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New quantum error-correcting codes for a bosonic mode New quantum error-correcting codes for a bosonic mode

New quantum error-correcting codes for a bosonic mode - PowerPoint Presentation

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New quantum error-correcting codes for a bosonic mode - PPT Presentation

Marios H Michael Matti Silveri R T Brierley Victor V Albert Philip Reinhold Juha Salmilehto Kyungjoo Noh Barbara M Terhal S M Girvin Liang Jiang AQIS Conference 2016 ID: 621680

binomial codes vva loss codes binomial loss vva code 2016 leghtas protect states pra 2013 prl prx subspace brierley

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Slide1

New quantum error-correcting codes for a bosonic mode

Marios

H. Michael

Matti

Silveri

R. T. Brierley

Victor V. AlbertPhilip ReinholdJuha SalmilehtoKyungjoo NohBarbara M. TerhalS. M. GirvinLiang Jiang

AQIS Conference 2016Taipei, TaiwanMost of the talk is about arXiv:1512.08079 (PRX)Slide2

Why do we want to encode in an oscillator?

Experimentally, microwave cavities have

longer (

ms) lifetimes

than (related) qubits.

Reagor, … Schoelkopf, APL2013, PRB2016

States in microwave cavities can be controlled.

Heeres, … Schoelkopf, PRL2015 Dominant error channels for oscillators are simpler than those for multi-qubit paradigm.E.g., less ancillas, measurements

 

 

 

Slide3

Why do we want to consider new codes?

Cavities/fibers

aren’t perfect

, have errors:

Photon loss

(with

)

Dephasing errors

Photon gain errors

“Standard” encodings (polarization, angular momentum, occupation number)

do not protect

from errors

. Multimode encodings require

more oscillators

. But Hilbert space of one oscillator is already big.

Why not utilize it?

We already have!

Full QEC has already been done on a code we will consider (

Ofek

, …

Schoelkopf

, Nature 2016

).

 Slide4

How do we encode in an oscillator?

A code

is a subspace of the full oscillator Hilbert space (i.e.,

Fock

space).

Logical states can be expressed in

Fock

state basis :

For example, the occupation number code states are Slide5

Codes that we consider:

GKP codes, e.g.,

Cat codes, e.g.,

Binomial codes (

new

), e.g.,

Numerical/optimized codes (ask computer what works)

Leghtas et al., PRL 2013Mirrahimi, Leghtas, VVA et al., NJP 2014

Gottesman, Kitaev, Preskill, PRA 2001

Michael, Silveri, Brierley, VVA et al., PRX 2016Slide6

Experimental code progress report

as of AQIS 2016

Code

En

/decoding

Gates

QEC

cat [1] [2] [3]bin

gkp

[1] Vlastakis

et al., Science 2013

[2]

Heeres et al., arXiv:1608.02430

[3]

Ofek

et al., Nature 2016

 Currently under investigation

 Possible using techniques from

Heeres et al., PRL 2015;

Krastanov

, VVA, et al., PRA.Slide7

Codes that we consider:

GKP codes, e.g.,

Cat codes, e.g.,

Binomial codes (

new

), e.g.,

Numerical/optimized codes (ask computer what works)

Leghtas et al., PRL 2013Mirrahimi, Leghtas, VVA et al., NJP 2014

Gottesman, Kitaev, Preskill, PRA 2001

Michael, Silveri, Brierley, VVA et al., PRX 2016Slide8

1. How binomial codes protect from loss

This particular binomial code protects from one lowering operator

. Upon undergoing the error,

Quantum information is preserved!

It just moved to the error subspace

, and can be moved back.

Since average occupation number of both states is 2, one loss event does not destroy the information.

 Slide9

2. How binomial codes protect from

no

loss

Unfortunately, no loss does not imply no damage? For

, the longer the system goes without a loss event

, the

more likely

it is to be in and the less likely it is to be in

.This effect --- the no-jump evolution --- is manifested by Kraus operator (with damping parameter )

that decays all states to

.

 Slide10

2. How binomial codes protect from

no

loss

Under the no-jump evolution (

is avg. occ. num.):

is not affected, but

is:

Within first order in

, one can rotate the information from

back to

without affecting

.

 Slide11

Summary: protection from loss and

no

loss.

Both events can thus be corrected as follows:

If a loss event

detected, apply rotation from

subspace to logical subspace.

While no loss detected, continuously apply -dependent rotation from to logical subspace.Therefore, amplitude damping channel

E + recovery R leave state invariant (within first order in ):

 Slide12

Why did we call them binomial codes?

Superimposing to write in the conjugate basis:

Coefficients are square roots of binomial coefficients; “1 2 1” from Pascal’s triangle.

This yields a natural generalization…Slide13

Binomial codes: general case

General formula for binomial codes:

Spacing

determines how many loss events

the code protects from.

Order

determines how many moments of

are equal for both logical states.

 Slide14

Binomial codes: general case

General formula for binomial codes:

Codes are customizable!

For example, picking

, one can show that binomial code can protect from amplitude damping up to order

:

 Slide15

So we have all these codes…

GKP codes, e.g.,

Cat codes, e.g.,

Binomial codes (

new

), e.g.,

Numerical/optimized codes (ask computer what works)

Leghtas et al., PRL 2013Mirrahimi, Leghtas, VVA et al., NJP 2014

Gottesman, Kitaev, Preskill, PRA 2001

Michael, Silveri, Brierley, VVA et al., PRX 2016Slide16

…How do we compare them?

We can use entanglement fidelity

, optimizable using a semi-definite program (

Fletcher, Shor, Win PRA2007

).

 

VVA et al., in preparationSlide17

Conclusion

Continuous variables, even using

only one mode

, offer long lifetimes, a high degree of experimental control, and tractable error channels. QEC has already been achieved*!

Binomial codes can have

lower mean occupation number

than previous codes. They are also customizable to protect against any combinations of and

.There are now several codes: It is a good time to benchmark

them, both analytically and numerically!

 

Reagor

et al., APL2013, PRB2016

Vlastakis

et al., Science 2013

Heeres et al., PRL2015

Krastanov

, et al., PRA

Heeres et al., arXiv:1608.02430

*

Ofek

et al., Nature 2016

gkp

Gottesman, Kitaev,

Preskill, PRA 2001

cat Leghtas et al., PRL 2013; Mirrahimi et al., NJP 2014

Michael, Silveri, Brierley, VVA, Salmilehto, Jiang,

Girvin

PRX

VVA et al., in preparation…