum

free classical communication

solvable!

?solvable!

Q. If an instance is classically solvable, then is quantum also under free classical communication?

Our Result

If there is a classical coding protocol for an instance,

then there is also a quantum coding protocol for the

corresponding quantum instance.

[Kobayashi-Le Gall-N-Roetteler 2011]

Our previous results was:

If there is a classical linear coding protocol for an instance,

then there is also a quantum coding protocol for the

corresponding quantum instance.

[Kobayashi-Le Gall-N-Roetteler 2009]

Idea of Our Protocol

Our protocol consists of three stages

1.Node-by-node simulation – 1 qubit for each edge

2.Removal of internal registers – 1 bit backward for each edge

3.Removal of initial registers – 1bit forward for each edge

Stage1: Node-by-node Simulation

u1

u2 um

R

v

Classical Quantum

P1

P2 Pm

Rv

1-1. Receive registers P1,...,Pm

1-2. Introduce fresh register R and apply the unitary transformation

on registers P1, P2,..., Pm and R.

1-3. Send R to w.

1 qubit for each edge

fyyyy mm ,...,0,..., 11

PmmPyy

11

RPmmPfyy

11

my2y

1y

),,(: 1 myyff

w

u1

u2 um

w

Stage1: Node-by-node Simulation

yx

PPyxPPyx

,21,2,1

yx

PRRPyx yxxx,

2211,43

,2211, RR

yxPRRPyx yyyxxx

543,

2211, RRRyx

PRRPyx yxyyyxxx

76543,

2211, RRRRRyx

PRRPyx yxyxyxyyyxxx

2176543,

2211, QQRRRRRyx

PRRPyx yxyxyxyxyyyxxx

P1R2 R4

R1

R5

R3

R6 R7

Q2 Q1

P2y

y

x

x

yx

x

x y

y

yx yx

xxxxCOPY 00:

yxyxyxXOR 0:

R2 R4

R1

R5

R3R6 R7

Stage2: Removal of Internal Registers

P1, P2,..., Pm

R

v

2-1. Apply the Hadamard transformation on R, and measure it.

a

zy

aymeasure

z

zyH

1

)1(

2-2. Send the measurement value backward

2-3. Erase ''phase error" using phase-shift transformation

Ignore R

since no correlation with other registers!!

2. Do the following for internal registers in the inverse topological order

1 bit backward for each edge

RPmmPfyy

11

RPmmP

yyaf ayym

11)( ,,11

PmmPyy

11

a

w

PmmP

yyaf

PmmPyyyy m

11),,(

1111

After 2-3

After 2-1


2176543,

2211, QQRRRRRyx

PRRPyx yxyxyxyxyyyxxx

P1

R2 R4

R1

R5

R3R6 R7

Q2 Q1

P2y

y

x

x

yx

x

x y

y

yx yx

2-1. Apply the Hadamard transformation on R7, and measure it

2176543

,2211

)(, 1

QQRRRRRyx

PRRP

yxayx yxayxyxyyyxxx

2-2. Send the measurement value backward.

a

2-3. Erase phase error by

5

)(

51

R

yxa

Ryxyx

216543,

2211, QQRRRRyx

PRRPyx yxyxyxyyyxxx ignore thisAfter 2-3

After 2-1


216543,

2211, QQRRRRyx

PRRPyx yxyxyxyyyxxx

P1

R2 R4

R1

R5

R3R6 R7

Q2 Q1

P2y

y

x

x

yx

x

x y

y

yx yx


216543

,2211

)(, 1

QQRRRRyx

PRRP

yxbyx yxbyxyyyxxx


b

2-3. Erase phase error by

5

)(

51

R

yxb

Ryxyx

21543,

2211, QQRRRyx

PRRPyx yxyxyyyxxx ignore this

After 2-1

After 2-3


21543,

2211, QQRRRyx

PRRPyx yxyxyyyxxx

P1

R2 R4

R1

R5

R3R6 R7

Q2 Q1

P2y

y

x

x

yx

x

x y

y

yx yx


21543

,2211

)(, 1

QQRRRyx

PRRP

yxcyx yxcyyyxxx


c2-3. Erase phase error by

42

)(

421

RR

yxc

RRyxyx

2143,

2211, QQRRyx

PRRPyx yxyyyxxx

By continuing these, we have

21,

21, QQyx

PPyx yxyx

After 2-3After 2-1

Stage3: Removal of Initial Registers1 bit forward for each edge

sources

sinks

QkkQxx

PkkPxx xxxxk

k11

,..,11,..,

1

1P1 P2 Pk

Q1 Q2 Qk

After Stage 2

3-1. Apply the Hadamard transformations

on the initial registers P1,....,Pk, and

measure them.

QkkQ

xxPkkP

xaxaxx xxaa

k

kk

k

11,..,

11,..,

1

11

11

After Stage 3.1

ignore these

3-2. Send the measurement values using the classical network coding protocol.

3-3. Erase ''phase error" using phase-shift transformation.

Stage3: Removal of Initial Registers

P1

Q2 Q1

P2

ba

a b

b

After Stage 2

21,

21, QQyx

PPyx yxyx

3-1. Apply the Hadamard transformations

on P1,P2, and measure them.

21

,21, 1

QQyx

PP

byaxyx yxba

ignore

a

a b

ba ba

b a3-2. Send the measurement values using

the classical network coding protocol.

3-3. Erase ''phase error" by

11

1Q

ax

Qxx

22

1Q

by

Qyy

21,

, QQyx

yx yx

Finally, we obtain

Comments for Free Classical Communication

• KLNR11 reduces the amount of classical communication compared to KLNR09 – k*m*#(node) where m:=max fan-in of all nodes

[KLNR09]– 1 bit forward +1 bit backward for each edge = total

2*#(edge) [KLNR11]

• Sending classical bits backward is necessary– Quantum butterfly is not solvable even for the case

where free classical communication is allowed in the direction of edges. [Leung-Oppenheim-Winter 2010]

Summary

• No additional assistance– Butterfly is not solvable [HINRY07, LOW10, H07]

– Routing is optimal for a few cases [LOW10]

• 2 source-sink pairs• shallow networks (including butterfly)

• Under free classical communication– If an instance is classically solvable, then the

corresponding instance is also quantumly solvable [KLNR11]

– Additional classical communication is efficient– Outer/inner bound for a few cases [LOW10]

Future Work

[No additional resources]

Q. If an instance is quantumly solvable with network coding,

then is it solvable with routing?

[LOW10]A. Unknown. Yes under only a few special cases

Generally;• Advantage from classical, say, for security or complexity • Lossy quantum channels• Application (such as wireless communication in classical case), etc.

An instance is classically solvable

The corresponding quantum instance is quantumly solvable?

[Under free classical communication]

[KLNR11]

Future Work


The corresponding quantum instance is quantumly solvableX


[KLNR11]

In case where underlying graphs are directed

s tx

classically not solvable

s t

quantumly solvable since sending two bits backward enables us to reverse the direction of the edge by quantum teleportation!!


The corresponding quantum instance is quantumly solvable?


[KLNR11]

In case where underlying graphs are undirected

Quantum Network Coding Harumichi Nishimura 西村治道 Graduate School of Science, Osaka Prefecture...

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