Decoupling - PowerPoint Presentation

Slide1

Decoupling with random diagonal-unitaries

Yoshifumi NakataUniversitat Autonoma de Barcelona & The University of Tokyo arXiv:1502.07514 & arXiv:1509.05155Joint work with C. Hirche, C. Morgan, and A. Winter

2 September 2016

Asian Quantum Information Processing Slide2

OutlineIntroductionDecouplingHaar random

unitaries and unitary t-designsDecoupling with random diagonal-unitaries (RDU)Basic ideaDecoupling with RDUsEfficient implementations of RDUsBy quantum circuitsBy Hamiltonian dynamicsConclusion and open questionsSlide3

What is Decoupling?

Choose a good unitary such that where

and

.

Choi-

Jamiolkowski

state of the CPTP map.

Reference

Alice

Bob

qubits

CPTP mapSlide4

What is Decoupling?

Decoupling is

NOT always possible:E.g.

is entangled, the CPTP map is an identity map

. 

Reference

Alice

Bob

qubits

CPTP map

When

and

How

can we achieve decoupling?Slide5

Why do we care?

Decoupling

provides a free decoder in Q. capacity theorems.plays key roles in

fundamental physics:Black hole information science & Thermalisation phenomenaMicroscopic dynamics of decoupling is NOT

fully understood.

Reference

Alice

Bob

qubits

CPTP map

When

and

How

can we achieve decoupling?

One of the questions in this talk.Slide6

What is Decoupling?

How?

Choose

to be a Haar random unitary.

When? 

Reference

Alice

Bob

qubits

CPTP map

When

and

How

can we achieve decoupling?

A unitary drawn

uniformly at random

w.r.t. the

Haar

measure.Slide7

What is Decoupling?

How?

Choose

to be a Haar random

unitary.When?When the decoupling rate is small.

Reference

Alice

Bob

qubits

CPTP map

When

and

How

can we achieve decoupling?

[

Duplis

et.al ’09

]

A

-approximate

unitary

2-design!

A

-approximate unitary 2-design

A random unitary that simulates the

2

nd

order statistical moments

(e.g. Variance) of a

Haar

random unitary

within an error

.

 Slide8

Decoupling Theorem

If

,

decoupling

is achieved at the rate of

.

Natural questions:

Is this result

tight

?

Do we really need

to achieve decoupling

?

(Larger

is easier to implement.)

Decoupling with approximate designs

[

Szehr

et.al ’11]

For a

-

approximate

unitary 2-design

where and

.

 Slide9

Questions in this talkQuestionsCan we achieve decoupling by

-approximate 2-designs where ?Physically natural realisation of decoupling? In this talkWe provide a

new construction of decoupling, based on random diagonal-unitaries

.We show that

is

NOT

necessary.

decoupling can be

realised

by

periodically changing spin-glass-type Hamiltonians

.

 Slide10

OutlineIntroductionDecouplingHaar random unitaries and unitary t-designs

Decoupling with random diagonal-unitaries (RDU)Basic ideaDecoupling with RDUsEfficient implementations of RDUsBy quantum circuitsBy Hamiltonian dynamicsConclusion and open questionsSlide11

Basic idea

To use random

unitaries diagonal in the Z- and X-basis

All of and

are randomly and independently chosen from

.

Repeat

and

many times:

Each

and

are independent and random

.

CPTP mapSlide12

Basic idea

Why

do we expect

it works

?

The

is

a unitary

t

-design,

if

(in preparation

).

Every time,

and

are chosen independently at random.

U(d)Slide13

Decoupling with  

qubits

CPTP map

Decoupling Theorem

[YN, CH, CM, and

AW:

arXiv

:

1509.05155

]

For

, the following holds:

is

sufficient to achieve decoupling.

 Slide14

Decoupling with  

qubits

Decoupling Theorem’

[YN, CH, CM, and AW:

arXiv

:

1509.05155

]

When the CPTP map is the

partial trace

, for

,

is

sufficient to achieve decoupling.

In

the

most important cases of the

partial

traces,

suffices.

 Slide15

Proof SketchAn upper bound of ?It’s obtained from the operator In terms of ,

Use the key lemma for (arXiv: 1502.07514).Still complicated, but durable 

qubits

CPTP map

Alice

Ãlice

Reference

M.E.S

ReferenceSlide16

How good approximate 2-designs are they?the

is

.the

is

.

Decoupling and 2-designs

Decoupling Theorem

[YN, CH, CM, and AW:

arXiv

:

1509.05155

]

The

for

(

) achieves decoupling at the rate of

for any CPTP (partial traces) map.

An

-app. 2-design is

NOT

necessary for decoupling.

Theorem

[YN, CH, CM, and AW,

arXiv

:

1502.07514

]

The

on

qubits is a

-

approximate unitary 2-design, where

.

 Slide17

Are these results optimal?Converse statement (weak)

Conjecture: () suffices for any CPTP (partial traces) map. 

CPTP map

CPTP map

Decoupling!

Decoupling!Slide18

Outline

Introduction

DecouplingHaar

random unitaries and unitary t-designsDecoupling with random diagonal-unitaries

(RDU)Basic ideaDecoupling with RDUs

Efficient implementations of RDUs

By quantum circuits

By Hamiltonian dynamics

Conclusion and open questionsSlide19

Implementation of  

Both and use exponentially many random numbers. in N-qubit systems. No way to implement efficiently...A way of approximating lower order properties of is known

[YN, Koashi, Murao ’14]

. 

 Slide20

Quantum circuits

Up to the

2

nd

order

(

(

.

All gates in the

part are

commuting

.

can be applied

simultaneously

=

Short

implementation.

Up to the 2

nd

orderSlide21

Hamiltonian dynamicsDecoupling by spin-glass type Hamiltonians.The time necessary to achieve decoupling is independent of the number of particles.All-to-all interactions are maybe feasible in cavity QED or in semiquantal spin gasses.

4

5

6

Time

Hamiltonian

7

Decoupled!!Slide22

ConclusionWe have presented a new construction of decoupling based on random X- and Z-diagonal unitaries.

We have shown that () suffices to achieve decoupling for any CPTP (partial traces) map, implying

precise designs are not needed.

Decoupling can be achieved by simple quantum circuits.Decoupling can be realised by

periodically changing spin-glass-type Hamiltonians.

CPTP mapSlide23

Open questionsAre 2-designs really needed to achieve decoupling at the rate of

?Many believe NO. Nobody knows how to show that.1-designs cannot, what is 1.5-designs?In decoupling with , how many repetitions are needed?Conjecture: suffices for any CPTP map.Is it possible to achieve decoupling by time-independent Hamiltonians?In closed systems, Hamiltonians should be time-indep…Thank you for your attention!