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
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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…